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Review ArticleOn Finsler Geometry and Applications in MechanicsReview and New Perspectives

J D Clayton12

1 Impact Physics US ARL Aberdeen MD 21005-5066 USA2A James Clark School of Engineering (Adjunct Faculty) University of Maryland College Park MD 20742 USA

Correspondence should be addressed to J D Clayton johndclayton1civmailmil

Received 21 November 2014 Accepted 18 January 2015

Academic Editor Mahouton N Hounkonnou

Copyright copy 2015 J D ClaytonThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In Finsler geometry each point of a base manifold can be endowed with coordinates describing its position as well as a set of one ormore vectors describing directions for exampleThe associatedmetric tensormay generally depend on direction as well as positionand a number of connections emerge associated with various covariant derivatives involving affine and nonlinear coefficientsFinsler geometry encompasses Riemannian Euclidean and Minkowskian geometries as special cases and thus it affords greatgenerality for describing a number of phenomena in physics Here descriptions of finite deformation of continuous media are ofprimary focus After a review of necessary mathematical definitions and derivations prior work involving application of Finslergeometry in continuum mechanics of solids is reviewed A new theoretical description of continua with microstructure is thenoutlined merging concepts from Finsler geometry and phase field theories of materials science

1 Introduction

Mechanical behavior of homogeneous isotropic elastic solidscan be described by constitutive models that depend onlyon local deformation for example some metric or straintensor that may generally vary with position in a bodyMaterials with microstructure require more elaborate consti-tutive models for example describing lattice orientation inanisotropic crystals dislocationmechanisms in elastic-plasticcrystals or cracks or voids in damaged brittle or ductilesolids In conventional continuum mechanics approachessuch models typically assign one or more time- and position-dependent vector(s) or higher-order tensor(s) in additionto total deformation or strain that describe physical mech-anisms associated with evolving internal structure

Mathematically in classical continuum physics [1ndash3]geometric field variables describing behavior of a simply con-nected region of a body depend fundamentally only on ref-erential and spatial coordinate charts 119883119860 and 119909

119886 (119860 119886 =

1 2 119899) related by a diffeomorphism119909 = 120593(119883 119905) with119909 and

119883 denoting corresponding points on the spatial and materialmanifolds covered by corresponding chart(s) and 119905 denotingtime State variables entering response functions dependultimately only on material points and relative changes intheir position (eg deformation gradients of first orderand possibly higher orders for strain gradient-type models[4]) Geometric objects such as metric tensors connectioncoefficients curvature tensors and anholonomic objects [5]also depend ultimately only on position This is true inconventional nonlinear elasticity and plasticity theories [1 6]as well as geometric theories incorporating torsion andorcurvature tensors associated with crystal defects for example[7ndash15] In these classical theories the metric tensor is alwaysRiemannian (ie essentially dependent only upon 119909 or 119883

in the spatial or material setting) meaning the length of adifferential line element depends only on position howevertorsion curvature andor covariant derivatives of the metricneed not always vanish if the material contains variouskinds of defects (non-Euclidean geometry) Connections arelinear (ie affine) Gauge field descriptions in the context of

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 828475 11 pageshttpdxdoiorg1011552015828475

2 Advances in Mathematical Physics

Riemannian metrics and affine connections include [16 17]Relevant references in geometry and mathematical physicsinclude [18ndash26] in addition to those already mentionedFinite deformation director theories of micropolar type areaddressed in the context of Riemannian (as opposed toFinslerian) metrics in [1 27]

Finsler geometry first attributed to Finsler in 1918 [28]is more general than Riemannian geometry in the sense thatthe fundamental (metric) tensor generally may depend onadditional independent variables labeled here as 119910 and 119884

in spatial and material configurations with correspondinggeneralized coordinates 119910

119886 and 119884

119860 Formal definitions

will be given later in this paper for the present immediatediscussion it suffices to mention that each point can be con-sidered endowedwith additional degrees-of-freedombeyond119909 or 119883 and that transformation laws among coordinates aswell as connection coefficients (ie covariant differentials)generally depend on 119910 or 119884 as well as 119909 or 119883 Relevantreferences in mathematics include [29ndash32] For descriptionsof mechanics of solids additional degrees-of-freedom can beassociated with evolving features of the microstructure of thematerial though more general physical interpretations arepossible

The use of Finsler geometry to describe continuummechanical behavior of solids was perhaps first noted byKroner in 1968 [33] and Eringen in 1971 [3] the latterreference incorporating some basic identities and definitionsderived primarily by Cartan [34] though neither devel-oped a Finsler-based framework more specifically directedtowards mechanics of continua The first theory of Finslergeometry applied to continuum mechanics of solids withmicrostructure appears to be the purely kinematic theory ofIkeda [35] in a generalization of Cosserat-type kinematicswhereby additional degrees-of-freedom are director vectorslinked to structure This theory was essentially extended byBejancu [30] to distinguish among horizontal and verticaldistributions of the fiber bundle of a deforming pseudo-Finslerian total space More complete theories incorporatinga Lagrangian functional (leading to physical balance orconservation laws) and couched in terms of Finsler geom-etry were developed by Saczuk Stumpf and colleagues fordescribing solids undergoing inelastic deformation mecha-nisms associated with plasticity andor damage [36ndash40] Tothe authorrsquos knowledge solution of a boundary value problemin solid mechanics using Finsler geometric theory has onlybeen reported once in [38] Finsler geometry has beenanalogously used to generalize fundamental descriptionsin other disciplines of physics such as electromagnetismquantum theory and gravitation [30 41ndash43]

This paper is organized as follows In Section 2 requisitemathematical background on Finsler geometry (sometimescalled Riemann-Finsler geometry [31]) is summarized InSection 3 the aforementioned theories from continuumphysics of solids [30 35ndash38 40] are reviewed and comparedIn Section 4 aspects of a new theory with a primary intentionof description of structural transformation processes in realmaterials are proposed and evaluated Conclusions follow inSection 5

2 Finsler Geometry Background

Notation used in the present section applies to a referentialdescription that is the initial state analogous formulae applyfor a spatial description that is a deformed body

21 Coordinates and Fundamentals Denote by 119872 an 119899-dimensional119862infinmanifold Each element (of support) of119872 isof the form (119883 119884) where 119883 isin 119872 and 119884 isin 119879119872 with 119879119872 thetangent bundle of 119872 A Finsler structure of 119872 is a function119871 119879119872 rarr [0infin) with the following three properties [31]

(i) The fundamental function 119871 is 119862infin on 119879119872 0(ii) 119871(119883 120582119884) = 120582119871(119883 119884) forall120582 gt 0 (ie 119871 is homogeneous

of degree one in 119884)

(iii) the fundamental tensor 119866119860119861

= (12)1205972(1198712)120597119884119860120597119884119861

is positive definite at every point of 119879119872 0

Restriction of 119871 to a particular tangent space 119879119883119872 gives rise

to a (local) Minkowski norm

1198712(119884) = 119866

119860119861 (119884) 119884119860119884119861 (1)

which follows from Eulerrsquos theorem and the identity

119866119860119861

=1198711205972119871

120597119884119860120597119884119861+ (

120597119871

120597119884119860)(

120597119871

120597119884119861) (2)

Specifically letting 119884119860

rarr d119883119860 the length of a differentialline element at119883 depends in general on both119883 and 119884 as

|dX (119883 119884)| = radicdX sdot dX = [119866119860119861 (119883 119884) d119883119860d119883119861]

12

(3)

A Finsler manifold (119872 119865) reduces to a Minkowskian man-ifold when 119871 does not depend on 119883 and to a Riemannianmanifold when 119871 does not depend on 119884 In the lattercase a Riemannian metric tensor is 119866

119860119861(119883)119889119883

119860otimes 119889119883

119861Cartanrsquos tensor with the following fully symmetric covariantcomponents is defined for use later

119862119860119861119862

=1

2

120597119866119860119861

120597119884119862=

1

4

1205973(1198712)

120597119884119860120597119884119861120597119884119862 (4)

Consider now a coordinate transformation to anotherchart on119872 for example

119883119860= 119883119860(1198831 1198832 119883

119899)

119860= (

120597119883119860

120597119883119861)119884119861 (5)

From the chain rule holonomic basis vectors on 119879119872 thentransform as [30 31]

120597

120597119883119860=

120597119883119861

120597119883119860

120597

120597119883119861+

1205972119883119861

120597119883119860120597119883119862119862 120597

120597119884119861 (6)

120597

120597119860=

120597119883119861

120597119883119860

120597

120597119884119861 (7)

Advances in Mathematical Physics 3

22 Connections and Differentiation Christoffel symbols ofthe second kind derived from the symmetric fundamentaltensor are

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863) (8)

Lowering and raising of indices are enabled via 119866119860119861

and itsinverse119866119860119861 Nonlinear connection coefficients on 1198791198720 aredefined as

119873119860

119861= 120574119860

119861119862119884119862minus 119862119860

119861119862120574119862

119863119864119884119863119884119864=

1

2

120597119866119860

120597119884119861 (9)

where 119866119860

= 120574119860

119861119862119884119861119884119862 The following nonholonomic bases

are then introduced

120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119884119861 120575119884

119860= 119889119884119860+ 119873119860

119861119889119883119861 (10)

It can be shown that unlike (6) these nonholonomicbases obey simple transformation laws like (7) The set120575120575119883

119860 120597120597119884

119860 serves as a convenient local basis for119879(119879119872

0) its dual set 119889119883119860 120575119884119860 applies for the cotangent bundle119879lowast(119879119872 0) A natural Riemannian metric can then be

introduced called a Sasaki metric [31]

G (119883 119884) = 119866119860119861

119889119883119860otimes 119889119883119861+ 119866119860119861

120575119884119860otimes 120575119884119861 (11)

The horizontal subspace spanned by 120575120575119883119860 is orthogonal

to the vertical subspace spanned by 120597120597119884119861 with respect to

thismetric Covariant derivativenabla or collectively connection1-forms120596119860

119861 define a linear connection on pulled-back bundle

120587lowast119879119872 over 119879119872 0 Letting 120592 denote an arbitrary direction

nabla120592

120597

120597119883119860= 120596119861

119860(120592)

120597

120597119883119861 nabla

120592119889119883119860= minus120596119860

119861(120592) 119889119883

119861 (12)

A number of linear connections have been introduced inthe Finsler literature [30 31] The Chern-Rund connection[29 44] is used most frequently in applications related tothe present paper It is a unique linear connection on 120587

lowast119879119872

characterized by the structural equations [31]

119889 (119889119883119860) minus 119889119883

119861and 120596119860

119861= 0

119889119866119860119861

minus 119866119861119862

120596119862

119860minus 119866119860119862

120596119862

119861= 2119862119860119861119862

120575119884119862

(13)

The first structure equation implies torsion freeness andresults in

120596119860

119861= Γ119860

119862119861119889119883119862 Γ

119860

119861119862= Γ119860

119862119861 (14)

The second leads to the connection coefficients

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863) (15)

When a Finsler manifold degenerates to a Riemannianmanifold119873119860

119861= 0 and Γ

119860

119861119862= 120574119860

119861119862 Cartanrsquos connection 1-forms

are defined by 120596119860

119861+ 119862119860

119863119861120575119884119863 where 120596

119860

119861correspond to (14)

its coordinate formulae and properties are listed in [3] It has

been shown [45] how components of Cartanrsquos connection ona Finsler manifold can be obtained as the induced connectionof an enveloping space (with torsion) of dimension 2119899 Whena Finsler manifold degenerates to a locally Minkowski space(119871 independent of 119883) then Γ

119860

119861119862= 120574119860

119861119862= 0 Gradients of

bases with respect to the Chern-Rund connection andCartantensor are

nabla120575120575119883119860

120575

120575119883119861= Γ119862

119860119861

120575

120575119883119862 nabla

120575120575119883119860

120597

120597119884119861= 119862119862

119860119861

120597

120597119884119862 (16)

As an example of covariant differentiation on a Finslermanifold with Chern-Rund connection nabla consider a (

1

1)

tensor field T = 119879119860

119861(120597120597119883

119860) otimes 119889119883

119861 on the manifold 119879119872 0The covariant differential of T(119883 119884) is

(nabla119879)119860

119861= 119889119879119860

119861+ 119879119862

119861120596119860

119862minus 119879119860

119862120596119862

119861

= 119879119860

119861|119862119889119883119862+ 119879119860

119861119862120575119884119862

= (nabla120575120575119883119862119879)119860

119861119889119883119862+ (nabla120597120597119884119862119879)119860

119861120575119884119862

= (120575119879119860

119861

120575119883119862+ 119879119863

119861Γ119860

119862119863minus 119879119860

119863Γ119863

119862119861)119889119883119862+ (

120597119879119860

119861

120597119884119862)120575119884119862

(17)

Notations (sdot)|119860

and (sdot)119860

denote respective horizontal andvertical covariant derivatives with respect to nabla

23 Geometric Quantities and Identities Focusing again onthe Chern-Rund connection nabla curvature 2-forms are

Ω119860

119861= 119889 (120596

119860

119861) minus 120596119862

119861and 120596119860

119862

=1

2119877119860

119861119862119863119889119883119862and 119889119883119863+ 119875119860

119861119862119863119889119883119862and 120575119884119863

+1

2119876119860

119861119862119863120575119884119862and 120575119884119863

(18)

with 119889(sdot) the exterior derivative and and the wedge product (nofactor of 12) HH- HV- and VV-curvature tensors of theChern-Rund connection have respective components

119877119860

119861119862119863=

120575Γ119860

119861119863

120575119883119862minus

120575Γ119860

119861119862

120575119883119863+ Γ119860

119864119862Γ119864

119861119863minus Γ119860

119864119863Γ119864

119861119862

119875119860

119861119862119863= minus

120597Γ119860

119861119862

120597119884119863 119876

119860

119861119862119863= 0

(19)

VV-curvature vanishes HV-curvature obeys 119875119860

119861119862119863= 119875119860

119862119861119863

and a Bianchi identity for HH-curvature is

119877119860

119861119862119863+ 119877119860

119862119863119861+ 119877119860

119863119861119862= 0 (20)

When a Finsler manifold degenerates to a Riemannianmanifold then 119877

119860

119861119862119863become the components of the usual

curvature tensor of Riemannian geometry constructed from120574119860

119861119862 and 119875

119860

119861119862119863= 0 All curvatures vanish in locally

Minkowski spaces It is not always possible to embed a Finsler


space in a Riemannian space without torsion but it is possibleto determine the metric and torsion tensors of a space ofdimension 2119899 minus 1 in such a way that any 119899-dimensionalFinsler space is a nonholonomic subspace of such a spacewithtorsion [46]

Nonholonomicity (ie nonintegrability) of the horizontaldistribution is measured by [32]

[120575

120575119883119860

120575

120575119883119861] = (

120575119873119862

119860

120575119883119861minus

120575119873119862

119861

120575119883119860)

120597

120597119884119862= Λ119862

119860119861

120597

120597119884119862 (21)

where [sdot sdot] is the Lie bracket and Λ119862

119860119861can be interpreted as

components of a torsion tensor [30] For the Chern-Rundconnection [31]

Λ119862

119860119861= minus119877119862

119863119860119861119884119863 (22)

Since Lie bracket (21) is strictly vertical the horizontaldistribution spanned by 120575120575119883

119860 is not involutive [31]

3 Applications in Solid Mechanics 1973ndash2003

31 Early Director Theory The first application of Finslergeometry to finite deformation continuum mechanics iscredited to Ikeda [35] who developed a director theory in thecontext of (pseudo-) Finslerian manifolds A slightly earlierwork [47] considered a generalized space (not necessarilyFinslerian) comprised of finitely deforming physical andgeometrical fields Paper [35] is focused on kinematics andgeometry descriptions of deformations of the continuumand the director vector fields and their possible interactionsmetric tensors (ie fundamental tensors) and gradients ofmotions Covariant differentials are defined that can be usedin field theories of Finsler space [41 48] Essential conceptsfrom [35] are reviewed and analyzed next

Let 119872 denote a pseudo-Finsler manifold in the sense of[35] representative of a material body with microstructureLet 119883 isin 119872 denote a material point with coordinate chart119883119860 (119860 = 1 2 3) covering the body in its undeformed state

A set of director vectors D(120572)

is attached to each 119883 where(120572 = 1 2 119901) In component form directors are written119863119860

(120572) In the context of notation in Section 2 119872 is similar

to a Finsler manifold with 119884119860

rarr 119863119860

(120572) though the director

theory involvesmore degrees-of-freedomwhen119901 gt 1 andnofundamental function is necessarily introduced Let 119909 denotethe spatial location in a deformed body of a point initiallyat 119883 and let d

(120572)denote a deformed director vector The

deformation process is described by (119886 = 1 2 3)

119909119886= 119909119886(119883119860) 119889

119886

(120572)= 119889119886

(120572)[119863119860

(120573)(119883119861)] = 119889

119886

(120572)(119883119861)

(23)

The deformation gradient and its inverse are

119865119886

119860=

120597119909119886

120597119883119860 119865

minus1119860

119886=

120597119883119860

120597119909119886 (24)

The following decoupled transformations are posited

d119909119886 = 119865119886

119860d119883119860 119889

119886

(120572)= 119861119886

119860119863119860

(120572) (25)

with 119861119886

119860= 119861119886

119860(119883) Differentiating the second of (25)

d119889119886(120572)

= 119864119886

(120572)119860d119883119860 = (119861

119886

119861119860119863119861

(120572)+ 119861119886

119861119863119861

(120572)119860) d119883119860 (26)

where (sdot)119860

denotes the total covariant derivative [20 26]Differential line and director elements can be related by

d119889119886(120572)

= 119891119886

(120572)119887d119909119887 = 119865

minus1119860

119887119864119886

(120572)119860d119909119887 (27)

Let 119866119860119861

(119883) denote the metric in the reference con-figuration such that d1198782 = d119883119860119866

119860119861d119883119861 is a measure

of length (119866119860119861

= 120575119860119861

for Cartesian coordinates 119883119860)

Fundamental tensors in the spatial frame describing strainsof the continuum and directors are

119862119886119887

= 119865minus1119860

119886119866119860119861

119865minus1119861

119887 119862

(120572120573)

119886119887= 119864minus1119860

(120572)119886119866119860119861

119864minus1119861

(120572)119887 (28)

Let Γ119860

119861119862and Γ

120572

120573120574denote coefficients of linear connections

associated with continuum and director fields related by

Γ119860

119861119862= 119863119860

(120572)119863(120573)

119861119863(120574)

119862Γ120572

120573120574+ 119863119860

(120572)119863(120572)

119862119861 (29)

where 119863119860

(120572)119863(120572)

119861= 120575119860

119861and 119863

119860

(120573)119863(120572)

119860= 120575120572

120573 The covariant

differential of a referential vector field V where locallyV(119883) isin 119879

119883119872 with respect to this connection is

(nabla119881)119860

= 119889119881119860+ Γ119860

119861119862119881119862d119883119861 (30)

the covariant differential of a spatial vector field 120592 wherelocally 120592(119909) isin 119879

119909119872 is defined as

(nabla120592)119886

= 119889120592119886+ Γ119886

119887119888120592119888d119909119887 + Γ

(120572)119886

119887119888120592119888d119909119887 (31)

with for example the differential of 120592119886(119909 d(120572)

) given by119889120592119886 =(120597120592119886120597119909119887)d119909119887 + (120597120592

119886120597119889119887

(120572))d119889119887(120572) Euler-Schouten tensors are

119867(120572)119886

119887119888= 119865119886

119860119864minus1119860

(120572)119888119887 119870

(120572)119886

119887119888= 119864119886

(120572)119860119865minus1119860

119888119887 (32)

Ikeda [35] implies that fundamental variables entering afield theory for directed media should include the set(FBEHK) Given the fields in (23) the kinematic-geometric theory is fully determined once Γ

120572

120573120574are defined

The latter coefficients can be related to defect content ina crystal For example setting Γ

120572

120573120574= 0 results in distant

parallelism associated with dislocation theory [7] in whichcase (29) gives the negative of the wryness tensor moregeneral theory is needed however to represent disclinationdefects [49] or other general sources of incompatibility [650] requiring a nonvanishing curvature tensor The directorsthemselves can be related to lattice directions slip vectors incrystal plasticity [51 52] preferred directions for twinning[6 53] or planes intrinsically prone to cleavage fracture [54]for example Applications of multifield theory [47] towardsmultiscale descriptions of crystal plasticity [50 55] are alsopossible


32 Kinematics and Gauge Theory on a Fiber Bundle Thesecond known application of Finsler geometry towards finitedeformation of solid bodies appears in Chapter 8 of the bookof Bejancu [30] Content in [30] extends and formalizes thedescription of Ikeda [35] using concepts of tensor calculus onthe fiber bundle of a (generalized pseudo-) Finsler manifoldGeometric quantities appropriate for use in gauge-invariantLagrangian functions are derived Relevant features of thetheory in [30] are reviewed and analyzed inwhat follows next

Define 120577 = (119885 120587119872119880) as a fiber bundle of total space119885 where 120587 119885 rarr 119872 is the projection to base manifold119872 and 119880 is the fiber Dimensions of 119872 and 119880 are 119899 (119899 = 3

for a solid volume) and 119901 respectively the dimension of 119885 is119903 = 119899 + 119901 Coordinates on 119885 are 119883

119860 119863120572 where 119883 isin 119872

is a point on the base body in its reference configurationand 119863 is a director of dimension 119901 that essentially replacesmultiple directors of dimension 3 considered in Section 31The natural basis on 119885 is the field of frames 120597120597119883119860 120597120597119863120572Let119873120572

119860(119883119863) denote nonlinear connection coefficients on119885

and introduce the nonholonomic bases

120575

120575119883119860=

120597

120597119883119860minus 119873120572

119860

120597

120597119863120572 120575119863

120572= 119889119863120572+ 119873120572

119860119889119883119860 (33)

Unlike 120597120597119883119860 these nonholonomic bases obey simple

transformation laws 120575120575119883119860 120597120597119863120572 serves as a convenientlocal basis for119879119885 adapted to a decomposition into horizontaland vertical distributions

119879119885 = 119867119885 oplus 119881119885 (34)

Dual set 119889119883119860 120575119863120572 is a local basis on 119879lowast119885 A fundamental

tensor (ie metric) for the undeformed state is

G (119883119863) = 119866119860119861 (119883119863) 119889119883

119860otimes 119889119883119861

+ 119866120572120573 (119883119863) 120575119863

120572otimes 120575119863120573

(35)

Let nabla denote covariant differentiation with respect to aconnection on the vector bundles119867119885 and 119881119885 with

nabla120575120575119883119860

120575

120575119883119861= Γ119862

119860119861

120575

120575119883119862 nabla

120575120575119883119860

120597

120597119863120572= 119862120573

119860120572

120597

120597119863120573 (36)

nabla120597120597119863120572

120575

120575119883119860= 119888119861

120572119860

120575

120575119883119861 nabla

120597120597119863120572

120597

120597119863120573= 120594120574

120572120573

120597

120597119863120574 (37)

Consider a horizontal vector field V = 119881119860(120575120575119883

119860) and a

vertical vector fieldW = 119882120572(120597120597119863

120572) Horizontal and vertical

covariant derivatives are defined as

119881119860

|119861=

120575119881119860

120575119883119861+ Γ119860

119861119862119881119862 119882

120572

|119861=

120575119882120572

120575119883119861+ 119862120572

119861120573119882120573

119881119860

||120573=

120575119881119860

120575119863120573+ 119888119860

120573119862119881119862 119882

120572

||120573=

120575119882120572

120575119863120573+ 120594120572

120573120575119882120575

(38)

Generalization to higher-order tensor fields is given in [30]In particular the following coefficients are assigned to the so-called gauge H-connection on 119885

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863)

119862120572

119860120573=

120597119873120572

119860

120597119863120573 119888

119860

120572119861= 0

120594120572

120573120574=

1

2119866120572120575

(120597119866120573120575

120597119863120574+

120597119866120574120575

120597119863120573minus

120597119866120573120574

120597119863120575)

(39)

Comparing with the formal theory of Finsler geometryoutlined in Section 2 coefficients Γ119860

119861119862are analogous to those

of the Chern-Rund connection and (36) is analogous to(16) The generalized pseudo-Finslerian description of [30]reduces to Finsler geometry of [31] when 119901 = 119899 and afundamental function 119871 exists fromwhichmetric tensors andnonlinear connection coefficients can be derived

Let119876119870(119883119863) denote a set of differentiable state variableswhere 119870 = 1 2 119903 A Lagrangian function L of thefollowing form is considered on 119885

L (119883119863) = L[119866119860119861

119866120572120573 119876119870120575119876119870

120575119883119860120597119876119870

120597119863120572] (40)

Let Ω be a compact domain of 119885 and define the functional(action integral)

119868 (Ω) = intΩ

L (119883119863) 1198891198831sdot sdot sdot and 119889119883

119899and 1198891198631sdot sdot sdot and 119889119863

119901

L = [10038161003816100381610038161003816det (119866

119860119861) det (119866

120572120573)10038161003816100381610038161003816]12

L

(41)

Euler-Lagrange equations referred to the reference configu-ration follow from the variational principle 120575119868 = 0

120597L

120597119876119870=

120597

120597119883119860[

120597L

120597 (120597119876119870120597119883119860)] +

120597

120597119863120572[

120597L

120597 (120597119876119870120597119863120572)]

(42)

These can be rewritten as invariant conservation laws involv-ing horizontal and vertical covariant derivatives with respectto the gauge-H connection [30]

Let 1205771015840

= (1198851015840 12058710158401198721015840 1198801015840) be the deformed image of

fiber bundle 120577 representative of deformed geometry of thebody for example Dimensions of 1198721015840 and 119880

1015840 are 119899 and 119901respectively the dimension of1198851015840 is 119903 = 119899+119901 Coordinates on1198851015840 are 119909119886 119889120572

1015840

where 119909 isin 1198721015840 is a point on the base body in

its current configuration and 119889 is a director of dimension 119901The natural basis on 119885

1015840 is the field of frames 120597120597119909119886 1205971205971198891205721015840

Let119873120572

1015840

119886(119909 119889) denote nonlinear connection coefficients on119885

1015840and introduce the nonholonomic bases

120575

120575119909119886=

120597

120597119909119886minus 1198731205721015840

119886

120597

1205971198891205721015840 120575119889

1205721015840

= 1198891198891205721015840

+ 1198731205721015840

119886119889119909119886 (43)

These nonholonomic bases obey simple transformation laws120575120575119909

119886 120597120597119889

1205721015840

serves as a convenient local basis for 1198791198851015840

where

1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (44)


Dual set 119889119909119886 1205751198891205721015840

is a local basis on 119879lowast1198851015840 Deformation

of (119885119872) to (11988510158401198721015840) is dictated by diffeomorphisms in local

coordinates

119909119886= 119909119886(119883119860) 119889

1205721015840

= 1198611205721015840

120573(119883)119863

120573 (45)

Let the usual (horizontal) deformation gradient and itsinverse have components

119865119886

119860=

120597119909119886

120597119883119860 119865

minus1119860

119886=

120597119883119860

120597119909119886 (46)

It follows from (45) that similar to (6) and (7)

120597

120597119883119860= 119865119886

119860

120597

120597119909119886+

1205971198611205721015840

120573

120597119883119860119863120573 120597

1205971198891205721015840

120597

120597119863120572= 1198611205731015840

120572

120597

1205971198891205731015840

(47)

Nonlinear connection coefficients on119885 and1198851015840 can be related

by

1198731205721015840

119886119865119886

119860= 119873120573

1198601198611205721015840

120573minus (

1205971198611205721015840

120573

120597119883119860)119863120573 (48)

Metric tensor components on 119885 and 1198851015840 can be related by

119866119886119887119865119886

119860119865119887

119861= 119866119860119861

119866120572101584012057310158401198611205721015840

1205751198611205731015840

120574= 119866120575120574 (49)

Linear connection coefficients on 119885 and 1198851015840 can be related by

Γ119886

119887119888119865119887

119861119865119888

119862= Γ119860

119861119862119865119886

119860minus

120597119865119886

119862

120597119883119861

1198621205721015840

1198861205731015840119865119886

1198601198611205731015840

120574= 119862120583

1198601205741198611205721015840

120583minus

1205971198611205721015840

120574

120597119883119860

119888119886

1205721015840119887119865119887

1198601198611205721015840

120573= 119888119861

120573119860119865119886

119861

1205941205721015840

120573101584012057510158401198611205731015840

1205741198611205751015840

120578= 120594120576

1205741205781198611205721015840

120576

(50)

Using the above transformations a complete gauge H-connection can be obtained for 1198851015840 from reference quantitieson 119885 if the deformation functions in (45) are known ALagrangian can then be constructed analogously to (40) andEuler-Lagrange equations for the current configuration ofthe body can be derived from a variational principle wherethe action integral is taken over the deformed space Inapplication of pseudo-Finslerian fiber bundle theory similarto that outlined above Fu et al [37] associate 119889 with thedirector of an oriented area element that may be degradedin strength due to damage processes such as fracture or voidgrowth in the material See also related work in [39]

33 Recent Theories in Damage Mechanics and Finite Plastic-ity Saczuk et al [36 38 40] adapted a generalized version ofpseudo-Finsler geometry similar to the fiber bundle approach

of [30] and Section 32 to describe mechanics of solids withmicrostructure undergoing finite elastic-plastic or elastic-damage deformations Key new contributions of these worksinclude definitions of total deformation gradients consist-ing of horizontal and vertical components and Lagrangianfunctions with corresponding energy functionals dependenton total deformations and possibly other state variablesConstitutive relations and balance equations are then derivedfrom variations of such functionals Essential details arecompared and analyzed in the following discussion Notationusually follows that of Section 32 with a few generalizationsdefined as they appear

Define 120577 = (119885 120587119872119880) as a fiber bundle of total space119885 where 120587 119885 rarr 119872 is the projection to base manifold119872 and 119880 is the fiber Dimensions of 119872 and 119880 are 3 and 119901respectively the dimension of 119885 is 119903 = 3 + 119901 Coordinateson 119885 are 119883119860 119863120572 where119883 isin 119872 is a point on the base bodyin its reference configuration and119863 is a vector of dimension119901 Let 1205771015840 = (119885

1015840 12058710158401198721015840 1198801015840) be the deformed image of fiber

bundle 120577 dimensions of1198721015840 and 1198801015840 are 3 and 119901 respectively

the dimension of 1198851015840 is 119903 = 3 + 119901 Coordinates on 1198851015840 are

119909119886 1198891205721015840


its current configuration and 119889 is a vector of dimension 119901Tangent bundles can be expressed as direct sums of horizontaland vertical distributions

119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (51)

Deformation of 119885 to 1198851015840 is locally represented by the smooth

and invertible coordinate transformations

119909119886= 119909119886(119883119860 119863120572) 119889

1205721015840

= 1198611205721015840

120573119863120573 (52)

where in general the director deformation map [38]

1198611205721015840

120573= 1198611205721015840

120573(119883119863) (53)

Total deformation gradientF 119879119885 rarr 1198791198851015840 is defined as

F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

1205721205751205731015840

119886

120597

1205971198891205731015840otimes 120575119863120572 (54)

In component form its horizontal and vertical parts are

119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119862

119888120575119886

119861119909119888

1198651205731015840

120572= 119909119886

1205721205751205731015840

119886= (

120597119909119886

120597119863120572+ 119862119861

119860119862120575119860

120572120575119886

119861120575119862

119888119909119888)1205751205731015840

119886

(55)

To eliminate further excessive use of Kronecker deltas let119901 = 3 119863

119860= 120575119860

120572119863120572 and 119889

119886= 120575119886

12057210158401198891205721015840

Introducinga fundamental Finsler function 119871(119883119863) with properties


described in Section 21 (119884 rarr 119863) the following definitionshold [38]

119866119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863)

120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861

(56)

Notice that Γ119860119861119862

correspond to the Chern-Rund connectionand 119862

119860119861119862to the Cartan tensor In [36] 119863 is presumed

stationary (1198611205721015840

120573rarr 120575

1205721015840

120573) and is associated with a residual

plastic disturbance 119862119860119861119862

is remarked to be associated withdislocation density and the HV-curvature tensor of Γ119860

119861119862(eg

119875119860

119861119862119863of (19)) is remarked to be associated with disclination

density In [38] B is associated with lattice distortion and 1198712

with residual strain energy density of the dislocation density[6 56] In [40] a fundamental function and connectioncoefficients are not defined explicitly leaving the theory opento generalization Note also that (52) is more general than(45) When the latter holds and Γ

119860

119861119862= 0 then F rarr F the

usual deformation gradient of continuum mechanicsRestricting attention to the time-independent case a

Lagrangian is posited of the form

L (119883119863)

= L [119883119863 119909 (119883119863) F (119883119863) Q (119883119863) nablaQ (119883119863)]

(57)

whereQ is a generic vector of state variables and nabla denotes ageneric gradient that may include partial horizontal andorvertical covariant derivatives in the reference configurationas physically and mathematically appropriate Let Ω be acompact domain of 119885 and define

119868 (Ω) = intΩ



119901

L = [1003816100381610038161003816det (119866119860119861)

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816det( 120597

120597119883119860sdot

120597

120597119883119861)10038161003816100381610038161003816100381610038161003816]

12

L

(58)

Euler-Lagrange equations referred to the reference configura-tion follow from the variational principle 120575119868 = 0 analogously

to (42) Conjugate forces to variations in kinematic and statevariables can be defined as derivatives of L with respectto these variables Time dependence dissipation first andsecond laws of thermodynamics and temperature effects arealso considered in [38 40] details are beyond the scope of thisreview In the only known application of Finsler geometry tosolve a boundary value problem in the context of mechanicsof solids with microstructure Stumpf and Saczuk [38] usethe theory outlined above to study localization of plastic slipin a bar loaded in tension with 119863

119860 specifying a preferredmaterial direction for slip In [40] various choices of Q andits gradient are considered in particular energy functions forexample state variables associated with gradients of damageparameters or void volume fractions

4 Towards a New Theory of Structured Media

41 Background and Scope In Section 4 it is shown howFinsler geometry can be applied to describe physical prob-lems in deformable continua with evolving microstructuresin a manner somewhat analogous to the phase field methodPhase field theory [57] encompasses various diffuse interfacemodels wherein the boundary between two (or more) phasesor states ofmaterial is distinguished by the gradient of a scalarfield called an order parameterThe order parameter denotedherein by 120578 typically varies continuously between values ofzero and unity in phases one and two with intermediatevalues in phase boundaries Mathematically

120578 (119883) = 0 forall119883 isin phase 1


120578 (119883) isin (0 1) forall119883 isin interface

(59)

Physically phases might correspond to liquid and solidin melting-solidification problems austenite and martensitein structure transformations vacuum and intact solid infracture mechanics or twin and parent crystal in twinningdescriptions Similarities between phase field theory andgradient-type theories of continuummechanics are describedin [58] both classes of theory benefit from regularizationassociated with a length scale dependence of solutions thatcan render numerical solutions to governing equations meshindependent Recent phase field theories incorporating finitedeformation kinematics include [59] for martensitic trans-formations [54] for fracture [60] for amorphization and[53] for twinning The latter (ie deformation twinning) isthe focus of a more specific description that follows later inSection 43 of this paper though it is anticipated that theFinsler-type description could be adapted straightforwardlyto describe other deformation physics

Deformation twinning involves shearing and lattice rota-tionreflection induced bymechanical stress in a solid crystalThe usual elastic driving force is a resolved shear stress onthe habit plane in the direction of twinning shear Twinningcan be reversible or irreversible depending on material andloading protocol the physics of deformation twinning isdescribed more fully in [61] and Chapter 8 of [6]Theory thatfollows in Section 42 is thought to be the first to recognize


analogies between phase field theory and Finsler geometryand that in Section 43 the first to apply Finsler geometricconcepts to model deformation twinning Notation followsthat of Section 32 and Section 33 with possible exceptionshighlighted as they appear

42 Finsler Geometry and Kinematics As in Section 33define 120577 = (119885 120587119872119880) as a fiber bundle of total space 119885

with 120587 119885 rarr 119872 Considered is a three-dimensional solidbody with state vector D dimensions of 119872 and 119880 are 3the dimension of 119885 is 6 Coordinates on 119885 are 119883

119860 119863119860

(119860 = 1 2 3) where 119883 isin 119872 is a point on the base bodyin its reference configuration Let 1205771015840 = (119885

1015840 12058710158401198721015840 1198801015840) be

the deformed image of fiber bundle 120577 dimensions of 1198721015840

and 1198801015840 are 3 and that of Z1015840 is 6 Coordinates on 119885

1015840 are119909119886 119889119886 where 119909 isin 119872

1015840 is a point on the base body inits current configuration and d is the updated state vectorTangent bundles are expressed as direct sums of horizontaland vertical distributions as in (51)

119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (60)



119909119886= 119909119886(119883119860 119863119860) 119889

119886= 119861119886

119860119863119860 (61)

where the director deformation function is in general

119861119886

119860= 119861119886

119860(119883119863) (62)

Introduce the smooth scalar order parameter field 120578 isin 119872

as in (59) where 120578 = 120578(119883) Here D is identified with thereference gradient of 120578

119863119860(119883) = 120575

119860119861 120597120578 (119883)

120597119883119861 (63)

For simplicity here 119883119860 is taken as a Cartesian coordinate

chart on 119872 such that 119883119860 = 120575119860119861

119883119861 and so forth Similarly

119909119886 is chosen Cartesian Generalization to curvilinear coor-

dinates is straightforward but involves additional notationDefine the partial deformation gradient

119865119886

119860(119883119863) =

120597119909119886(119883119863)

120597119883119860 (64)

By definition and then from the chain rule

119889119886[119909 (119883119863) 119863] = 120575

119886119887 120597120578 (119883)

120597119909119887

= 120575119886119887

(120597119883119860

120597119909119887)(

120597120578

120597119883119860)

= 119865minus1119860

119887120575119886119887120575119860119861

119863119861

(65)

leading to

119861119886

119860(119883119863) = 119865

minus1119861

119887(X 119863) 120575

119886119887120575119860119861

(66)

From (63) and (65) local integrability (null curl) conditions120597119863119860120597119883119861

= 120597119863119861120597119883119860 and 120597119889

119886120597119909119887

= 120597119889119887120597119909119886 hold

As in Section 3 119873119860

119861(119883119863) denote nonlinear connection

coefficients on 119885 and the Finsler-type nonholonomic bases

120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861 (67)

Let 119873119886119887= 119873119860

119861120575119886

119860120575119861

119887denote nonlinear connection coefficients

on 1198851015840 [38] and define the nonholonomic spatial bases as

120575

120575119909119886=

120597

120597119909119886minus 119873119887

119886

120597

120597119889119887 120575119889

119886= 119889119889119886+ 119873119886

119887119889119909119887 (68)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

119860

120597

120597119889119886otimes 120575119863119860 (69)

In component form its horizontal and vertical parts are as in(55) and the theory of [38]

119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119886

119861120575119862

119888119909119888

119865119886

119860= 119909119886

119860=

120597119909119886

120597119863119860+ 119862119861

119860119862120575119886

119861120575119862

119888119909119888

(70)

As in Section 33 a fundamental Finsler function 119871(119883119863)homogeneous of degree one in119863 is introduced Then

G119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861 (71)

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863) (72)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862 (73)

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862 (74)

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863) (75)

Specifically for application of the theory in the context ofinitially homogeneous single crystals let

119871 = [120581119860119861

119863119860119863119861]12

= [120581119860119861

(120597120578

120597119883119860)(

120597120578

120597119883119861)]

12

(76)

120581119860119861

= 119866119860119861

= 120575119860119862

120575119861119863

119866119862119863

= constant (77)

The metric tensor with Cartesian components 120581119860119861

is iden-tified with the gradient energy contribution to the surfaceenergy term in phase field theory [53] as will bemade explicitlater in Section 43 From (76) and (77) reference geometry 120577is now Minkowskian since the fundamental Finsler function


119871 = 119871(119863) is independent of 119883 In this case (72)ndash(75) and(67)-(68) reduce to

Γ119860

119861119862= 120574119860

119861119862= 119862119860

119861119862= 0 119873

119860

119861= 0 119899

119886

119887= 0

120575

120575119883119860=

120597

120597119883119860 120575119863

119860= 119889119863119860

120575

120575119909119886=

120597

120597119909119886 120575119889

119886= 119889119889119886

(78)

Horizontal and vertical covariant derivatives in (69) reduceto partial derivatives with respect to119883

119860 and119863119860 respectively

leading to

F =120597119909119886

120597119883119860120597

120597119909119886otimes 119889119883119860+

120597119909119886

120597119863119860120597

120597119889119886otimes 119889119863119860

= 119865119886

119860

120597

120597119909119886otimes 119889119883119860+ 119881119886

119860

120597

120597119889119886otimes 119889119863119860

(79)

When 119909 = 119909(119883) then vertical deformation components119865119886

119860= 119881119886

119860= 0 In a more general version of (76) applicable

to heterogeneousmaterial properties 120581119860119861

= 120581119860119861

(119883119863) and ishomogeneous of degree zero with respect to119863 and the abovesimplifications (ie vanishing connection coefficients) neednot apply

43 Governing Equations Twinning Application Consider acrystal with a single potentially active twin system Applying(59) let 120578(119883) = 1 forall119883 isin twinned domains 120578(119883) = 0 forall119883 isin

parent (original) crystal domains and 120578(119883) isin (0 1) forall119883 isin intwin boundary domains As defined in [53] let 120574

0denote the

twinning eigenshear (a scalar constant)m = 119898119860119889119883119860 denote

the unit normal to the habit plane (ie the normal covectorto the twin boundary) and s = 119904

119860(120597120597119883

119860) the direction of

twinning shear In the context of the geometric frameworkof Section 42 and with simplifying assumptions (76)ndash(79)applied throughout the present application s isin 119879119872 andm isin 119879

lowast119872 are constant fields that obey the orthonormality

conditions

⟨sm⟩ = 119904119860119898119861⟨

120597

120597119883119860 119889119883119861⟩ = 119904

119860119898119860= 0 (80)

Twinning deformation is defined as the (11) tensor field

Ξ = Ξ119860

119861

120597

120597119883119860otimes 119889119883119861= 1 + 120574

0120593s otimesm

Ξ119860

119861[120578 (119883)] = 120575

119860

119861+ 1205740120593 [120578 (119883)] 119904

119860119898119861

(81)

Note that detΞ = 1 + 1205740120593⟨sm⟩ = 1 Scalar interpolation

function 120593(120578) isin [0 1] monotonically increases between itsendpoints with increasing 120578 and it satisfies 120593(0) = 0 120593(1) =

1 and 1205931015840(0) = 120593

1015840(1) = 0 A typical example is the cubic

polynomial [53 59]

120593 [120578 (119883)] = 3 [120578 (119883)]2minus 2 [120578 (119883)]

3 (82)

The horizontal part of the deformation gradient in (79) obeysa multiplicative decomposition

F = AΞ 119865119886

119860= 119860119886

119861Ξ119861

119860 (83)

The elastic lattice deformation is the two-point tensor A

A = 119860119886

119860

120597

120597119909119886otimes 119889119883119861= FΞminus1 = F (1 minus 120574

0120593s otimesm) (84)

Note that (83) can be considered a version of the Bilby-Kroner decomposition proposed for elastic-plastic solids [862] and analyzed at length in [5 11 26] from perspectives ofdifferential geometry of anholonomic space (neither A norΞ is necessarily integrable to a vector field) The followingenergy potentials measured per unit reference volume aredefined

120595 (F 120578D) = 119882(F 120578) + 119891 (120578D) (85)

119882 = 119882[A (F 120578)] 119891 (120578D) = 1205721205782(1 minus 120578)

2+ 1198712(D)

(86)

Here 119882 is the strain energy density that depends on elasticlattice deformation A (assuming 119881

119886

119860= 119909119886

119860= 0 in (86))

119891 is the total interfacial energy that includes a double-wellfunction with constant coefficient 120572 and 119871

2= 120581119860119861

119863119860119863119861

is the square of the fundamental Finsler function given in(76) For isotropic twin boundary energy 120581

119860119861= 1205810120575119860119861 and

equilibrium surface energy Γ0and regularization width 119897

0

obey 1205810= (34)Γ

01198970and 120572 = 12Γ

01198970[53] The total energy

potential per unit volume is 120595 Let Ω be a compact domainof119885 which can be identified as a region of the material bodyand define the total potential energy functional

Ψ [119909 (119883) 120578 (119883)] = intΩ

120595dΩ

dΩ =1003816100381610038161003816det (119866119860119861)

100381610038161003816100381612

1198891198831sdot sdot sdot and 119889119883


119901

(87)

Recall that for solid bodies 119899 = 119901 = 3 For quasi-static con-ditions (null kinetic energy) the Lagrangian energy densityis L = minus120595 The null first variation of the action integralappropriate for essential (Dirichlet) boundary conditions onboundary 120597Ω is

120575119868 (Ω) = minus120575Ψ = minusintΩ

120575120595dΩ = 0 (88)

where the first variation of potential energy density is definedhere by varying 119909 and 120578 within Ω holding reference coordi-nates119883 and reference volume form dΩ fixed

120575120595 =120597119882

120597119865119886119860

120597120575119909119886

120597119883119860+ (

120597119882

120597120578+

120597119891

120597120578) 120575120578 +

120597119891

120597119863119860120597120575120578

120597119883119860 (89)

Application of the divergence theorem here with vanishingvariations of 119909 and 120578 on 120597Ω leads to the Euler-Lagrangeequations [53]

120597119875119860

119886

120597119883119860=

120597 (120597119882120597119909119886

|119860)

120597119883119860= 0

120589 +120597119891

120597120578= 2120581119860119861

120597119863119860

120597119883119861

= 2120581119860119861

1205972120578

120597119883119860120597119883119861

(90)


The first Piola-Kirchhoff stress tensor is P = 120597119882120597F andthe elastic driving force for twinning is 120589 = 120597119882120597120578 Theseequations which specifymechanical and phase equilibria areidentical to those derived in [53] but have arrived here viause of Finsler geometry on fiber bundle 120577 In order to achievesuch correspondence simplifications 119871(119883119863) rarr 119871(119863) and119909(119883119863) rarr 119909(119883) have been applied the first reducingthe fundamental Finsler function to one of Minkowskiangeometry to describe energetics of twinning in an initiallyhomogeneous single crystal body (120581

119860B = constant)Amore general and potentially powerful approach would

be to generalize fundamental function 119871 and deformedcoordinates 119909 to allow for all possible degrees-of-freedomFor example a fundamental function 119871(119883119863) correspondingto nonuniform values of 120581

119860119861(119883) in the vicinity of grain

or phase boundaries wherein properties change rapidlywith position 119883 could be used instead of a Minkowskian(position-independent) fundamental function 119871(119863) Suchgeneralization would lead to enriched kinematics and nonva-nishing connection coefficients and itmay yield new physicaland mathematical insight into equilibrium equationsmdashforexample when expressed in terms of horizontal and verticalcovariant derivatives [30]mdashused to describe mechanics ofinterfaces and heterogeneities such as inclusions or otherdefects Further study to be pursued in the future is neededto relate such a general geometric description to physicalprocesses in real heterogeneous materials An analogoustheoretical description could be derived straightforwardly todescribe stress-induced amorphization or cleavage fracturein crystalline solids extending existing phase field models[54 60] of such phenomena

5 Conclusions

Finsler geometry and its prior applications towards contin-uum physics of materials with microstructure have beenreviewed A new theory in general considering a deformablevector bundle of Finsler character has been posited whereinthe director vector of Finsler space is associated with agradient of a scalar order parameter It has been shownhow a particular version of the new theory (Minkowskiangeometry) can reproduce governing equations for phasefield modeling of twinning in initially homogeneous singlecrystals A more general approach allowing the fundamentalfunction to depend explicitly on material coordinates hasbeen posited that would offer enriched description of inter-facial mechanics in polycrystals or materials with multiplephases

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] C A Truesdell and R A Toupin ldquoThe classical field theoriesrdquoin Handbuch der Physik S Flugge Ed vol 31 pp 226ndash793Springer Berlin Germany 1960

[2] A C EringenNonlinearTheory of ContinuousMediaMcGraw-Hill New York NY USA 1962

[3] A C Eringen ldquoTensor analysisrdquo in Continuum Physics A CEringen Ed vol 1 pp 1ndash155 Academic Press New York NYUSA 1971

[4] R A Toupin ldquoTheories of elasticity with couple-stressrdquoArchivefor Rational Mechanics and Analysis vol 17 pp 85ndash112 1964

[5] J D Clayton ldquoOn anholonomic deformation geometry anddifferentiationrdquoMathematics andMechanics of Solids vol 17 no7 pp 702ndash735 2012

[6] J D Clayton Nonlinear Mechanics of Crystals Springer Dor-drecht The Netherlands 2011

[7] B A Bilby R Bullough and E Smith ldquoContinuous distribu-tions of dislocations a new application of the methods of non-Riemannian geometryrdquo Proceedings of the Royal Society A vol231 pp 263ndash273 1955

[8] E Kroner ldquoAllgemeine kontinuumstheorie der versetzungenund eigenspannungenrdquo Archive for Rational Mechanics andAnalysis vol 4 pp 273ndash334 1960

[9] K Kondo ldquoOn the analytical and physical foundations of thetheory of dislocations and yielding by the differential geometryof continuardquo International Journal of Engineering Science vol 2pp 219ndash251 1964

[10] B A Bilby L R T Gardner A Grinberg and M ZorawskildquoContinuous distributions of dislocations VI Non-metricconnexionsrdquo Proceedings of the Royal Society of London AMathematical Physical and Engineering Sciences vol 292 no1428 pp 105ndash121 1966

[11] W Noll ldquoMaterially uniform simple bodies with inhomo-geneitiesrdquo Archive for Rational Mechanics and Analysis vol 27no 1 pp 1ndash32 1967

[12] K Kondo ldquoFundamentals of the theory of yielding elemen-tary and more intrinsic expositions riemannian and non-riemannian terminologyrdquoMatrix and Tensor Quarterly vol 34pp 55ndash63 1984

[13] J D Clayton D J Bammann andD LMcDowell ldquoA geometricframework for the kinematics of crystals with defectsrdquo Philo-sophical Magazine vol 85 no 33ndash35 pp 3983ndash4010 2005

[14] J D Clayton ldquoDefects in nonlinear elastic crystals differentialgeometry finite kinematics and second-order analytical solu-tionsrdquo Zeitschrift fur Angewandte Mathematik und Mechanik2013

[15] A Yavari and A Goriely ldquoThe geometry of discombinationsand its applications to semi-inverse problems in anelasticityrdquoProceedings of the Royal Society of London A vol 470 article0403 2014

[16] D G B Edelen and D C Lagoudas Gauge Theory andDefects in Solids North-Holland Publishing Amsterdam TheNetherlands 1988

[17] I A Kunin ldquoKinematics of media with continuously changingtopologyrdquo International Journal of Theoretical Physics vol 29no 11 pp 1167ndash1176 1990

[18] H Weyl Space-Time-Matter Dover New York NY USA 4thedition 1952

[19] J A Schouten Ricci Calculus Springer Berlin Germany 1954[20] J L Ericksen ldquoTensor Fieldsrdquo in Handbuch der Physik S

Flugge Ed vol 3 pp 794ndash858 Springer BerlinGermany 1960[21] T Y Thomas Tensor Analysis and Differential Geometry Aca-

demic Press New York NY USA 2nd edition 1965[22] M A Grinfeld Thermodynamic Methods in the Theory of

Heterogeneous Systems Longman Sussex UK 1991


[23] H Stumpf and U Hoppe ldquoThe application of tensor algebra onmanifolds to nonlinear continuum mechanicsmdashinvited surveyarticlerdquo Zeitschrift fur Angewandte Mathematik und Mechanikvol 77 no 5 pp 327ndash339 1997

[24] P Grinfeld Introduction to Tensor Analysis and the Calculus ofMoving Surfaces Springer New York NY USA 2013

[25] P Steinmann ldquoOn the roots of continuum mechanics indifferential geometrymdasha reviewrdquo in Generalized Continua fromthe Theory to Engineering Applications H Altenbach and VA Eremeyev Eds vol 541 of CISM International Centre forMechanical Sciences pp 1ndash64 Springer Udine Italy 2013

[26] J D ClaytonDifferential Geometry and Kinematics of ContinuaWorld Scientific Singapore 2014

[27] V A Eremeyev L P Lebedev andH Altenbach Foundations ofMicropolar Mechanics Springer Briefs in Applied Sciences andTechnology Springer Heidelberg Germany 2013

[28] P Finsler Uber Kurven und Flachen in allgemeinen Raumen[Dissertation] Gottingen Germany 1918

[29] H Rund The Differential Geometry of Finsler Spaces SpringerBerlin Germany 1959

[30] A Bejancu Finsler Geometry and Applications Ellis HorwoodNew York NY USA 1990

[31] D Bao S-S Chern and Z Shen An Introduction to Riemann-Finsler Geometry Springer New York NY USA 2000

[32] A Bejancu and H R Farran Geometry of Pseudo-FinslerSubmanifolds Kluwer Academic Publishers Dordrecht TheNetherlands 2000

[33] E Kroner ldquoInterrelations between various branches of con-tinuum mechanicsrdquo in Mechanics of Generalized Continua pp330ndash340 Springer Berlin Germany 1968

[34] E Cartan Les Espaces de Finsler Hermann Paris France 1934[35] S Ikeda ldquoA physico-geometrical consideration on the theory of

directors in the continuum mechanics of oriented mediardquo TheTensor Society Tensor New Series vol 27 pp 361ndash368 1973

[36] J Saczuk ldquoOn the role of the Finsler geometry in the theory ofelasto-plasticityrdquo Reports onMathematical Physics vol 39 no 1pp 1ndash17 1997

[37] M F Fu J Saczuk andH Stumpf ldquoOn fibre bundle approach toa damage analysisrdquo International Journal of Engineering Sciencevol 36 no 15 pp 1741ndash1762 1998

[38] H Stumpf and J Saczuk ldquoA generalized model of oriented con-tinuum with defectsrdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 80 no 3 pp 147ndash169 2000

[39] J Saczuk ldquoContinua with microstructure modelled by thegeometry of higher-order contactrdquo International Journal ofSolids and Structures vol 38 no 6-7 pp 1019ndash1044 2001

[40] J Saczuk K Hackl and H Stumpf ldquoRate theory of nonlocalgradient damage-gradient viscoinelasticityrdquo International Jour-nal of Plasticity vol 19 no 5 pp 675ndash706 2003

[41] S Ikeda ldquoOn the theory of fields in Finsler spacesrdquo Journal ofMathematical Physics vol 22 no 6 pp 1215ndash1218 1981

[42] H E Brandt ldquoDifferential geometry of spacetime tangentbundlerdquo International Journal of Theoretical Physics vol 31 no3 pp 575ndash580 1992

[43] I Suhendro ldquoA new Finslerian unified field theory of physicalinteractionsrdquo Progress in Physics vol 4 pp 81ndash90 2009

[44] S-S Chern ldquoLocal equivalence and Euclidean connectionsin Finsler spacesrdquo Scientific Reports of National Tsing HuaUniversity Series A vol 5 pp 95ndash121 1948

[45] K Yano and E T Davies ldquoOn the connection in Finsler spaceas an induced connectionrdquo Rendiconti del CircoloMatematico diPalermo Serie II vol 3 pp 409ndash417 1954

[46] A Deicke ldquoFinsler spaces as non-holonomic subspaces of Rie-mannian spacesrdquo Journal of the London Mathematical Societyvol 30 pp 53ndash58 1955

[47] S Ikeda ldquoA geometrical construction of the physical interactionfield and its application to the rheological deformation fieldrdquoTensor vol 24 pp 60ndash68 1972

[48] Y Takano ldquoTheory of fields in Finsler spaces Irdquo Progress ofTheoretical Physics vol 40 pp 1159ndash1180 1968

[49] J D Clayton D L McDowell and D J Bammann ldquoModelingdislocations and disclinations with finite micropolar elastoplas-ticityrdquo International Journal of Plasticity vol 22 no 2 pp 210ndash256 2006

[50] T Hasebe ldquoInteraction fields based on incompatibility tensorin field theory of plasticityndashpart I theoryrdquo Interaction andMultiscale Mechanics vol 2 no 1 pp 1ndash14 2009

[51] J D Clayton ldquoDynamic plasticity and fracture in high densitypolycrystals constitutive modeling and numerical simulationrdquoJournal of the Mechanics and Physics of Solids vol 53 no 2 pp261ndash301 2005

[52] J RMayeur D LMcDowell andD J Bammann ldquoDislocation-based micropolar single crystal plasticity comparison of multi-and single criterion theoriesrdquo Journal of the Mechanics andPhysics of Solids vol 59 no 2 pp 398ndash422 2011

[53] J D Clayton and J Knap ldquoA phase field model of deformationtwinning nonlinear theory and numerical simulationsrdquo PhysicaD Nonlinear Phenomena vol 240 no 9-10 pp 841ndash858 2011

[54] J D Clayton and J Knap ldquoA geometrically nonlinear phase fieldtheory of brittle fracturerdquo International Journal of Fracture vol189 no 2 pp 139ndash148 2014

[55] J D Clayton and D L McDowell ldquoA multiscale multiplicativedecomposition for elastoplasticity of polycrystalsrdquo InternationalJournal of Plasticity vol 19 no 9 pp 1401ndash1444 2003

[56] J D Clayton ldquoAn alternative three-term decomposition forsingle crystal deformation motivated by non-linear elasticdislocation solutionsrdquo The Quarterly Journal of Mechanics andApplied Mathematics vol 67 no 1 pp 127ndash158 2014

[57] H Emmerich The Diffuse Interface Approach in MaterialsScience Thermodynamic Concepts and Applications of Phase-Field Models Springer Berlin Germany 2003

[58] G Z Voyiadjis and N Mozaffari ldquoNonlocal damage modelusing the phase field method theory and applicationsrdquo Inter-national Journal of Solids and Structures vol 50 no 20-21 pp3136ndash3151 2013

[59] V I Levitas V A Levin K M Zingerman and E I FreimanldquoDisplacive phase transitions at large strains phase-field theoryand simulationsrdquo Physical Review Letters vol 103 no 2 ArticleID 025702 2009

[60] J D Clayton ldquoPhase field theory and analysis of pressure-shearinduced amorphization and failure in boron carbide ceramicrdquoAIMS Materials Science vol 1 no 3 pp 143ndash158 2014

[61] V S Boiko R I Garber and A M Kosevich Reversible CrystalPlasticity AIP Press New York NY USA 1994

[62] B A Bilby L R T Gardner and A N Stroh ldquoContinuousdistributions of dislocations and the theory of plasticityrdquo in Pro-ceedings of the 9th International Congress of Applied Mechanicsvol 8 pp 35ndash44 University de Bruxelles Brussels Belgium1957

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Riemannian metrics and affine connections include [16 17]Relevant references in geometry and mathematical physicsinclude [18ndash26] in addition to those already mentionedFinite deformation director theories of micropolar type areaddressed in the context of Riemannian (as opposed toFinslerian) metrics in [1 27]

Finsler geometry first attributed to Finsler in 1918 [28]is more general than Riemannian geometry in the sense thatthe fundamental (metric) tensor generally may depend onadditional independent variables labeled here as 119910 and 119884

in spatial and material configurations with correspondinggeneralized coordinates 119910

119886 and 119884

119860 Formal definitions

will be given later in this paper for the present immediatediscussion it suffices to mention that each point can be con-sidered endowedwith additional degrees-of-freedombeyond119909 or 119883 and that transformation laws among coordinates aswell as connection coefficients (ie covariant differentials)generally depend on 119910 or 119884 as well as 119909 or 119883 Relevantreferences in mathematics include [29ndash32] For descriptionsof mechanics of solids additional degrees-of-freedom can beassociated with evolving features of the microstructure of thematerial though more general physical interpretations arepossible

The use of Finsler geometry to describe continuummechanical behavior of solids was perhaps first noted byKroner in 1968 [33] and Eringen in 1971 [3] the latterreference incorporating some basic identities and definitionsderived primarily by Cartan [34] though neither devel-oped a Finsler-based framework more specifically directedtowards mechanics of continua The first theory of Finslergeometry applied to continuum mechanics of solids withmicrostructure appears to be the purely kinematic theory ofIkeda [35] in a generalization of Cosserat-type kinematicswhereby additional degrees-of-freedom are director vectorslinked to structure This theory was essentially extended byBejancu [30] to distinguish among horizontal and verticaldistributions of the fiber bundle of a deforming pseudo-Finslerian total space More complete theories incorporatinga Lagrangian functional (leading to physical balance orconservation laws) and couched in terms of Finsler geom-etry were developed by Saczuk Stumpf and colleagues fordescribing solids undergoing inelastic deformation mecha-nisms associated with plasticity andor damage [36ndash40] Tothe authorrsquos knowledge solution of a boundary value problemin solid mechanics using Finsler geometric theory has onlybeen reported once in [38] Finsler geometry has beenanalogously used to generalize fundamental descriptionsin other disciplines of physics such as electromagnetismquantum theory and gravitation [30 41ndash43]

This paper is organized as follows In Section 2 requisitemathematical background on Finsler geometry (sometimescalled Riemann-Finsler geometry [31]) is summarized InSection 3 the aforementioned theories from continuumphysics of solids [30 35ndash38 40] are reviewed and comparedIn Section 4 aspects of a new theory with a primary intentionof description of structural transformation processes in realmaterials are proposed and evaluated Conclusions follow inSection 5

2 Finsler Geometry Background

Notation used in the present section applies to a referentialdescription that is the initial state analogous formulae applyfor a spatial description that is a deformed body

21 Coordinates and Fundamentals Denote by 119872 an 119899-dimensional119862infinmanifold Each element (of support) of119872 isof the form (119883 119884) where 119883 isin 119872 and 119884 isin 119879119872 with 119879119872 thetangent bundle of 119872 A Finsler structure of 119872 is a function119871 119879119872 rarr [0infin) with the following three properties [31]

(i) The fundamental function 119871 is 119862infin on 119879119872 0(ii) 119871(119883 120582119884) = 120582119871(119883 119884) forall120582 gt 0 (ie 119871 is homogeneous

of degree one in 119884)

(iii) the fundamental tensor 119866119860119861

= (12)1205972(1198712)120597119884119860120597119884119861

is positive definite at every point of 119879119872 0

Restriction of 119871 to a particular tangent space 119879119883119872 gives rise

to a (local) Minkowski norm

1198712(119884) = 119866

119860119861 (119884) 119884119860119884119861 (1)

which follows from Eulerrsquos theorem and the identity

119866119860119861

=1198711205972119871

120597119884119860120597119884119861+ (

120597119871

120597119884119860)(

120597119871

120597119884119861) (2)

Specifically letting 119884119860

rarr d119883119860 the length of a differentialline element at119883 depends in general on both119883 and 119884 as

|dX (119883 119884)| = radicdX sdot dX = [119866119860119861 (119883 119884) d119883119860d119883119861]

12

(3)

A Finsler manifold (119872 119865) reduces to a Minkowskian man-ifold when 119871 does not depend on 119883 and to a Riemannianmanifold when 119871 does not depend on 119884 In the lattercase a Riemannian metric tensor is 119866

119860119861(119883)119889119883

119860otimes 119889119883

119861Cartanrsquos tensor with the following fully symmetric covariantcomponents is defined for use later

119862119860119861119862

=1

2

120597119866119860119861

120597119884119862=

1

4

1205973(1198712)

120597119884119860120597119884119861120597119884119862 (4)

Consider now a coordinate transformation to anotherchart on119872 for example

119883119860= 119883119860(1198831 1198832 119883

119899)

119860= (

120597119883119860

120597119883119861)119884119861 (5)

From the chain rule holonomic basis vectors on 119879119872 thentransform as [30 31]

120597

120597119883119860=

120597119883119861

120597119883119860

120597

120597119883119861+

1205972119883119861

120597119883119860120597119883119862119862 120597

120597119884119861 (6)

120597

120597119860=

120597119883119861

120597119883119860

120597

120597119884119861 (7)



120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863) (8)



119873119860

119861= 120574119860

119861119862119884119862minus 119862119860

119861119862120574119862

119863119864119884119863119884119864=

1

2

120597119866119860

120597119884119861 (9)

where 119866119860

= 120574119860


are then introduced

120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119884119861 120575119884

119860= 119889119884119860+ 119873119860

119861119889119883119861 (10)


119860 120597120597119884




G (119883 119884) = 119866119860119861

119889119883119860otimes 119889119883119861+ 119866119860119861

120575119884119860otimes 120575119884119861 (11)






nabla120592

120597

120597119883119860= 120596119861

119860(120592)

120597

120597119883119861 nabla

120592119889119883119860= minus120596119860

119861(120592) 119889119883

119861 (12)


lowast119879119872


119889 (119889119883119860) minus 119889119883

119861and 120596119860

119861= 0

119889119866119860119861

minus 119866119861119862

120596119862

119860minus 119866119860119862

120596119862

119861= 2119862119860119861119862

120575119884119862

(13)


120596119860

119861= Γ119860

119862119861119889119883119862 Γ

119860

119861119862= Γ119860

119862119861 (14)


Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863) (15)


119861= 0 and Γ

119860

119861119862= 120574119860



119861+ 119862119860

119863119861120575119884119863 where 120596

119860




119860

119861119862= 120574119860

119861119862= 0 Gradients of


nabla120575120575119883119860

120575

120575119883119861= Γ119862

119860119861

120575

120575119883119862 nabla

120575120575119883119860

120597

120597119884119861= 119862119862

119860119861

120597

120597119884119862 (16)


1

1)


119861(120597120597119883

119860) otimes 119889119883


(nabla119879)119860

119861= 119889119879119860

119861+ 119879119862

119861120596119860

119862minus 119879119860

119862120596119862

119861

= 119879119860

119861|119862119889119883119862+ 119879119860

119861119862120575119884119862

= (nabla120575120575119883119862119879)119860

119861119889119883119862+ (nabla120597120597119884119862119879)119860

119861120575119884119862

= (120575119879119860

119861

120575119883119862+ 119879119863

119861Γ119860

119862119863minus 119879119860

119863Γ119863

119862119861)119889119883119862+ (

120597119879119860

119861

120597119884119862)120575119884119862

(17)


and (sdot)119860



Ω119860

119861= 119889 (120596

119860

119861) minus 120596119862

119861and 120596119860

119862

=1

2119877119860

119861119862119863119889119883119862and 119889119883119863+ 119875119860

119861119862119863119889119883119862and 120575119884119863

+1

2119876119860

119861119862119863120575119884119862and 120575119884119863

(18)


119877119860

119861119862119863=

120575Γ119860

119861119863

120575119883119862minus

120575Γ119860

119861119862

120575119883119863+ Γ119860

119864119862Γ119864

119861119863minus Γ119860

119864119863Γ119864

119861119862

119875119860

119861119862119863= minus

120597Γ119860

119861119862

120597119884119863 119876

119860

119861119862119863= 0

(19)


119861119862119863= 119875119860

119862119861119863


119877119860

119861119862119863+ 119877119860

119862119863119861+ 119877119860

119863119861119862= 0 (20)


119860



119861119862 and 119875

119860






[120575

120575119883119860

120575

120575119883119861] = (

120575119873119862

119860

120575119883119861minus

120575119873119862

119861

120575119883119860)

120597

120597119884119862= Λ119862

119860119861

120597

120597119884119862 (21)




Λ119862

119860119861= minus119877119862

119863119860119861119884119863 (22)










rarr 119863119860





119909119886= 119909119886(119883119860) 119889

119886

(120572)= 119889119886

(120572)[119863119860

(120573)(119883119861)] = 119889

119886

(120572)(119883119861)

(23)


119865119886

119860=

120597119909119886

120597119883119860 119865

minus1119860

119886=

120597119883119860

120597119909119886 (24)


d119909119886 = 119865119886

119860d119883119860 119889

119886

(120572)= 119861119886

119860119863119860

(120572) (25)

with 119861119886

119860= 119861119886


d119889119886(120572)

= 119864119886

(120572)119860d119883119860 = (119861

119886

119861119860119863119861

(120572)+ 119861119886

119861119863119861

(120572)119860) d119883119860 (26)

where (sdot)119860


d119889119886(120572)

= 119891119886

(120572)119887d119909119887 = 119865

minus1119860

119887119864119886

(120572)119860d119909119887 (27)

Let 119866119860119861


119860119861d119883119861 is a measure

of length (119866119860119861

= 120575119860119861



119862119886119887

= 119865minus1119860

119886119866119860119861

119865minus1119861

119887 119862

(120572120573)

119886119887= 119864minus1119860

(120572)119886119866119860119861

119864minus1119861

(120572)119887 (28)

Let Γ119860

119861119862and Γ

120572



Γ119860

119861119862= 119863119860

(120572)119863(120573)

119861119863(120574)

119862Γ120572

120573120574+ 119863119860

(120572)119863(120572)

119862119861 (29)

where 119863119860

(120572)119863(120572)

119861= 120575119860

119861and 119863

119860

(120573)119863(120572)

119860= 120575120572




(nabla119881)119860

= 119889119881119860+ Γ119860

119861119862119881119862d119883119861 (30)



(nabla120592)119886

= 119889120592119886+ Γ119886

119887119888120592119888d119909119887 + Γ

(120572)119886

119887119888120592119888d119909119887 (31)


) given by119889120592119886 =(120597120592119886120597119909119887)d119909119887 + (120597120592

119886120597119889119887


119867(120572)119886

119887119888= 119865119886

119860119864minus1119860

(120572)119888119887 119870

(120572)119886

119887119888= 119864119886

(120572)119860119865minus1119860

119888119887 (32)


120572



120572







119860 119863120572 where 119883 isin 119872




120575

120575119883119860=

120597

120597119883119860minus 119873120572

119860

120597

120597119863120572 120575119863

120572= 119889119863120572+ 119873120572

119860119889119883119860 (33)



119879119885 = 119867119885 oplus 119881119885 (34)



G (119883119863) = 119866119860119861 (119883119863) 119889119883

119860otimes 119889119883119861

+ 119866120572120573 (119883119863) 120575119863

120572otimes 120575119863120573

(35)


nabla120575120575119883119860

120575

120575119883119861= Γ119862

119860119861

120575

120575119883119862 nabla

120575120575119883119860

120597

120597119863120572= 119862120573

119860120572

120597

120597119863120573 (36)

nabla120597120597119863120572

120575

120575119883119860= 119888119861

120572119860

120575

120575119883119861 nabla

120597120597119863120572

120597

120597119863120573= 120594120574

120572120573

120597

120597119863120574 (37)


119860) and a




119881119860

|119861=

120575119881119860

120575119883119861+ Γ119860

119861119862119881119862 119882

120572

|119861=

120575119882120572

120575119883119861+ 119862120572

119861120573119882120573

119881119860

||120573=

120575119881119860

120575119863120573+ 119888119860

120573119862119881119862 119882

120572

||120573=

120575119882120572

120575119863120573+ 120594120572

120573120575119882120575

(38)


Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863)

119862120572

119860120573=

120597119873120572

119860

120597119863120573 119888

119860

120572119861= 0

120594120572

120573120574=

1

2119866120572120575

(120597119866120573120575

120597119863120574+

120597119866120574120575

120597119863120573minus

120597119866120573120574

120597119863120575)

(39)





L (119883119863) = L[119866119860119861

119866120572120573 119876119870120575119876119870

120575119883119860120597119876119870

120597119863120572] (40)


119868 (Ω) = intΩ



119901

L = [10038161003816100381610038161003816det (119866

119860119861) det (119866

120572120573)10038161003816100381610038161003816]12

L

(41)


120597L

120597119876119870=

120597

120597119883119860[

120597L

120597 (120597119876119870120597119883119860)] +

120597

120597119863120572[

120597L

120597 (120597119876119870120597119863120572)]

(42)


Let 1205771015840




1015840




Let119873120572

1015840



120575

120575119909119886=

120597

120597119909119886minus 1198731205721015840

119886

120597

1205971198891205721015840 120575119889

1205721015840

= 1198891198891205721015840

+ 1198731205721015840

119886119889119909119886 (43)


119886 120597120597119889

1205721015840


where

1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (44)


Dual set 119889119909119886 1205751198891205721015840



coordinates

119909119886= 119909119886(119883119860) 119889

1205721015840

= 1198611205721015840

120573(119883)119863

120573 (45)


119865119886

119860=

120597119909119886

120597119883119860 119865

minus1119860

119886=

120597119883119860

120597119909119886 (46)


120597

120597119883119860= 119865119886

119860

120597

120597119909119886+

1205971198611205721015840

120573

120597119883119860119863120573 120597

1205971198891205721015840

120597

120597119863120572= 1198611205731015840

120572

120597

1205971198891205731015840

(47)


by

1198731205721015840

119886119865119886

119860= 119873120573

1198601198611205721015840

120573minus (

1205971198611205721015840

120573

120597119883119860)119863120573 (48)


119866119886119887119865119886

119860119865119887

119861= 119866119860119861

119866120572101584012057310158401198611205721015840

1205751198611205731015840

120574= 119866120575120574 (49)


Γ119886

119887119888119865119887

119861119865119888

119862= Γ119860

119861119862119865119886

119860minus

120597119865119886

119862

120597119883119861

1198621205721015840

1198861205731015840119865119886

1198601198611205731015840

120574= 119862120583

1198601205741198611205721015840

120583minus

1205971198611205721015840

120574

120597119883119860

119888119886

1205721015840119887119865119887

1198601198611205721015840

120573= 119888119861

120573119860119865119886

119861

1205941205721015840

120573101584012057510158401198611205731015840

1205741198611205751015840

120578= 120594120576

1205741205781198611205721015840

120576

(50)








119909119886 1198891205721015840



119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (51)



119909119886= 119909119886(119883119860 119863120572) 119889

1205721015840

= 1198611205721015840

120573119863120573 (52)


1198611205721015840

120573= 1198611205721015840

120573(119883119863) (53)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

1205721205751205731015840

119886

120597

1205971198891205731015840otimes 120575119863120572 (54)


119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119862

119888120575119886

119861119909119888

1198651205731015840

120572= 119909119886

1205721205751205731015840

119886= (

120597119909119886

120597119863120572+ 119862119861

119860119862120575119860

120572120575119886

119861120575119862

119888119909119888)1205751205731015840

119886

(55)


119860= 120575119860

120572119863120572 and 119889

119886= 120575119886

12057210158401198891205721015840




119866119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863)

120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861

(56)

Notice that Γ119860119861119862



stationary (1198611205721015840

120573rarr 120575

1205721015840




119861119862(eg

119875119860




119860

119861119862= 0 then F rarr F the



L (119883119863)

= L [119883119863 119909 (119883119863) F (119883119863) Q (119883119863) nablaQ (119883119863)]

(57)


119868 (Ω) = intΩ



119901

L = [1003816100381610038161003816det (119866119860119861)

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816det( 120597

120597119883119860sdot

120597

120597119883119861)10038161003816100381610038161003816100381610038161003816]

12

L

(58)









(59)







119860 119863119860


1015840 12058710158401198721015840 1198801015840) be



1015840 are119909119886 119889119886 where 119909 isin 119872


119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (60)



119909119886= 119909119886(119883119860 119863119860) 119889

119886= 119861119886

119860119863119860 (61)


119861119886

119860= 119861119886

119860(119883119863) (62)



119863119860(119883) = 120575

119860119861 120597120578 (119883)

120597119883119861 (63)






119865119886

119860(119883119863) =

120597119909119886(119883119863)

120597119883119860 (64)


119889119886[119909 (119883119863) 119863] = 120575

119886119887 120597120578 (119883)

120597119909119887

= 120575119886119887

(120597119883119860

120597119909119887)(

120597120578

120597119883119860)

= 119865minus1119860

119887120575119886119887120575119860119861

119863119861

(65)

leading to

119861119886

119860(119883119863) = 119865

minus1119861

119887(X 119863) 120575

119886119887120575119860119861

(66)


= 120597119863119861120597119883119860 and 120597119889

119886120597119909119887

= 120597119889119887120597119909119886 hold

As in Section 3 119873119860



120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861 (67)

Let 119873119886119887= 119873119860

119861120575119886

119860120575119861



120575

120575119909119886=

120597

120597119909119886minus 119873119887

119886

120597

120597119889119887 120575119889

119886= 119889119889119886+ 119873119886

119887119889119909119887 (68)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

119860

120597

120597119889119886otimes 120575119863119860 (69)


119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119886

119861120575119862

119888119909119888

119865119886

119860= 119909119886

119860=

120597119909119886

120597119863119860+ 119862119861

119860119862120575119886

119861120575119862

119888119909119888

(70)


G119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861 (71)

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863) (72)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862 (73)

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862 (74)

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863) (75)


119871 = [120581119860119861

119863119860119863119861]12

= [120581119860119861

(120597120578

120597119883119860)(

120597120578

120597119883119861)]

12

(76)

120581119860119861

= 119866119860119861

= 120575119860119862

120575119861119863

119866119862119863

= constant (77)





Γ119860

119861119862= 120574119860

119861119862= 119862119860

119861119862= 0 119873

119860

119861= 0 119899

119886

119887= 0

120575

120575119883119860=

120597

120597119883119860 120575119863

119860= 119889119863119860

120575

120575119909119886=

120597

120597119909119886 120575119889

119886= 119889119889119886

(78)



leading to

F =120597119909119886

120597119883119860120597

120597119909119886otimes 119889119883119860+

120597119909119886

120597119863119860120597

120597119889119886otimes 119889119863119860

= 119865119886

119860

120597

120597119909119886otimes 119889119883119860+ 119881119886

119860

120597

120597119889119886otimes 119889119863119860

(79)


119860= 119881119886



= 120581119860119861




0denote the



119860(120597120597119883




conditions

⟨sm⟩ = 119904119860119898119861⟨

120597

120597119883119860 119889119883119861⟩ = 119904

119860119898119860= 0 (80)


Ξ = Ξ119860

119861

120597

120597119883119860otimes 119889119883119861= 1 + 120574

0120593s otimesm

Ξ119860

119861[120578 (119883)] = 120575

119860

119861+ 1205740120593 [120578 (119883)] 119904

119860119898119861

(81)



1 and 1205931015840(0) = 120593


polynomial [53 59]

120593 [120578 (119883)] = 3 [120578 (119883)]2minus 2 [120578 (119883)]

3 (82)


F = AΞ 119865119886

119860= 119860119886

119861Ξ119861

119860 (83)


A = 119860119886

119860

120597


0120593s otimesm) (84)


120595 (F 120578D) = 119882(F 120578) + 119891 (120578D) (85)

119882 = 119882[A (F 120578)] 119891 (120578D) = 1205721205782(1 minus 120578)

2+ 1198712(D)

(86)


119886

119860= 119909119886

119860= 0 in (86))


2= 120581119860119861

119863119860119863119861


119860119861= 1205810120575119860119861 and


0

obey 1205810= (34)Γ

01198970and 120572 = 12Γ



Ψ [119909 (119883) 120578 (119883)] = intΩ

120595dΩ

dΩ =1003816100381610038161003816det (119866119860119861)

100381610038161003816100381612



119901

(87)



120575120595dΩ = 0 (88)


120575120595 =120597119882

120597119865119886119860

120597120575119909119886

120597119883119860+ (

120597119882

120597120578+

120597119891

120597120578) 120575120578 +

120597119891

120597119863119860120597120575120578

120597119883119860 (89)


120597119875119860

119886

120597119883119860=

120597 (120597119882120597119909119886

|119860)

120597119883119860= 0

120589 +120597119891

120597120578= 2120581119860119861

120597119863119860

120597119883119861

= 2120581119860119861

1205972120578

120597119883119860120597119883119861

(90)







5 Conclusions




References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863) (8)



119873119860

119861= 120574119860

119861119862119884119862minus 119862119860

119861119862120574119862

119863119864119884119863119884119864=

1

2

120597119866119860

120597119884119861 (9)

where 119866119860

= 120574119860


are then introduced

120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119884119861 120575119884

119860= 119889119884119860+ 119873119860

119861119889119883119861 (10)


119860 120597120597119884




G (119883 119884) = 119866119860119861

119889119883119860otimes 119889119883119861+ 119866119860119861

120575119884119860otimes 120575119884119861 (11)






nabla120592

120597

120597119883119860= 120596119861

119860(120592)

120597

120597119883119861 nabla

120592119889119883119860= minus120596119860

119861(120592) 119889119883

119861 (12)


lowast119879119872


119889 (119889119883119860) minus 119889119883

119861and 120596119860

119861= 0

119889119866119860119861

minus 119866119861119862

120596119862

119860minus 119866119860119862

120596119862

119861= 2119862119860119861119862

120575119884119862

(13)


120596119860

119861= Γ119860

119862119861119889119883119862 Γ

119860

119861119862= Γ119860

119862119861 (14)


Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863) (15)


119861= 0 and Γ

119860

119861119862= 120574119860



119861+ 119862119860

119863119861120575119884119863 where 120596

119860




119860

119861119862= 120574119860

119861119862= 0 Gradients of


nabla120575120575119883119860

120575

120575119883119861= Γ119862

119860119861

120575

120575119883119862 nabla

120575120575119883119860

120597

120597119884119861= 119862119862

119860119861

120597

120597119884119862 (16)


1

1)


119861(120597120597119883

119860) otimes 119889119883


(nabla119879)119860

119861= 119889119879119860

119861+ 119879119862

119861120596119860

119862minus 119879119860

119862120596119862

119861

= 119879119860

119861|119862119889119883119862+ 119879119860

119861119862120575119884119862

= (nabla120575120575119883119862119879)119860

119861119889119883119862+ (nabla120597120597119884119862119879)119860

119861120575119884119862

= (120575119879119860

119861

120575119883119862+ 119879119863

119861Γ119860

119862119863minus 119879119860

119863Γ119863

119862119861)119889119883119862+ (

120597119879119860

119861

120597119884119862)120575119884119862

(17)


and (sdot)119860



Ω119860

119861= 119889 (120596

119860

119861) minus 120596119862

119861and 120596119860

119862

=1

2119877119860

119861119862119863119889119883119862and 119889119883119863+ 119875119860

119861119862119863119889119883119862and 120575119884119863

+1

2119876119860

119861119862119863120575119884119862and 120575119884119863

(18)


119877119860

119861119862119863=

120575Γ119860

119861119863

120575119883119862minus

120575Γ119860

119861119862

120575119883119863+ Γ119860

119864119862Γ119864

119861119863minus Γ119860

119864119863Γ119864

119861119862

119875119860

119861119862119863= minus

120597Γ119860

119861119862

120597119884119863 119876

119860

119861119862119863= 0

(19)


119861119862119863= 119875119860

119862119861119863


119877119860

119861119862119863+ 119877119860

119862119863119861+ 119877119860

119863119861119862= 0 (20)


119860



119861119862 and 119875

119860






[120575

120575119883119860

120575

120575119883119861] = (

120575119873119862

119860

120575119883119861minus

120575119873119862

119861

120575119883119860)

120597

120597119884119862= Λ119862

119860119861

120597

120597119884119862 (21)




Λ119862

119860119861= minus119877119862

119863119860119861119884119863 (22)










rarr 119863119860





119909119886= 119909119886(119883119860) 119889

119886

(120572)= 119889119886

(120572)[119863119860

(120573)(119883119861)] = 119889

119886

(120572)(119883119861)

(23)


119865119886

119860=

120597119909119886

120597119883119860 119865

minus1119860

119886=

120597119883119860

120597119909119886 (24)


d119909119886 = 119865119886

119860d119883119860 119889

119886

(120572)= 119861119886

119860119863119860

(120572) (25)

with 119861119886

119860= 119861119886


d119889119886(120572)

= 119864119886

(120572)119860d119883119860 = (119861

119886

119861119860119863119861

(120572)+ 119861119886

119861119863119861

(120572)119860) d119883119860 (26)

where (sdot)119860


d119889119886(120572)

= 119891119886

(120572)119887d119909119887 = 119865

minus1119860

119887119864119886

(120572)119860d119909119887 (27)

Let 119866119860119861


119860119861d119883119861 is a measure

of length (119866119860119861

= 120575119860119861



119862119886119887

= 119865minus1119860

119886119866119860119861

119865minus1119861

119887 119862

(120572120573)

119886119887= 119864minus1119860

(120572)119886119866119860119861

119864minus1119861

(120572)119887 (28)

Let Γ119860

119861119862and Γ

120572



Γ119860

119861119862= 119863119860

(120572)119863(120573)

119861119863(120574)

119862Γ120572

120573120574+ 119863119860

(120572)119863(120572)

119862119861 (29)

where 119863119860

(120572)119863(120572)

119861= 120575119860

119861and 119863

119860

(120573)119863(120572)

119860= 120575120572




(nabla119881)119860

= 119889119881119860+ Γ119860

119861119862119881119862d119883119861 (30)



(nabla120592)119886

= 119889120592119886+ Γ119886

119887119888120592119888d119909119887 + Γ

(120572)119886

119887119888120592119888d119909119887 (31)


) given by119889120592119886 =(120597120592119886120597119909119887)d119909119887 + (120597120592

119886120597119889119887


119867(120572)119886

119887119888= 119865119886

119860119864minus1119860

(120572)119888119887 119870

(120572)119886

119887119888= 119864119886

(120572)119860119865minus1119860

119888119887 (32)


120572



120572







119860 119863120572 where 119883 isin 119872




120575

120575119883119860=

120597

120597119883119860minus 119873120572

119860

120597

120597119863120572 120575119863

120572= 119889119863120572+ 119873120572

119860119889119883119860 (33)



119879119885 = 119867119885 oplus 119881119885 (34)



G (119883119863) = 119866119860119861 (119883119863) 119889119883

119860otimes 119889119883119861

+ 119866120572120573 (119883119863) 120575119863

120572otimes 120575119863120573

(35)


nabla120575120575119883119860

120575

120575119883119861= Γ119862

119860119861

120575

120575119883119862 nabla

120575120575119883119860

120597

120597119863120572= 119862120573

119860120572

120597

120597119863120573 (36)

nabla120597120597119863120572

120575

120575119883119860= 119888119861

120572119860

120575

120575119883119861 nabla

120597120597119863120572

120597

120597119863120573= 120594120574

120572120573

120597

120597119863120574 (37)


119860) and a




119881119860

|119861=

120575119881119860

120575119883119861+ Γ119860

119861119862119881119862 119882

120572

|119861=

120575119882120572

120575119883119861+ 119862120572

119861120573119882120573

119881119860

||120573=

120575119881119860

120575119863120573+ 119888119860

120573119862119881119862 119882

120572

||120573=

120575119882120572

120575119863120573+ 120594120572

120573120575119882120575

(38)


Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863)

119862120572

119860120573=

120597119873120572

119860

120597119863120573 119888

119860

120572119861= 0

120594120572

120573120574=

1

2119866120572120575

(120597119866120573120575

120597119863120574+

120597119866120574120575

120597119863120573minus

120597119866120573120574

120597119863120575)

(39)





L (119883119863) = L[119866119860119861

119866120572120573 119876119870120575119876119870

120575119883119860120597119876119870

120597119863120572] (40)


119868 (Ω) = intΩ



119901

L = [10038161003816100381610038161003816det (119866

119860119861) det (119866

120572120573)10038161003816100381610038161003816]12

L

(41)


120597L

120597119876119870=

120597

120597119883119860[

120597L

120597 (120597119876119870120597119883119860)] +

120597

120597119863120572[

120597L

120597 (120597119876119870120597119863120572)]

(42)


Let 1205771015840




1015840




Let119873120572

1015840



120575

120575119909119886=

120597

120597119909119886minus 1198731205721015840

119886

120597

1205971198891205721015840 120575119889

1205721015840

= 1198891198891205721015840

+ 1198731205721015840

119886119889119909119886 (43)


119886 120597120597119889

1205721015840


where

1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (44)


Dual set 119889119909119886 1205751198891205721015840



coordinates

119909119886= 119909119886(119883119860) 119889

1205721015840

= 1198611205721015840

120573(119883)119863

120573 (45)


119865119886

119860=

120597119909119886

120597119883119860 119865

minus1119860

119886=

120597119883119860

120597119909119886 (46)


120597

120597119883119860= 119865119886

119860

120597

120597119909119886+

1205971198611205721015840

120573

120597119883119860119863120573 120597

1205971198891205721015840

120597

120597119863120572= 1198611205731015840

120572

120597

1205971198891205731015840

(47)


by

1198731205721015840

119886119865119886

119860= 119873120573

1198601198611205721015840

120573minus (

1205971198611205721015840

120573

120597119883119860)119863120573 (48)


119866119886119887119865119886

119860119865119887

119861= 119866119860119861

119866120572101584012057310158401198611205721015840

1205751198611205731015840

120574= 119866120575120574 (49)


Γ119886

119887119888119865119887

119861119865119888

119862= Γ119860

119861119862119865119886

119860minus

120597119865119886

119862

120597119883119861

1198621205721015840

1198861205731015840119865119886

1198601198611205731015840

120574= 119862120583

1198601205741198611205721015840

120583minus

1205971198611205721015840

120574

120597119883119860

119888119886

1205721015840119887119865119887

1198601198611205721015840

120573= 119888119861

120573119860119865119886

119861

1205941205721015840

120573101584012057510158401198611205731015840

1205741198611205751015840

120578= 120594120576

1205741205781198611205721015840

120576

(50)








119909119886 1198891205721015840



119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (51)



119909119886= 119909119886(119883119860 119863120572) 119889

1205721015840

= 1198611205721015840

120573119863120573 (52)


1198611205721015840

120573= 1198611205721015840

120573(119883119863) (53)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

1205721205751205731015840

119886

120597

1205971198891205731015840otimes 120575119863120572 (54)


119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119862

119888120575119886

119861119909119888

1198651205731015840

120572= 119909119886

1205721205751205731015840

119886= (

120597119909119886

120597119863120572+ 119862119861

119860119862120575119860

120572120575119886

119861120575119862

119888119909119888)1205751205731015840

119886

(55)


119860= 120575119860

120572119863120572 and 119889

119886= 120575119886

12057210158401198891205721015840




119866119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863)

120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861

(56)

Notice that Γ119860119861119862



stationary (1198611205721015840

120573rarr 120575

1205721015840




119861119862(eg

119875119860




119860

119861119862= 0 then F rarr F the



L (119883119863)

= L [119883119863 119909 (119883119863) F (119883119863) Q (119883119863) nablaQ (119883119863)]

(57)


119868 (Ω) = intΩ



119901

L = [1003816100381610038161003816det (119866119860119861)

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816det( 120597

120597119883119860sdot

120597

120597119883119861)10038161003816100381610038161003816100381610038161003816]

12

L

(58)









(59)







119860 119863119860


1015840 12058710158401198721015840 1198801015840) be



1015840 are119909119886 119889119886 where 119909 isin 119872


119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (60)



119909119886= 119909119886(119883119860 119863119860) 119889

119886= 119861119886

119860119863119860 (61)


119861119886

119860= 119861119886

119860(119883119863) (62)



119863119860(119883) = 120575

119860119861 120597120578 (119883)

120597119883119861 (63)






119865119886

119860(119883119863) =

120597119909119886(119883119863)

120597119883119860 (64)


119889119886[119909 (119883119863) 119863] = 120575

119886119887 120597120578 (119883)

120597119909119887

= 120575119886119887

(120597119883119860

120597119909119887)(

120597120578

120597119883119860)

= 119865minus1119860

119887120575119886119887120575119860119861

119863119861

(65)

leading to

119861119886

119860(119883119863) = 119865

minus1119861

119887(X 119863) 120575

119886119887120575119860119861

(66)


= 120597119863119861120597119883119860 and 120597119889

119886120597119909119887

= 120597119889119887120597119909119886 hold

As in Section 3 119873119860



120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861 (67)

Let 119873119886119887= 119873119860

119861120575119886

119860120575119861



120575

120575119909119886=

120597

120597119909119886minus 119873119887

119886

120597

120597119889119887 120575119889

119886= 119889119889119886+ 119873119886

119887119889119909119887 (68)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

119860

120597

120597119889119886otimes 120575119863119860 (69)


119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119886

119861120575119862

119888119909119888

119865119886

119860= 119909119886

119860=

120597119909119886

120597119863119860+ 119862119861

119860119862120575119886

119861120575119862

119888119909119888

(70)


G119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861 (71)

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863) (72)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862 (73)

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862 (74)

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863) (75)


119871 = [120581119860119861

119863119860119863119861]12

= [120581119860119861

(120597120578

120597119883119860)(

120597120578

120597119883119861)]

12

(76)

120581119860119861

= 119866119860119861

= 120575119860119862

120575119861119863

119866119862119863

= constant (77)





Γ119860

119861119862= 120574119860

119861119862= 119862119860

119861119862= 0 119873

119860

119861= 0 119899

119886

119887= 0

120575

120575119883119860=

120597

120597119883119860 120575119863

119860= 119889119863119860

120575

120575119909119886=

120597

120597119909119886 120575119889

119886= 119889119889119886

(78)



leading to

F =120597119909119886

120597119883119860120597

120597119909119886otimes 119889119883119860+

120597119909119886

120597119863119860120597

120597119889119886otimes 119889119863119860

= 119865119886

119860

120597

120597119909119886otimes 119889119883119860+ 119881119886

119860

120597

120597119889119886otimes 119889119863119860

(79)


119860= 119881119886



= 120581119860119861




0denote the



119860(120597120597119883




conditions

⟨sm⟩ = 119904119860119898119861⟨

120597

120597119883119860 119889119883119861⟩ = 119904

119860119898119860= 0 (80)


Ξ = Ξ119860

119861

120597

120597119883119860otimes 119889119883119861= 1 + 120574

0120593s otimesm

Ξ119860

119861[120578 (119883)] = 120575

119860

119861+ 1205740120593 [120578 (119883)] 119904

119860119898119861

(81)



1 and 1205931015840(0) = 120593


polynomial [53 59]

120593 [120578 (119883)] = 3 [120578 (119883)]2minus 2 [120578 (119883)]

3 (82)


F = AΞ 119865119886

119860= 119860119886

119861Ξ119861

119860 (83)


A = 119860119886

119860

120597


0120593s otimesm) (84)


120595 (F 120578D) = 119882(F 120578) + 119891 (120578D) (85)

119882 = 119882[A (F 120578)] 119891 (120578D) = 1205721205782(1 minus 120578)

2+ 1198712(D)

(86)


119886

119860= 119909119886

119860= 0 in (86))


2= 120581119860119861

119863119860119863119861


119860119861= 1205810120575119860119861 and


0

obey 1205810= (34)Γ

01198970and 120572 = 12Γ



Ψ [119909 (119883) 120578 (119883)] = intΩ

120595dΩ

dΩ =1003816100381610038161003816det (119866119860119861)

100381610038161003816100381612



119901

(87)



120575120595dΩ = 0 (88)


120575120595 =120597119882

120597119865119886119860

120597120575119909119886

120597119883119860+ (

120597119882

120597120578+

120597119891

120597120578) 120575120578 +

120597119891

120597119863119860120597120575120578

120597119883119860 (89)


120597119875119860

119886

120597119883119860=

120597 (120597119882120597119909119886

|119860)

120597119883119860= 0

120589 +120597119891

120597120578= 2120581119860119861

120597119863119860

120597119883119861

= 2120581119860119861

1205972120578

120597119883119860120597119883119861

(90)







5 Conclusions




References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












[120575

120575119883119860

120575

120575119883119861] = (

120575119873119862

119860

120575119883119861minus

120575119873119862

119861

120575119883119860)

120597

120597119884119862= Λ119862

119860119861

120597

120597119884119862 (21)




Λ119862

119860119861= minus119877119862

119863119860119861119884119863 (22)










rarr 119863119860





119909119886= 119909119886(119883119860) 119889

119886

(120572)= 119889119886

(120572)[119863119860

(120573)(119883119861)] = 119889

119886

(120572)(119883119861)

(23)


119865119886

119860=

120597119909119886

120597119883119860 119865

minus1119860

119886=

120597119883119860

120597119909119886 (24)


d119909119886 = 119865119886

119860d119883119860 119889

119886

(120572)= 119861119886

119860119863119860

(120572) (25)

with 119861119886

119860= 119861119886


d119889119886(120572)

= 119864119886

(120572)119860d119883119860 = (119861

119886

119861119860119863119861

(120572)+ 119861119886

119861119863119861

(120572)119860) d119883119860 (26)

where (sdot)119860


d119889119886(120572)

= 119891119886

(120572)119887d119909119887 = 119865

minus1119860

119887119864119886

(120572)119860d119909119887 (27)

Let 119866119860119861


119860119861d119883119861 is a measure

of length (119866119860119861

= 120575119860119861



119862119886119887

= 119865minus1119860

119886119866119860119861

119865minus1119861

119887 119862

(120572120573)

119886119887= 119864minus1119860

(120572)119886119866119860119861

119864minus1119861

(120572)119887 (28)

Let Γ119860

119861119862and Γ

120572



Γ119860

119861119862= 119863119860

(120572)119863(120573)

119861119863(120574)

119862Γ120572

120573120574+ 119863119860

(120572)119863(120572)

119862119861 (29)

where 119863119860

(120572)119863(120572)

119861= 120575119860

119861and 119863

119860

(120573)119863(120572)

119860= 120575120572




(nabla119881)119860

= 119889119881119860+ Γ119860

119861119862119881119862d119883119861 (30)



(nabla120592)119886

= 119889120592119886+ Γ119886

119887119888120592119888d119909119887 + Γ

(120572)119886

119887119888120592119888d119909119887 (31)


) given by119889120592119886 =(120597120592119886120597119909119887)d119909119887 + (120597120592

119886120597119889119887


119867(120572)119886

119887119888= 119865119886

119860119864minus1119860

(120572)119888119887 119870

(120572)119886

119887119888= 119864119886

(120572)119860119865minus1119860

119888119887 (32)


120572



120572







119860 119863120572 where 119883 isin 119872




120575

120575119883119860=

120597

120597119883119860minus 119873120572

119860

120597

120597119863120572 120575119863

120572= 119889119863120572+ 119873120572

119860119889119883119860 (33)



119879119885 = 119867119885 oplus 119881119885 (34)



G (119883119863) = 119866119860119861 (119883119863) 119889119883

119860otimes 119889119883119861

+ 119866120572120573 (119883119863) 120575119863

120572otimes 120575119863120573

(35)


nabla120575120575119883119860

120575

120575119883119861= Γ119862

119860119861

120575

120575119883119862 nabla

120575120575119883119860

120597

120597119863120572= 119862120573

119860120572

120597

120597119863120573 (36)

nabla120597120597119863120572

120575

120575119883119860= 119888119861

120572119860

120575

120575119883119861 nabla

120597120597119863120572

120597

120597119863120573= 120594120574

120572120573

120597

120597119863120574 (37)


119860) and a




119881119860

|119861=

120575119881119860

120575119883119861+ Γ119860

119861119862119881119862 119882

120572

|119861=

120575119882120572

120575119883119861+ 119862120572

119861120573119882120573

119881119860

||120573=

120575119881119860

120575119863120573+ 119888119860

120573119862119881119862 119882

120572

||120573=

120575119882120572

120575119863120573+ 120594120572

120573120575119882120575

(38)


Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863)

119862120572

119860120573=

120597119873120572

119860

120597119863120573 119888

119860

120572119861= 0

120594120572

120573120574=

1

2119866120572120575

(120597119866120573120575

120597119863120574+

120597119866120574120575

120597119863120573minus

120597119866120573120574

120597119863120575)

(39)





L (119883119863) = L[119866119860119861

119866120572120573 119876119870120575119876119870

120575119883119860120597119876119870

120597119863120572] (40)


119868 (Ω) = intΩ



119901

L = [10038161003816100381610038161003816det (119866

119860119861) det (119866

120572120573)10038161003816100381610038161003816]12

L

(41)


120597L

120597119876119870=

120597

120597119883119860[

120597L

120597 (120597119876119870120597119883119860)] +

120597

120597119863120572[

120597L

120597 (120597119876119870120597119863120572)]

(42)


Let 1205771015840




1015840




Let119873120572

1015840



120575

120575119909119886=

120597

120597119909119886minus 1198731205721015840

119886

120597

1205971198891205721015840 120575119889

1205721015840

= 1198891198891205721015840

+ 1198731205721015840

119886119889119909119886 (43)


119886 120597120597119889

1205721015840


where

1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (44)


Dual set 119889119909119886 1205751198891205721015840



coordinates

119909119886= 119909119886(119883119860) 119889

1205721015840

= 1198611205721015840

120573(119883)119863

120573 (45)


119865119886

119860=

120597119909119886

120597119883119860 119865

minus1119860

119886=

120597119883119860

120597119909119886 (46)


120597

120597119883119860= 119865119886

119860

120597

120597119909119886+

1205971198611205721015840

120573

120597119883119860119863120573 120597

1205971198891205721015840

120597

120597119863120572= 1198611205731015840

120572

120597

1205971198891205731015840

(47)


by

1198731205721015840

119886119865119886

119860= 119873120573

1198601198611205721015840

120573minus (

1205971198611205721015840

120573

120597119883119860)119863120573 (48)


119866119886119887119865119886

119860119865119887

119861= 119866119860119861

119866120572101584012057310158401198611205721015840

1205751198611205731015840

120574= 119866120575120574 (49)


Γ119886

119887119888119865119887

119861119865119888

119862= Γ119860

119861119862119865119886

119860minus

120597119865119886

119862

120597119883119861

1198621205721015840

1198861205731015840119865119886

1198601198611205731015840

120574= 119862120583

1198601205741198611205721015840

120583minus

1205971198611205721015840

120574

120597119883119860

119888119886

1205721015840119887119865119887

1198601198611205721015840

120573= 119888119861

120573119860119865119886

119861

1205941205721015840

120573101584012057510158401198611205731015840

1205741198611205751015840

120578= 120594120576

1205741205781198611205721015840

120576

(50)








119909119886 1198891205721015840



119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (51)



119909119886= 119909119886(119883119860 119863120572) 119889

1205721015840

= 1198611205721015840

120573119863120573 (52)


1198611205721015840

120573= 1198611205721015840

120573(119883119863) (53)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

1205721205751205731015840

119886

120597

1205971198891205731015840otimes 120575119863120572 (54)


119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119862

119888120575119886

119861119909119888

1198651205731015840

120572= 119909119886

1205721205751205731015840

119886= (

120597119909119886

120597119863120572+ 119862119861

119860119862120575119860

120572120575119886

119861120575119862

119888119909119888)1205751205731015840

119886

(55)


119860= 120575119860

120572119863120572 and 119889

119886= 120575119886

12057210158401198891205721015840




119866119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863)

120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861

(56)

Notice that Γ119860119861119862



stationary (1198611205721015840

120573rarr 120575

1205721015840




119861119862(eg

119875119860




119860

119861119862= 0 then F rarr F the



L (119883119863)

= L [119883119863 119909 (119883119863) F (119883119863) Q (119883119863) nablaQ (119883119863)]

(57)


119868 (Ω) = intΩ



119901

L = [1003816100381610038161003816det (119866119860119861)

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816det( 120597

120597119883119860sdot

120597

120597119883119861)10038161003816100381610038161003816100381610038161003816]

12

L

(58)









(59)







119860 119863119860


1015840 12058710158401198721015840 1198801015840) be



1015840 are119909119886 119889119886 where 119909 isin 119872


119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (60)



119909119886= 119909119886(119883119860 119863119860) 119889

119886= 119861119886

119860119863119860 (61)


119861119886

119860= 119861119886

119860(119883119863) (62)



119863119860(119883) = 120575

119860119861 120597120578 (119883)

120597119883119861 (63)






119865119886

119860(119883119863) =

120597119909119886(119883119863)

120597119883119860 (64)


119889119886[119909 (119883119863) 119863] = 120575

119886119887 120597120578 (119883)

120597119909119887

= 120575119886119887

(120597119883119860

120597119909119887)(

120597120578

120597119883119860)

= 119865minus1119860

119887120575119886119887120575119860119861

119863119861

(65)

leading to

119861119886

119860(119883119863) = 119865

minus1119861

119887(X 119863) 120575

119886119887120575119860119861

(66)


= 120597119863119861120597119883119860 and 120597119889

119886120597119909119887

= 120597119889119887120597119909119886 hold

As in Section 3 119873119860



120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861 (67)

Let 119873119886119887= 119873119860

119861120575119886

119860120575119861



120575

120575119909119886=

120597

120597119909119886minus 119873119887

119886

120597

120597119889119887 120575119889

119886= 119889119889119886+ 119873119886

119887119889119909119887 (68)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

119860

120597

120597119889119886otimes 120575119863119860 (69)


119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119886

119861120575119862

119888119909119888

119865119886

119860= 119909119886

119860=

120597119909119886

120597119863119860+ 119862119861

119860119862120575119886

119861120575119862

119888119909119888

(70)


G119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861 (71)

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863) (72)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862 (73)

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862 (74)

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863) (75)


119871 = [120581119860119861

119863119860119863119861]12

= [120581119860119861

(120597120578

120597119883119860)(

120597120578

120597119883119861)]

12

(76)

120581119860119861

= 119866119860119861

= 120575119860119862

120575119861119863

119866119862119863

= constant (77)





Γ119860

119861119862= 120574119860

119861119862= 119862119860

119861119862= 0 119873

119860

119861= 0 119899

119886

119887= 0

120575

120575119883119860=

120597

120597119883119860 120575119863

119860= 119889119863119860

120575

120575119909119886=

120597

120597119909119886 120575119889

119886= 119889119889119886

(78)



leading to

F =120597119909119886

120597119883119860120597

120597119909119886otimes 119889119883119860+

120597119909119886

120597119863119860120597

120597119889119886otimes 119889119863119860

= 119865119886

119860

120597

120597119909119886otimes 119889119883119860+ 119881119886

119860

120597

120597119889119886otimes 119889119863119860

(79)


119860= 119881119886



= 120581119860119861




0denote the



119860(120597120597119883




conditions

⟨sm⟩ = 119904119860119898119861⟨

120597

120597119883119860 119889119883119861⟩ = 119904

119860119898119860= 0 (80)


Ξ = Ξ119860

119861

120597

120597119883119860otimes 119889119883119861= 1 + 120574

0120593s otimesm

Ξ119860

119861[120578 (119883)] = 120575

119860

119861+ 1205740120593 [120578 (119883)] 119904

119860119898119861

(81)



1 and 1205931015840(0) = 120593


polynomial [53 59]

120593 [120578 (119883)] = 3 [120578 (119883)]2minus 2 [120578 (119883)]

3 (82)


F = AΞ 119865119886

119860= 119860119886

119861Ξ119861

119860 (83)


A = 119860119886

119860

120597


0120593s otimesm) (84)


120595 (F 120578D) = 119882(F 120578) + 119891 (120578D) (85)

119882 = 119882[A (F 120578)] 119891 (120578D) = 1205721205782(1 minus 120578)

2+ 1198712(D)

(86)


119886

119860= 119909119886

119860= 0 in (86))


2= 120581119860119861

119863119860119863119861


119860119861= 1205810120575119860119861 and


0

obey 1205810= (34)Γ

01198970and 120572 = 12Γ



Ψ [119909 (119883) 120578 (119883)] = intΩ

120595dΩ

dΩ =1003816100381610038161003816det (119866119860119861)

100381610038161003816100381612



119901

(87)



120575120595dΩ = 0 (88)


120575120595 =120597119882

120597119865119886119860

120597120575119909119886

120597119883119860+ (

120597119882

120597120578+

120597119891

120597120578) 120575120578 +

120597119891

120597119863119860120597120575120578

120597119883119860 (89)


120597119875119860

119886

120597119883119860=

120597 (120597119882120597119909119886

|119860)

120597119883119860= 0

120589 +120597119891

120597120578= 2120581119860119861

120597119863119860

120597119883119861

= 2120581119860119861

1205972120578

120597119883119860120597119883119861

(90)







5 Conclusions




References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119860 119863120572 where 119883 isin 119872




120575

120575119883119860=

120597

120597119883119860minus 119873120572

119860

120597

120597119863120572 120575119863

120572= 119889119863120572+ 119873120572

119860119889119883119860 (33)



119879119885 = 119867119885 oplus 119881119885 (34)



G (119883119863) = 119866119860119861 (119883119863) 119889119883

119860otimes 119889119883119861

+ 119866120572120573 (119883119863) 120575119863

120572otimes 120575119863120573

(35)


nabla120575120575119883119860

120575

120575119883119861= Γ119862

119860119861

120575

120575119883119862 nabla

120575120575119883119860

120597

120597119863120572= 119862120573

119860120572

120597

120597119863120573 (36)

nabla120597120597119863120572

120575

120575119883119860= 119888119861

120572119860

120575

120575119883119861 nabla

120597120597119863120572

120597

120597119863120573= 120594120574

120572120573

120597

120597119863120574 (37)


119860) and a




119881119860

|119861=

120575119881119860

120575119883119861+ Γ119860

119861119862119881119862 119882

120572

|119861=

120575119882120572

120575119883119861+ 119862120572

119861120573119882120573

119881119860

||120573=

120575119881119860

120575119863120573+ 119888119860

120573119862119881119862 119882

120572

||120573=

120575119882120572

120575119863120573+ 120594120572

120573120575119882120575

(38)


Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863)

119862120572

119860120573=

120597119873120572

119860

120597119863120573 119888

119860

120572119861= 0

120594120572

120573120574=

1

2119866120572120575

(120597119866120573120575

120597119863120574+

120597119866120574120575

120597119863120573minus

120597119866120573120574

120597119863120575)

(39)





L (119883119863) = L[119866119860119861

119866120572120573 119876119870120575119876119870

120575119883119860120597119876119870

120597119863120572] (40)


119868 (Ω) = intΩ



119901

L = [10038161003816100381610038161003816det (119866

119860119861) det (119866

120572120573)10038161003816100381610038161003816]12

L

(41)


120597L

120597119876119870=

120597

120597119883119860[

120597L

120597 (120597119876119870120597119883119860)] +

120597

120597119863120572[

120597L

120597 (120597119876119870120597119863120572)]

(42)


Let 1205771015840




1015840




Let119873120572

1015840



120575

120575119909119886=

120597

120597119909119886minus 1198731205721015840

119886

120597

1205971198891205721015840 120575119889

1205721015840

= 1198891198891205721015840

+ 1198731205721015840

119886119889119909119886 (43)


119886 120597120597119889

1205721015840


where

1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (44)


Dual set 119889119909119886 1205751198891205721015840



coordinates

119909119886= 119909119886(119883119860) 119889

1205721015840

= 1198611205721015840

120573(119883)119863

120573 (45)


119865119886

119860=

120597119909119886

120597119883119860 119865

minus1119860

119886=

120597119883119860

120597119909119886 (46)


120597

120597119883119860= 119865119886

119860

120597

120597119909119886+

1205971198611205721015840

120573

120597119883119860119863120573 120597

1205971198891205721015840

120597

120597119863120572= 1198611205731015840

120572

120597

1205971198891205731015840

(47)


by

1198731205721015840

119886119865119886

119860= 119873120573

1198601198611205721015840

120573minus (

1205971198611205721015840

120573

120597119883119860)119863120573 (48)


119866119886119887119865119886

119860119865119887

119861= 119866119860119861

119866120572101584012057310158401198611205721015840

1205751198611205731015840

120574= 119866120575120574 (49)


Γ119886

119887119888119865119887

119861119865119888

119862= Γ119860

119861119862119865119886

119860minus

120597119865119886

119862

120597119883119861

1198621205721015840

1198861205731015840119865119886

1198601198611205731015840

120574= 119862120583

1198601205741198611205721015840

120583minus

1205971198611205721015840

120574

120597119883119860

119888119886

1205721015840119887119865119887

1198601198611205721015840

120573= 119888119861

120573119860119865119886

119861

1205941205721015840

120573101584012057510158401198611205731015840

1205741198611205751015840

120578= 120594120576

1205741205781198611205721015840

120576

(50)








119909119886 1198891205721015840



119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (51)



119909119886= 119909119886(119883119860 119863120572) 119889

1205721015840

= 1198611205721015840

120573119863120573 (52)


1198611205721015840

120573= 1198611205721015840

120573(119883119863) (53)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

1205721205751205731015840

119886

120597

1205971198891205731015840otimes 120575119863120572 (54)


119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119862

119888120575119886

119861119909119888

1198651205731015840

120572= 119909119886

1205721205751205731015840

119886= (

120597119909119886

120597119863120572+ 119862119861

119860119862120575119860

120572120575119886

119861120575119862

119888119909119888)1205751205731015840

119886

(55)


119860= 120575119860

120572119863120572 and 119889

119886= 120575119886

12057210158401198891205721015840




119866119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863)

120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861

(56)

Notice that Γ119860119861119862



stationary (1198611205721015840

120573rarr 120575

1205721015840




119861119862(eg

119875119860




119860

119861119862= 0 then F rarr F the



L (119883119863)

= L [119883119863 119909 (119883119863) F (119883119863) Q (119883119863) nablaQ (119883119863)]

(57)


119868 (Ω) = intΩ



119901

L = [1003816100381610038161003816det (119866119860119861)

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816det( 120597

120597119883119860sdot

120597

120597119883119861)10038161003816100381610038161003816100381610038161003816]

12

L

(58)









(59)







119860 119863119860


1015840 12058710158401198721015840 1198801015840) be



1015840 are119909119886 119889119886 where 119909 isin 119872


119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (60)



119909119886= 119909119886(119883119860 119863119860) 119889

119886= 119861119886

119860119863119860 (61)


119861119886

119860= 119861119886

119860(119883119863) (62)



119863119860(119883) = 120575

119860119861 120597120578 (119883)

120597119883119861 (63)






119865119886

119860(119883119863) =

120597119909119886(119883119863)

120597119883119860 (64)


119889119886[119909 (119883119863) 119863] = 120575

119886119887 120597120578 (119883)

120597119909119887

= 120575119886119887

(120597119883119860

120597119909119887)(

120597120578

120597119883119860)

= 119865minus1119860

119887120575119886119887120575119860119861

119863119861

(65)

leading to

119861119886

119860(119883119863) = 119865

minus1119861

119887(X 119863) 120575

119886119887120575119860119861

(66)


= 120597119863119861120597119883119860 and 120597119889

119886120597119909119887

= 120597119889119887120597119909119886 hold

As in Section 3 119873119860



120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861 (67)

Let 119873119886119887= 119873119860

119861120575119886

119860120575119861



120575

120575119909119886=

120597

120597119909119886minus 119873119887

119886

120597

120597119889119887 120575119889

119886= 119889119889119886+ 119873119886

119887119889119909119887 (68)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

119860

120597

120597119889119886otimes 120575119863119860 (69)


119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119886

119861120575119862

119888119909119888

119865119886

119860= 119909119886

119860=

120597119909119886

120597119863119860+ 119862119861

119860119862120575119886

119861120575119862

119888119909119888

(70)


G119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861 (71)

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863) (72)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862 (73)

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862 (74)

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863) (75)


119871 = [120581119860119861

119863119860119863119861]12

= [120581119860119861

(120597120578

120597119883119860)(

120597120578

120597119883119861)]

12

(76)

120581119860119861

= 119866119860119861

= 120575119860119862

120575119861119863

119866119862119863

= constant (77)





Γ119860

119861119862= 120574119860

119861119862= 119862119860

119861119862= 0 119873

119860

119861= 0 119899

119886

119887= 0

120575

120575119883119860=

120597

120597119883119860 120575119863

119860= 119889119863119860

120575

120575119909119886=

120597

120597119909119886 120575119889

119886= 119889119889119886

(78)



leading to

F =120597119909119886

120597119883119860120597

120597119909119886otimes 119889119883119860+

120597119909119886

120597119863119860120597

120597119889119886otimes 119889119863119860

= 119865119886

119860

120597

120597119909119886otimes 119889119883119860+ 119881119886

119860

120597

120597119889119886otimes 119889119863119860

(79)


119860= 119881119886



= 120581119860119861




0denote the



119860(120597120597119883




conditions

⟨sm⟩ = 119904119860119898119861⟨

120597

120597119883119860 119889119883119861⟩ = 119904

119860119898119860= 0 (80)


Ξ = Ξ119860

119861

120597

120597119883119860otimes 119889119883119861= 1 + 120574

0120593s otimesm

Ξ119860

119861[120578 (119883)] = 120575

119860

119861+ 1205740120593 [120578 (119883)] 119904

119860119898119861

(81)



1 and 1205931015840(0) = 120593


polynomial [53 59]

120593 [120578 (119883)] = 3 [120578 (119883)]2minus 2 [120578 (119883)]

3 (82)


F = AΞ 119865119886

119860= 119860119886

119861Ξ119861

119860 (83)


A = 119860119886

119860

120597


0120593s otimesm) (84)


120595 (F 120578D) = 119882(F 120578) + 119891 (120578D) (85)

119882 = 119882[A (F 120578)] 119891 (120578D) = 1205721205782(1 minus 120578)

2+ 1198712(D)

(86)


119886

119860= 119909119886

119860= 0 in (86))


2= 120581119860119861

119863119860119863119861


119860119861= 1205810120575119860119861 and


0

obey 1205810= (34)Γ

01198970and 120572 = 12Γ



Ψ [119909 (119883) 120578 (119883)] = intΩ

120595dΩ

dΩ =1003816100381610038161003816det (119866119860119861)

100381610038161003816100381612



119901

(87)



120575120595dΩ = 0 (88)


120575120595 =120597119882

120597119865119886119860

120597120575119909119886

120597119883119860+ (

120597119882

120597120578+

120597119891

120597120578) 120575120578 +

120597119891

120597119863119860120597120575120578

120597119883119860 (89)


120597119875119860

119886

120597119883119860=

120597 (120597119882120597119909119886

|119860)

120597119883119860= 0

120589 +120597119891

120597120578= 2120581119860119861

120597119863119860

120597119883119861

= 2120581119860119861

1205972120578

120597119883119860120597119883119861

(90)







5 Conclusions




References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Dual set 119889119909119886 1205751198891205721015840



coordinates

119909119886= 119909119886(119883119860) 119889

1205721015840

= 1198611205721015840

120573(119883)119863

120573 (45)


119865119886

119860=

120597119909119886

120597119883119860 119865

minus1119860

119886=

120597119883119860

120597119909119886 (46)


120597

120597119883119860= 119865119886

119860

120597

120597119909119886+

1205971198611205721015840

120573

120597119883119860119863120573 120597

1205971198891205721015840

120597

120597119863120572= 1198611205731015840

120572

120597

1205971198891205731015840

(47)


by

1198731205721015840

119886119865119886

119860= 119873120573

1198601198611205721015840

120573minus (

1205971198611205721015840

120573

120597119883119860)119863120573 (48)


119866119886119887119865119886

119860119865119887

119861= 119866119860119861

119866120572101584012057310158401198611205721015840

1205751198611205731015840

120574= 119866120575120574 (49)


Γ119886

119887119888119865119887

119861119865119888

119862= Γ119860

119861119862119865119886

119860minus

120597119865119886

119862

120597119883119861

1198621205721015840

1198861205731015840119865119886

1198601198611205731015840

120574= 119862120583

1198601205741198611205721015840

120583minus

1205971198611205721015840

120574

120597119883119860

119888119886

1205721015840119887119865119887

1198601198611205721015840

120573= 119888119861

120573119860119865119886

119861

1205941205721015840

120573101584012057510158401198611205731015840

1205741198611205751015840

120578= 120594120576

1205741205781198611205721015840

120576

(50)








119909119886 1198891205721015840



119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (51)



119909119886= 119909119886(119883119860 119863120572) 119889

1205721015840

= 1198611205721015840

120573119863120573 (52)


1198611205721015840

120573= 1198611205721015840

120573(119883119863) (53)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

1205721205751205731015840

119886

120597

1205971198891205731015840otimes 120575119863120572 (54)


119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119862

119888120575119886

119861119909119888

1198651205731015840

120572= 119909119886

1205721205751205731015840

119886= (

120597119909119886

120597119863120572+ 119862119861

119860119862120575119860

120572120575119886

119861120575119862

119888119909119888)1205751205731015840

119886

(55)


119860= 120575119860

120572119863120572 and 119889

119886= 120575119886

12057210158401198891205721015840




119866119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863)

120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861

(56)

Notice that Γ119860119861119862



stationary (1198611205721015840

120573rarr 120575

1205721015840




119861119862(eg

119875119860




119860

119861119862= 0 then F rarr F the



L (119883119863)

= L [119883119863 119909 (119883119863) F (119883119863) Q (119883119863) nablaQ (119883119863)]

(57)


119868 (Ω) = intΩ



119901

L = [1003816100381610038161003816det (119866119860119861)

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816det( 120597

120597119883119860sdot

120597

120597119883119861)10038161003816100381610038161003816100381610038161003816]

12

L

(58)









(59)







119860 119863119860


1015840 12058710158401198721015840 1198801015840) be



1015840 are119909119886 119889119886 where 119909 isin 119872


119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (60)



119909119886= 119909119886(119883119860 119863119860) 119889

119886= 119861119886

119860119863119860 (61)


119861119886

119860= 119861119886

119860(119883119863) (62)



119863119860(119883) = 120575

119860119861 120597120578 (119883)

120597119883119861 (63)






119865119886

119860(119883119863) =

120597119909119886(119883119863)

120597119883119860 (64)


119889119886[119909 (119883119863) 119863] = 120575

119886119887 120597120578 (119883)

120597119909119887

= 120575119886119887

(120597119883119860

120597119909119887)(

120597120578

120597119883119860)

= 119865minus1119860

119887120575119886119887120575119860119861

119863119861

(65)

leading to

119861119886

119860(119883119863) = 119865

minus1119861

119887(X 119863) 120575

119886119887120575119860119861

(66)


= 120597119863119861120597119883119860 and 120597119889

119886120597119909119887

= 120597119889119887120597119909119886 hold

As in Section 3 119873119860



120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861 (67)

Let 119873119886119887= 119873119860

119861120575119886

119860120575119861



120575

120575119909119886=

120597

120597119909119886minus 119873119887

119886

120597

120597119889119887 120575119889

119886= 119889119889119886+ 119873119886

119887119889119909119887 (68)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

119860

120597

120597119889119886otimes 120575119863119860 (69)


119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119886

119861120575119862

119888119909119888

119865119886

119860= 119909119886

119860=

120597119909119886

120597119863119860+ 119862119861

119860119862120575119886

119861120575119862

119888119909119888

(70)


G119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861 (71)

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863) (72)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862 (73)

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862 (74)

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863) (75)


119871 = [120581119860119861

119863119860119863119861]12

= [120581119860119861

(120597120578

120597119883119860)(

120597120578

120597119883119861)]

12

(76)

120581119860119861

= 119866119860119861

= 120575119860119862

120575119861119863

119866119862119863

= constant (77)





Γ119860

119861119862= 120574119860

119861119862= 119862119860

119861119862= 0 119873

119860

119861= 0 119899

119886

119887= 0

120575

120575119883119860=

120597

120597119883119860 120575119863

119860= 119889119863119860

120575

120575119909119886=

120597

120597119909119886 120575119889

119886= 119889119889119886

(78)



leading to

F =120597119909119886

120597119883119860120597

120597119909119886otimes 119889119883119860+

120597119909119886

120597119863119860120597

120597119889119886otimes 119889119863119860

= 119865119886

119860

120597

120597119909119886otimes 119889119883119860+ 119881119886

119860

120597

120597119889119886otimes 119889119863119860

(79)


119860= 119881119886



= 120581119860119861




0denote the



119860(120597120597119883




conditions

⟨sm⟩ = 119904119860119898119861⟨

120597

120597119883119860 119889119883119861⟩ = 119904

119860119898119860= 0 (80)


Ξ = Ξ119860

119861

120597

120597119883119860otimes 119889119883119861= 1 + 120574

0120593s otimesm

Ξ119860

119861[120578 (119883)] = 120575

119860

119861+ 1205740120593 [120578 (119883)] 119904

119860119898119861

(81)



1 and 1205931015840(0) = 120593


polynomial [53 59]

120593 [120578 (119883)] = 3 [120578 (119883)]2minus 2 [120578 (119883)]

3 (82)


F = AΞ 119865119886

119860= 119860119886

119861Ξ119861

119860 (83)


A = 119860119886

119860

120597


0120593s otimesm) (84)


120595 (F 120578D) = 119882(F 120578) + 119891 (120578D) (85)

119882 = 119882[A (F 120578)] 119891 (120578D) = 1205721205782(1 minus 120578)

2+ 1198712(D)

(86)


119886

119860= 119909119886

119860= 0 in (86))


2= 120581119860119861

119863119860119863119861


119860119861= 1205810120575119860119861 and


0

obey 1205810= (34)Γ

01198970and 120572 = 12Γ



Ψ [119909 (119883) 120578 (119883)] = intΩ

120595dΩ

dΩ =1003816100381610038161003816det (119866119860119861)

100381610038161003816100381612



119901

(87)



120575120595dΩ = 0 (88)


120575120595 =120597119882

120597119865119886119860

120597120575119909119886

120597119883119860+ (

120597119882

120597120578+

120597119891

120597120578) 120575120578 +

120597119891

120597119863119860120597120575120578

120597119883119860 (89)


120597119875119860

119886

120597119883119860=

120597 (120597119882120597119909119886

|119860)

120597119883119860= 0

120589 +120597119891

120597120578= 2120581119860119861

120597119863119860

120597119883119861

= 2120581119860119861

1205972120578

120597119883119860120597119883119861

(90)







5 Conclusions




References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119866119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863)

120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861

(56)

Notice that Γ119860119861119862



stationary (1198611205721015840

120573rarr 120575

1205721015840




119861119862(eg

119875119860




119860

119861119862= 0 then F rarr F the



L (119883119863)

= L [119883119863 119909 (119883119863) F (119883119863) Q (119883119863) nablaQ (119883119863)]

(57)


119868 (Ω) = intΩ



119901

L = [1003816100381610038161003816det (119866119860119861)

1003816100381610038161003816

10038161003816100381610038161003816100381610038161003816det( 120597

120597119883119860sdot

120597

120597119883119861)10038161003816100381610038161003816100381610038161003816]

12

L

(58)









(59)







119860 119863119860


1015840 12058710158401198721015840 1198801015840) be



1015840 are119909119886 119889119886 where 119909 isin 119872


119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (60)



119909119886= 119909119886(119883119860 119863119860) 119889

119886= 119861119886

119860119863119860 (61)


119861119886

119860= 119861119886

119860(119883119863) (62)



119863119860(119883) = 120575

119860119861 120597120578 (119883)

120597119883119861 (63)






119865119886

119860(119883119863) =

120597119909119886(119883119863)

120597119883119860 (64)


119889119886[119909 (119883119863) 119863] = 120575

119886119887 120597120578 (119883)

120597119909119887

= 120575119886119887

(120597119883119860

120597119909119887)(

120597120578

120597119883119860)

= 119865minus1119860

119887120575119886119887120575119860119861

119863119861

(65)

leading to

119861119886

119860(119883119863) = 119865

minus1119861

119887(X 119863) 120575

119886119887120575119860119861

(66)


= 120597119863119861120597119883119860 and 120597119889

119886120597119909119887

= 120597119889119887120597119909119886 hold

As in Section 3 119873119860



120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861 (67)

Let 119873119886119887= 119873119860

119861120575119886

119860120575119861



120575

120575119909119886=

120597

120597119909119886minus 119873119887

119886

120597

120597119889119887 120575119889

119886= 119889119889119886+ 119873119886

119887119889119909119887 (68)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

119860

120597

120597119889119886otimes 120575119863119860 (69)


119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119886

119861120575119862

119888119909119888

119865119886

119860= 119909119886

119860=

120597119909119886

120597119863119860+ 119862119861

119860119862120575119886

119861120575119862

119888119909119888

(70)


G119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861 (71)

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863) (72)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862 (73)

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862 (74)

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863) (75)


119871 = [120581119860119861

119863119860119863119861]12

= [120581119860119861

(120597120578

120597119883119860)(

120597120578

120597119883119861)]

12

(76)

120581119860119861

= 119866119860119861

= 120575119860119862

120575119861119863

119866119862119863

= constant (77)





Γ119860

119861119862= 120574119860

119861119862= 119862119860

119861119862= 0 119873

119860

119861= 0 119899

119886

119887= 0

120575

120575119883119860=

120597

120597119883119860 120575119863

119860= 119889119863119860

120575

120575119909119886=

120597

120597119909119886 120575119889

119886= 119889119889119886

(78)



leading to

F =120597119909119886

120597119883119860120597

120597119909119886otimes 119889119883119860+

120597119909119886

120597119863119860120597

120597119889119886otimes 119889119863119860

= 119865119886

119860

120597

120597119909119886otimes 119889119883119860+ 119881119886

119860

120597

120597119889119886otimes 119889119863119860

(79)


119860= 119881119886



= 120581119860119861




0denote the



119860(120597120597119883




conditions

⟨sm⟩ = 119904119860119898119861⟨

120597

120597119883119860 119889119883119861⟩ = 119904

119860119898119860= 0 (80)


Ξ = Ξ119860

119861

120597

120597119883119860otimes 119889119883119861= 1 + 120574

0120593s otimesm

Ξ119860

119861[120578 (119883)] = 120575

119860

119861+ 1205740120593 [120578 (119883)] 119904

119860119898119861

(81)



1 and 1205931015840(0) = 120593


polynomial [53 59]

120593 [120578 (119883)] = 3 [120578 (119883)]2minus 2 [120578 (119883)]

3 (82)


F = AΞ 119865119886

119860= 119860119886

119861Ξ119861

119860 (83)


A = 119860119886

119860

120597


0120593s otimesm) (84)


120595 (F 120578D) = 119882(F 120578) + 119891 (120578D) (85)

119882 = 119882[A (F 120578)] 119891 (120578D) = 1205721205782(1 minus 120578)

2+ 1198712(D)

(86)


119886

119860= 119909119886

119860= 0 in (86))


2= 120581119860119861

119863119860119863119861


119860119861= 1205810120575119860119861 and


0

obey 1205810= (34)Γ

01198970and 120572 = 12Γ



Ψ [119909 (119883) 120578 (119883)] = intΩ

120595dΩ

dΩ =1003816100381610038161003816det (119866119860119861)

100381610038161003816100381612



119901

(87)



120575120595dΩ = 0 (88)


120575120595 =120597119882

120597119865119886119860

120597120575119909119886

120597119883119860+ (

120597119882

120597120578+

120597119891

120597120578) 120575120578 +

120597119891

120597119863119860120597120575120578

120597119883119860 (89)


120597119875119860

119886

120597119883119860=

120597 (120597119882120597119909119886

|119860)

120597119883119860= 0

120589 +120597119891

120597120578= 2120581119860119861

120597119863119860

120597119883119861

= 2120581119860119861

1205972120578

120597119883119860120597119883119861

(90)







5 Conclusions




References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119860 119863119860


1015840 12058710158401198721015840 1198801015840) be



1015840 are119909119886 119889119886 where 119909 isin 119872


119879119885 = 119867119885 oplus 119881119885 1198791198851015840= 119867119885

1015840oplus 1198811198851015840 (60)



119909119886= 119909119886(119883119860 119863119860) 119889

119886= 119861119886

119860119863119860 (61)


119861119886

119860= 119861119886

119860(119883119863) (62)



119863119860(119883) = 120575

119860119861 120597120578 (119883)

120597119883119861 (63)






119865119886

119860(119883119863) =

120597119909119886(119883119863)

120597119883119860 (64)


119889119886[119909 (119883119863) 119863] = 120575

119886119887 120597120578 (119883)

120597119909119887

= 120575119886119887

(120597119883119860

120597119909119887)(

120597120578

120597119883119860)

= 119865minus1119860

119887120575119886119887120575119860119861

119863119861

(65)

leading to

119861119886

119860(119883119863) = 119865

minus1119861

119887(X 119863) 120575

119886119887120575119860119861

(66)


= 120597119863119861120597119883119860 and 120597119889

119886120597119909119887

= 120597119889119887120597119909119886 hold

As in Section 3 119873119860



120575

120575119883119860=

120597

120597119883119860minus 119873119861

119860

120597

120597119863119861 120575119863

119860= 119889119863119860+ 119873119860

119861119889119883119861 (67)

Let 119873119886119887= 119873119860

119861120575119886

119860120575119861



120575

120575119909119886=

120597

120597119909119886minus 119873119887

119886

120597

120597119889119887 120575119889

119886= 119889119889119886+ 119873119886

119887119889119909119887 (68)


F = F + F = 119909119886

|119860

120575

120575119909119886otimes 119889119883119860+ 119909119886

119860

120597

120597119889119886otimes 120575119863119860 (69)


119865119886

119860= 119909119886

|119860=

120575119909119886

120575119883119860+ Γ119861

119860119862120575119886

119861120575119862

119888119909119888

119865119886

119860= 119909119886

119860=

120597119909119886

120597119863119860+ 119862119861

119860119862120575119886

119861120575119862

119888119909119888

(70)


G119860119861

=1

2

1205972(1198712)

120597119863119860120597119863119861 (71)

120574119860

119861119862=

1

2119866119860119863

(120597119866119861119863

120597119883119862+

120597119866119862119863

120597119883119861minus

120597119866119861119862

120597119883119863) (72)

119873119860

119861=

1

2

120597119866119860

120597119863119861 119866119860= 120574119860

119861119862119863119861119863119862 (73)

119862119860119861119862

=1

2

120597119866119860119861

120597119863119862=

1

4

1205973(1198712)

120597119863119860120597119863119861120597119863119862 (74)

Γ119860

119861119862=

1

2119866119860119863

(120575119866119861119863

120575119883119862+

120575119866119862119863

120575119883119861minus

120575119866119861119862

120575119883119863) (75)


119871 = [120581119860119861

119863119860119863119861]12

= [120581119860119861

(120597120578

120597119883119860)(

120597120578

120597119883119861)]

12

(76)

120581119860119861

= 119866119860119861

= 120575119860119862

120575119861119863

119866119862119863

= constant (77)





Γ119860

119861119862= 120574119860

119861119862= 119862119860

119861119862= 0 119873

119860

119861= 0 119899

119886

119887= 0

120575

120575119883119860=

120597

120597119883119860 120575119863

119860= 119889119863119860

120575

120575119909119886=

120597

120597119909119886 120575119889

119886= 119889119889119886

(78)



leading to

F =120597119909119886

120597119883119860120597

120597119909119886otimes 119889119883119860+

120597119909119886

120597119863119860120597

120597119889119886otimes 119889119863119860

= 119865119886

119860

120597

120597119909119886otimes 119889119883119860+ 119881119886

119860

120597

120597119889119886otimes 119889119863119860

(79)


119860= 119881119886



= 120581119860119861




0denote the



119860(120597120597119883




conditions

⟨sm⟩ = 119904119860119898119861⟨

120597

120597119883119860 119889119883119861⟩ = 119904

119860119898119860= 0 (80)


Ξ = Ξ119860

119861

120597

120597119883119860otimes 119889119883119861= 1 + 120574

0120593s otimesm

Ξ119860

119861[120578 (119883)] = 120575

119860

119861+ 1205740120593 [120578 (119883)] 119904

119860119898119861

(81)



1 and 1205931015840(0) = 120593


polynomial [53 59]

120593 [120578 (119883)] = 3 [120578 (119883)]2minus 2 [120578 (119883)]

3 (82)


F = AΞ 119865119886

119860= 119860119886

119861Ξ119861

119860 (83)


A = 119860119886

119860

120597


0120593s otimesm) (84)


120595 (F 120578D) = 119882(F 120578) + 119891 (120578D) (85)

119882 = 119882[A (F 120578)] 119891 (120578D) = 1205721205782(1 minus 120578)

2+ 1198712(D)

(86)


119886

119860= 119909119886

119860= 0 in (86))


2= 120581119860119861

119863119860119863119861


119860119861= 1205810120575119860119861 and


0

obey 1205810= (34)Γ

01198970and 120572 = 12Γ



Ψ [119909 (119883) 120578 (119883)] = intΩ

120595dΩ

dΩ =1003816100381610038161003816det (119866119860119861)

100381610038161003816100381612



119901

(87)



120575120595dΩ = 0 (88)


120575120595 =120597119882

120597119865119886119860

120597120575119909119886

120597119883119860+ (

120597119882

120597120578+

120597119891

120597120578) 120575120578 +

120597119891

120597119863119860120597120575120578

120597119883119860 (89)


120597119875119860

119886

120597119883119860=

120597 (120597119882120597119909119886

|119860)

120597119883119860= 0

120589 +120597119891

120597120578= 2120581119860119861

120597119863119860

120597119883119861

= 2120581119860119861

1205972120578

120597119883119860120597119883119861

(90)







5 Conclusions




References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Γ119860

119861119862= 120574119860

119861119862= 119862119860

119861119862= 0 119873

119860

119861= 0 119899

119886

119887= 0

120575

120575119883119860=

120597

120597119883119860 120575119863

119860= 119889119863119860

120575

120575119909119886=

120597

120597119909119886 120575119889

119886= 119889119889119886

(78)



leading to

F =120597119909119886

120597119883119860120597

120597119909119886otimes 119889119883119860+

120597119909119886

120597119863119860120597

120597119889119886otimes 119889119863119860

= 119865119886

119860

120597

120597119909119886otimes 119889119883119860+ 119881119886

119860

120597

120597119889119886otimes 119889119863119860

(79)


119860= 119881119886



= 120581119860119861




0denote the



119860(120597120597119883




conditions

⟨sm⟩ = 119904119860119898119861⟨

120597

120597119883119860 119889119883119861⟩ = 119904

119860119898119860= 0 (80)


Ξ = Ξ119860

119861

120597

120597119883119860otimes 119889119883119861= 1 + 120574

0120593s otimesm

Ξ119860

119861[120578 (119883)] = 120575

119860

119861+ 1205740120593 [120578 (119883)] 119904

119860119898119861

(81)



1 and 1205931015840(0) = 120593


polynomial [53 59]

120593 [120578 (119883)] = 3 [120578 (119883)]2minus 2 [120578 (119883)]

3 (82)


F = AΞ 119865119886

119860= 119860119886

119861Ξ119861

119860 (83)


A = 119860119886

119860

120597


0120593s otimesm) (84)


120595 (F 120578D) = 119882(F 120578) + 119891 (120578D) (85)

119882 = 119882[A (F 120578)] 119891 (120578D) = 1205721205782(1 minus 120578)

2+ 1198712(D)

(86)


119886

119860= 119909119886

119860= 0 in (86))


2= 120581119860119861

119863119860119863119861


119860119861= 1205810120575119860119861 and


0

obey 1205810= (34)Γ

01198970and 120572 = 12Γ



Ψ [119909 (119883) 120578 (119883)] = intΩ

120595dΩ

dΩ =1003816100381610038161003816det (119866119860119861)

100381610038161003816100381612



119901

(87)



120575120595dΩ = 0 (88)


120575120595 =120597119882

120597119865119886119860

120597120575119909119886

120597119883119860+ (

120597119882

120597120578+

120597119891

120597120578) 120575120578 +

120597119891

120597119863119860120597120575120578

120597119883119860 (89)


120597119875119860

119886

120597119883119860=

120597 (120597119882120597119909119886

|119860)

120597119883119860= 0

120589 +120597119891

120597120578= 2120581119860119861

120597119863119860

120597119883119861

= 2120581119860119861

1205972120578

120597119883119860120597119883119861

(90)







5 Conclusions




References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















5 Conclusions




References







































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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