Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 247

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 248 IV.2 VECTOR AND TENSOR ANALYSIS IV.2.1. Tensor function Let

1 2 ni i i nA V∈

be an thn order tensor. Then the tensor function of a scalar variable is defined as a map

( )

1 2 ni i i nA t : V→

(42) The scalar variable (parameter) t∈ can represent time, path, etc.

For n 0= , 0V = and we have a real-valued function of a real variable:

( )y a t : I= ⊂ → vector function For n 1= , 1 2V = and we have the vector function of a scalar variable is a

vector-valued function defined as a map from the set of real numbers to the space of vectors 2 :

( )i 2x t : I ⊂ → (42a)

( ) 2t : I ⊂ →r (42b)

The vector function provides a convenient method for the definition of curves in space by tracing the points by the position vector . The change of

parameter also provides information about the position of the point on the curve for different moments of time. This definition of curves by vector functions is equivalent to the parametric definition of curves:

or (43)

where are real valued functions. The derivative of a tensor function with respect to a scalar variable is defined as

Derivative of a tensor function ( ) ( ) ( )1 2 n 1 2 n

1 2 n

i i i i i ii i i t 0

A t t A td A t limdt t∆

∆∆→

+ −=

(43)

If the limit exists, then the derivative is a tensor of the same order

( )1 2 ni i i n n

d A t :V Vdt

→

Repeatedly, the higher order derivatives can be defined.

( ) [ ]t , t a,b∈r

[ ]t a,b∈

( )( )( )( )

x tt y t

z t

=

r( )( )( )

x x t y y t

z z t

= = =

[ ] t a,b∈

( ) ( ) ( ) [ ]x t , y t ,z t , t a,b∈

0

( )( )( )( )

x tt y t

z t

=

r

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 249

IV.2.2. Tensor field Let 3∈r be a position vector which specifies a location in Euclidian space 3E A tensor field is defined as a map ( )

1 2 ni i i 3 nA :V V→r

In particular, we have ( )s r scalar field (temperature, density, etc)

( )v r vector field (velocity, force, etc)

( )ijA r tensor field (deformation tensor, etc) defined at all points of space 3V∈r . A non-stationary tensor field is defined as a map ( )

1 2 ni i i 3 nA ,t :V V× →r

All considered functions are assumed to be continuous: ( ) ( )

1 2 n 1 2 n0

i i i i i i 0lim A A→

=r r

r r

scalar field A scalar field is defined as a real valued function of a vector variable:

By this function, a scalar value is specified for any point of space .

A scalar field can describe distribution in space of temperature, density, concentration, etc.

Function defines a scalar field on a plane. vector field A vector field is defined as a vector valued function of a vector variable:

By this function, a vector value is specified at any point 3V∈r .

A vector field can be described by a distribution in space of velocity, acceleration, force, etc.

A non-stationary scalar or vector field are defined as time-dependent maps

All operations defined for tensors can be applied for tensor fields point-wise.

( ) 3:ϕ →r

( ) 3x, y,z :ϕ →

r

( ) ( ) 2x, y :ϕ ϕ≡ →r

( ) 3 3: →v r

( ) ( ) ( ) ( )P x, y,z Q x, y,z R x, y,z= + +v r i j k

( ) 3,t :ϕ × →r

( ) 3 3,t : × →v r

( )ϕ r

r

r

( )v r

( ),tv r

r

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 250 IV.3.3. Space curves Consider a bounded space curve C in 3V with the end points A and B .

The graph of the space curve C is traced by a vector function C : ( )tr [ ]t a,b∈ with ( )a =r A , ( )b =r B (44) or in the tensor notations: C : ( )ix t [ ]t a,b∈ with ( )i ix a A= , ( )i ix b B= (45) which is just another form of a traditional parametric definition of the curve: C : ( )1 1x x t=

( )2 2x x t= [ ]t a,b∈ (46)

( )3 3x x t= Natural parameterization ( )sr with the help of arc length s has some useful properties. A differential element on the space curve is tangent to the curve:

( )ii

dxx t

dt′= ( )i idx x t dt′= (47)

d d dsdt ds dt

=r r ds

dt= T d ds=r T (48)

Types of some particular space curves: C is a closed curve if ( ) ( )a b=r r or ( ) ( )i ix a x b= C is a smooth curve if ( ) [ ]ix t C a,b′ ∈ (derivatives are continuous)

C is a piece-wise smooth if kk

C C=

, where kC are smooth curves

C is a simple curve if ( ) ( )i i 2x t x t≠ if 1 2t t≠ (w/o self-intersection)

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 251 IV.2.4. Level curves and surfaces Let ( ) 2f :V →r be a scalar field, then equation

( )f c=r or

describes the curves in the plane 1 2x x called the level curves: ( ) ( ){ }1 2 1 2x ,x f x ,x c, c= ∈

Let be a scalar field, then equation

( ) cϕ =r or ( )1 2 3x ,x ,x cϕ =

describes the surface in 3 called the level surface:

( ) ( ){ }1 2 3 1 2 3x ,x ,x x ,x ,x c, cϕ = ∈

For different values of the constant c∈ , we obtain the families of un-intersected level curves and surfaces (why do level curves not intersect?). IV.2.5. Operator “nabla”, gradient and directional derivative nabla ( )1

3 3: C V V∇ → is a differential vector operator defined as

∇ 1 2 3

, , x x x∂ ∂ ∂

=∂ ∂ ∂

i

x∂

=∂

tensor notation (49)

i = ∂ short-hand tensor notation Sometimes operator nabla is also called the Hamilton operator. gradient Operator nabla applied to a scalar-valued vector function yields a vector called the gradient of the scalar field

ϕ∇1 2 3

, ,x x xϕ ϕ ϕ∂ ∂ ∂

=∂ ∂ ∂

1 2 3

1 2 3

+ +x x xϕ ϕ ϕ∂ ∂ ∂

=∂ ∂ ∂

i i i (50)

which is a vector orthogonal to the level surface of the scalar field (or to the level curve in the 2-D case).

ϕ∇i

xϕ∂

=∂

i ϕ= ∂ short-hand tensor notation

( )1 2f x ,x c=

( ) 3:Vϕ →r

( )ϕ r

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 252 Directional derivative of the scalar field Let function defines a scalar field, and let be a unit vector,

1=s . Vector hs with a small h 0> is an increment in the direction .

Then a derivative of function in the direction s at the point in space r is defined as

D ϕs ( ) ( )h 0

hlim

hϕ ϕ

→

+ −=

r s r (51)

It determines the rate of change of the scalar field at the point r in the direction s . Using the differentiation rules for the multivariable functions and the definition of the operator nabla, let us write the other representations of the directional derivative:

sϕ∂∂

31 2

1 2 3

xx x + +x s x s x sϕ ϕ ϕ ∂∂ ∂∂ ∂ ∂

=∂ ∂ ∂ ∂ ∂ ∂

( ) ( ) ( )1 1 11 2 3

cos , + cos , + cos ,x x xϕ ϕ ϕ∂ ∂ ∂

=∂ ∂ ∂

s i s i s i

1 2 3

1 2 3

s + s + sx x xϕ ϕ ϕ∂ ∂ ∂

=∂ ∂ ∂

ϕ= ∇ ⋅s

ii

sxϕ∂

=∂

(52)

Therefore, the derivative of ϕ in any direction is equal to the projection of the

gradient ϕ∇ onto this direction:

sϕ∂∂

ϕ= ∇ ⋅s ( )cos ,ϕ ϕ= ∇ ∇ s (53)

It follows from this equation that the maximum value of the directional

derivative is achieved in the direction of the gradient of the scalar field at this point. So we can conclude that the gradient of the scalar field is a vector which has a direction of the greatest increase and its magnitude is equal to the directional derivative in this direction. The opposite direction ϕ−∇ corresponds to the direction of the greatest decrease.

From equation (53) yields also that d dϕ ϕ= ∇ ⋅ r . If n is the unit normal vector to the level surface of ϕ , then

nϕϕ ∂

∇ =∂

n (54)

Gradient of scalar field is orthogonal to level surface.

( ) 3:Vϕ →r s

s

( )ϕ r

sϕ∂

=∂

( )ϕ r

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 253 Directional derivative of the vector field Let function ( ) 3 3:V V→a r define a vector field, and let be a unit vector,

1=s . Vector hs with a small h 0> is an increment in the direction .

Then the derivative of function ( )a r in a direction s at the point of space r is defined as

Dsa s∂

=∂a ( ) ( )

h 0

hlim

h→

+ −=

a r s a r (55)

provided that the limit exists. It determines the rate of change of the vector field ( )a r at the point r in the direction s :

s∂∂a 31 2

1 2 3

xx x + +x s x s x s

∂∂ ∂∂ ∂ ∂=

∂ ∂ ∂ ∂ ∂ ∂a a a

( ) ( ) ( )1 1 11 2 3

cos , + cos , + cos ,x x x∂ ∂ ∂

=∂ ∂ ∂

a a as i s i s i

1 2 3

1 2 3

s + s + sx x x∂ ∂ ∂

=∂ ∂ ∂

a a a

( ) = ∇ ⋅s a (56) In tensor notation, the directional derivative is written as

ias

∂∂

ik

k

a s

x∂

=∂

(57)

Directional derivative of the 2nd order tensor field The directional derivative of the 2nd order tensor field is the natural

generalization of the directional derivatives of the scalar and vector fields in terms of tensors given by equations (52) and (57). Let function ( )ik 9 9A :V V→r define a 2nd order tensor field, and let be a unit

vector, 1=s . Vector hs with a small h 0> is an increment in the direction

. Then a derivative of tensor function ( )ikA r in a direction s at the point of space r is defined as

ikD As ikAs

∂=

∂ ( ) ( )ik ik

h 0

A h Alim

h→

+ −=

r s r (58)

provided that the limit exists. Consider the components of the directional derivative

ikAs

∂∂

ik ik ik 31 2

1 2 3

A A A xx x + +x s x s x s

∂ ∂ ∂ ∂∂ ∂=

∂ ∂ ∂ ∂ ∂ ∂ik ik ik

1 2 31 2 3

A A A s + s + s

x x x∂ ∂ ∂

=∂ ∂ ∂

or as a contruction of the 3rd order tensor

ikAs

∂∂

ikj

j

A s

x∂

=∂

(59)

s

s

s

s

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 254 IV.2.6. Flux Consider a vector field ( )a r . Let us define a flux of a vector field through the surface S as a surface integral

of the vector function ( )a r :

S

flux through the surface S flux = d⋅∫a r (60)

Let S be a surface and let a unit vector n determine the unit direction to surface

S (for the direction of the normal vector, we will agree to take the exterior direction for closed surfaces, and one of two directions for non-closed surfaces and stick to this direction when changing position on the surface).

The dot product ( )i i na cos , a⋅ = =a n n i

is the magnitude of the projection of vector on the normal direction n . Subdivide surface S into subsurfaces with the areas S∆ which can be assumed

to be flat and be characterized by the normal vector ∆S with the magnitude equal to the area S∆ : S∆ ∆=S n . Then the surface integral can be defined as a limit of the sums

(61)

Using the differential relation d dS=S n , we can express d⋅a S ( )dS= ⋅a n na dS= ( ) ( ) ( )

1 2 3x 1 x 2 x 3a cos , a cos , a cos , dS = + + n i n i n i

( ) ( ) ( )

1 2 3x 1 x 2 x 3a cos , dS a cos , dS a cos , dS = + + n i n i n i

1 2 3x 2 3 x 1 3 x 1 2a dx dx a dx dx a dx dx= + + (62) Then the flux of the vector field through the surface S can be written in the

traditional form of the surface integral:

( )1 2 3x 2 3 x 1 3 x 1 2S S S

d = dS = a dx dx a dx dx a dx dx⋅ ⋅ + +∫ ∫ ∫a S a n (63)

The flux of the vector field through the closed surface S is denoted by

Useful fact: the flux through any closed surface in the constant vector field ( ) 0=a r a is zero (conservation law):

0

S

d = 0⋅∫ a S

( )a r

S S

d = dS ⋅ ⋅∫ ∫a S a n

kS 0 kS

d = lim∆

∆→

⋅ ⋅∑∫a S a S

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 255 Flux of the 2nd order tensor Let function ( )ik 9 9A :V V→r define a 2nd order tensor field and let n be a

variable normal vector to the smooth surface S . Then the flux of the 2nd order tensor field through the surface S is defined as a vector f with the components

i ik kS

f A n dS= ∫ (64)

For example, the flux of the stress tensor ikp in the elastic medium through the

surface S defines the total stress vector P acting on the surface which has the components:

i ik kS

P p n dS= ∫

IV.2.7. Divergence Divergence of the vector field ( )a r at the point of space r is defined as a limit of the averaged flux through the surface of the arbitrary volume

containing point r :

diva (65)

Use a parallelepiped for the arbitrary volume V with one corner located at the point ( )1 2 3x ,x ,x=r , sides ix∆ and faces perpendicular to the coordinate axes ,

and 1 2 3V x x x∆ ∆ ∆= with the surfaces 1 2 3S x x∆ ∆ ∆= ,… (see picture). Then

diva

k

k

V 0= lim

V

∆

→

⋅∑a S

k k

k

V 0

S= lim

V

∆

→

⋅∑a n

( ) ( )1 1 2 3 1 2 3 1 1

V 0

x x ,x ,x x ,x ,x S ...= lim

V

∆ ∆→

+ − ⋅ + a a i

( ) ( )

1 2 3

1 1 2 3 1 2 3 1 2 3

x x x 01 2 3

x x ,x ,x x ,x ,x x x ...= lim

x x x∆ ∆ ∆

∆ ∆ ∆

∆ ∆ ∆→

+ − ⋅ + a a i

( ) ( )

1 2 3

1 1 2 3 1 1 2 3 1

x x x 01

x x ,x ,x x ,x ,x ...= lim

x∆ ∆ ∆

∆

∆→

+ ⋅ − ⋅ + a i a i

S

V 0

d lim

V→

⋅=

∫ a S

S

V 0

d lim

V→

⋅=

∫ a S

S

V 0

dS = lim

V→

⋅∫ a n

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 256

( ) ( )

1 1

1 2 3

x 1 1 2 3 x 1 2 3

x x x 01

a x x ,x ,x a x ,x ,x ...= lim

x∆ ∆ ∆

∆

∆→

+ − +

( ) ( )

1 1

1 2 3

x 1 1 2 3 x 1 2 3

x x x 01

a x x ,x ,x a x ,x ,x ...= lim

x∆ ∆ ∆

∆

∆→

+ − +

31 2 xx x

1 2 3

aa a=

x x x∂∂ ∂

+ +∂ ∂ ∂

k

k

a=

x∂∂

kk

= x∂⋅∂ai

= ∇ ⋅a (66) Physical meaning of the divergence [Z-519, K-145]:

div a = ∇ ⋅a For incompressible fluid, the divergence of the velocity vector field div =v 0 Divergence of the 2nd order tensor field is defined as a limit

( )ik jdivA

jm mS

V 0

A n dS = lim

V→

∫ (67)

It can be shown that the divergence of the 2nd order tensor is a vector with the

components

( )ik jdivA jm

m

A

x∂

=∂

(68)

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 257 IV.2.8. Curl The other physical characteristic of a tensor field is given by the curl of the

tensor field. The curl of the vector field ( )a r at the point of space r is defined as a limit of

the averaged flux through the surface of the arbitrary volume containing point r

curl a S

V 0

dS lim

V→

×=

∫ n a

(69)

Compare with divergence (equation (66)):

diva S

V 0

dS = lim

V→

⋅∫ n a

= ∇ ⋅a

Application of the operator nabla yields a similar representation of the curl curl a = ∇×a (70)

Physical meaning: curl a measures how fast vector field rotates.

For irrotational fluid, the curl of the velocity vector field is zero curl =v 0 Useful formulas:

curl a = ∇×a

1 2 3

1 2 3

1 2 3

x x x

a a a

∂ ∂ ∂=

∂ ∂ ∂

i i i

(71)

( )icurla jk

j k

aa

x x∂∂

= −∂ ∂

where indices i, j ,k are a cyclic

permutation of the numbers With the help of alternating unit tensor, it can be written as

( )icurla k

ijkj

a

xε

∂=

∂ (72)

In the vector form:

curl a 3 32 1 2 1

2 3 3 1 1 2

a aa a a a x x x x x x

∂ ∂∂ ∂ ∂ ∂= − + − + − ∂ ∂ ∂ ∂ ∂ ∂

i j k

3 2

2 3

31

3 1

2 1

1 2

a ax x

aa x xa ax x

∂ ∂− ∂ ∂

∂∂= −

∂ ∂ ∂ ∂ −∂ ∂

1,2,3

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 258 IV.2.9. OPERATOR NABLA AND RELATED DIFFERENTIAL OPERATORS

nabla ∇ ( )i ∇ i

x∂

=∂

1

2

3

x x x

∂ ∂ ∂

= ∂ ∂ ∂

grad ϕ S

V 0

dSlim

V

ϕ

→=

∫ n

ϕ= ∇ ( )iϕ∇ i

xϕ∂

=∂

1

2

3

x

x

x

ϕ

ϕ

ϕ

∂ ∂ ∂

= ∂ ∂ ∂

div a diva k

k

a

x∂

=∂

31 2

1 2 3

aa ax x x

∂∂ ∂= + +∂ ∂ ∂

curl a ( )icurla k

ijkj

a

xε

∂=

∂

curl a 3 32 1 2 1

2 3 3 1 1 2

a aa a a a x x x x x x

∂ ∂∂ ∂ ∂ ∂= − + − + − ∂ ∂ ∂ ∂ ∂ ∂

i j k

3 2

2 3

31

3 1

2 1

1 2

a ax x

aa x xa ax x

∂ ∂− ∂ ∂

∂∂= −

∂ ∂ ∂ ∂ −∂ ∂

2 2 2 22

1 2 3 k k

div grad x x x x xϕ ϕ ϕ ϕ∆ϕ ϕ ϕ ϕ ∂ ∂ ∂ ∂

≡ ∇ ⋅∇ ≡ ∇ = = + + =∂ ∂ ∂ ∂ ∂

, Laplacian of scalar field

ij i ii j i i

x x x x

∆ϕ δ ϕ ϕ ϕ∂ ∂ ∂ ∂= = = ∂ ∂

∂ ∂ ∂ ∂ short-hand notation for Laplacian 2 ∆∇ ⋅∇ = ∇ =

S

V 0

dS = lim

V→

⋅∫ n a

= ∇ ⋅a

S

V 0

dS lim

V→

×=

∫ n a

= ∇×a jk

j k

aa

x x∂∂

= −∂ ∂

= ∇×a

1 2 3

1 2 3

1 2 3

x x x

x x x

a a a

∂ ∂ ∂=

∂ ∂ ∂

i i i

( ) 3:ϕ →r

notations for the Laplacian operator

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 259

Let ( ) ( ) ( ) 3 3, , : →a r b r F r be vector fields, ( ) ( ) 3,u :ϕ →r r be scalar fields, [B&T, p.168]

ϕ∇ grad ϕ=

∇⋅a div = a

∇×a curl = a

( )∇ ∇ ⋅a grad div = a

ϕ∇ ⋅ ∇ 2 div grad ϕ ∆ϕ ϕ= ≡ ≡ ∇ Laplacian operator

( )∇ ⋅ ∇×a div curl = a 0= vanishes identically

ϕ∇× ∇ curl grad ϕ= = 0 vanishes identically

( )∇× ∇×a curl curl = a

1. ( )∆ ϕ ψ ∆ϕ ∆ψ+ = +

2. ( )ϕ ψ ϕ ψ∇ + = ∇ +∇ ( )grad grad gradϕ ψ ϕ ψ+ = +

3. ( ) ( ) ( )∇ ∇ ⋅ + = ∇ ∇⋅ +∇ ∇ ⋅ a b a b ( )grad div grad div grad div + = +a b a b

4.

5.

6.

7. ( )( ) ( ) ( )∇× ∇× + = ∇× ∇× +∇× ∇×a b a b ( ) ( ) ( )curl curl curl curl curl curl+ = +a b a b

8.

9. ( )ϕψ ψ ϕ ϕ ψ∇ = ∇ + ∇ ( )grad grad gradϕψ ψ ϕ ϕ ψ= +

10.

11.

12.

13. ( ) ( ) ( )2 ∆∇ =

∇× ∇× = ∇ ∇⋅ − ∇ ⋅∇a a a

( ) ( )curl curl grad div ∆= −a a a

For composite functions ( )fϕ r and ( )f a r , the chain rule is applied

14. ( ) df fdfϕϕ∇ = ∇ r ( ) dgrad f grad f

dfϕϕ = r

15. ( ) ( ) df fdf

∇⋅ = ∇ ⋅ aa r ( ) ddiv f grad f

df= ⋅

aa r

16. ( ) ( ) df fdf

∇× = ∇ × aa r ( ) dcurl f grad f

df= ×

aa r

c∈

( )∇ ⋅ + = ∇ ⋅ +∇ ⋅a b a b ( )div div div+ = +a b a b

( )c c∇⋅ = ∇ ⋅a a ( )div c cdiv=a a

( )∇× + = ∇× +∇×a b a b ( )curl curl curl+ = +a b a b

( ) ( )ϕ∇× ∇× +∇ = ∇× ∇×a a ( ) ( )curl curl grad curl curlϕ+ =a a

( )ϕ ϕ ϕ∇ ⋅ = ∇ ⋅ + ⋅∇a a a ( )div div gradϕ ϕ ϕ= + ⋅a a a

( ) ( ) ( )∇ ⋅ × = ⋅ ∇× − ⋅ ∇×a b b a a b ( )div curl curl× = ⋅ − ⋅a b b a a b

( ) ( )ϕ ϕ ϕ∇× = ∇× +∇ ×a a a ( )curl curl gradϕ ϕ ϕ= + ×a a a

( ) ( ) ( )mijk klmi i i

j l

a x x

ε ε∗ ∂∂∇× ∇× = = ∇ ∇⋅ − ∇ ⋅∇ ∂ ∂

a a a

∗

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 260 Cartesian coordinates ( )x, y,z Cylindrical coordinates ( )r, ,zθ Spherical coordinates ( )r, ,φ θ

x r cosθ= x r cos sinφ θ=

y r sinθ= y r sin sinφ θ= z z= z r cosθ=

2 2 2r x y= + 2 2 2 2r x y z= + +

ytanx

θ = ytanx

φ = , 2 2 2

z zcosr x y z

θ = =+ +

Basis vectors ( ) = 1,0,0i r = cos sinθ θ+e i j r = cos sin sin sin cosφ θ φ θ θ+ +e i j k

( ) = 0,1,0j = - sin cosθ θ θ+e i j = - sin cosθ θ φ+e i j

( ) = 0,0,1k z = e k = cos cos sin cos sinφ φ θ φ θ θ+ −e i j k

,

Line elements dx,dy,dz dr,rd ,dzθ dr, r sin d , rdθ φ θ

Differential areas

xdA = dydz rdA = rd dzθ 2rdA = r sin d dθ φ θ

ydA = dxdz dA = drdzθ dA = r sin d drφ θ φ

zdA = dxdy zdA = rd drθ dA = d dφ ρ φ ρ

Differential volume

dV = dxdydz dV = rdrd dzθ 2dV = r sin d d drθ φ θ

Arc length

2 2 2 2ds dx dy dz= + + 2 2 2 2 2ds = dr r d dzθ+ + 2 2 2 2 2 2ds = dr r sin d r dθ φ θ+ +

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 261 scalar field ( )u r ( )u x, y,z ( )u r, ,zθ ( )u r, ,φ θ

Gradient u∇ u u uu = , ,x y z

∂ ∂ ∂∇ ∂ ∂ ∂

u 1 u uu = , ,r r zθ∂ ∂ ∂ ∇ ∂ ∂ ∂

u 1 u 1 uu = , ,r r sin rθ φ θ

∂ ∂ ∂∇ ∂ ∂ ∂

u u u= x y z∂ ∂ ∂

+ +∂ ∂ ∂

i j k r zu 1 u u= r r zθθ∂ ∂ ∂

+ +∂ ∂ ∂

e e e ru 1 u 1 u= r r sin rφ θθ φ θ∂ ∂ ∂

+ +∂ ∂ ∂

e e e

Laplacian 2u∇ 2 2 2

22 2 2

u u uu = x y z∂ ∂ ∂

∇ + +∂ ∂ ∂

2 1 uu = rr r r∂ ∂ ∇ + ∂ ∂

2 22

1 uu = rr rr∂ ∂ ∇ + ∂ ∂

2 2

2 2 2

1 u ur zθ

∂ ∂+ +

∂ ∂

2

2 2 2 2

1 u 1 usinr sin r sin

θθ θθ φ θ

∂ ∂ ∂ + + ∂ ∂∂

vector field ( )F r ( )x y zF ,F ,F ( )r zF ,F ,Fθ ( )rF ,F ,Fφ θ

r x yF F cos F sinθ θ= + r x y zF F cos sin F sin sin F cosφ θ φ θ θ= + +

x yF F sin F cosθ θ θ= + x yF F sin F cosφ φ φ= − +

z zF F= x y zF F cos cos F sin cos F sinθ φ θ φ θ θ= + −

x rF F cos F sinθθ θ= − x rF F cos sin F sin F cos cosφ θφ θ φ φ θ= − +

y rF F sin F cosθθ θ= + y rF F sin sin F cos F sin cosφ θφ θ φ φ θ= + +

z zF F= z rF F cos F sinθθ θ= −

Divergence

div = ∇ ⋅F F yx zFF Fx y z

∂∂ ∂+ +

∂ ∂ ∂ ( ) z

rF F1 1rF

r r r zθ

θ∂ ∂∂

+ +∂ ∂ ∂

( )2r2

F1 1r Fr r sinr

φ

θ φ∂∂

+ +∂ ∂

( )1 sin Fr sin θθ

θ θ∂

+ ∂

curl = ∇×F F

x y z

x y zF F F

∂ ∂ ∂=

∂ ∂ ∂

i j k

r z

r z

r1r r z

F rF F

θ

θ

θ∂ ∂ ∂

=∂ ∂ ∂

e e e

r

2

r

r r sin1

rr sinF rF r sin F

θ φ

θ φ

θ

θ φθθ

∂ ∂ ∂=

∂ ∂ ∂

e e e

yz FFy z

∂ ∂= − + ∂ ∂

i zr

FF1r z

θ

θ∂∂ = − + ∂ ∂

e ( )

r

F sin F1r sin

φ θθ

θ θ φ

∂ ∂ = − +

∂ ∂ e

x zF Fz x

∂ ∂ + − + ∂ ∂ j r zF F

z r θ∂ ∂ + − + ∂ ∂

e ( ) rrF F1

r rθ

φθ ∂ ∂

+ − + ∂ ∂ e

y xF Fx y

∂ ∂+ − ∂ ∂

k ( ) r

z

rF F1r r

θ

θ ∂ ∂

− ∂ ∂ e

( )rrFF1 sin

r sin rφ

θθθ φ

∂∂ + −∂ ∂

e

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 262 Exercises: 1) Navier-Stokes equations (governing equation for fluid motion):

( ) 2V V V P g Vt

ρ ρ µ∂+ ⋅∇ = −∇ + + ∇

∂

Using tensor notations show that for incompressible, irrotational, steady flow this simplifies to Bernoulli’s equation:

2V P g

2ρ ρ∇ = −∇ +

2) Maxwell’s equations [B&T, 246]

Electromagnetic field in a medium of dialectric constant ε , magnetic permeability µ , and conductivity σ ( 0ρ = , no free chardge inconducting medium, and σ=j E (Ohm’s law) ) is described by a system of Maxwelll’s equations: div 0=E Gauss’s law

div 0=H Gauss’s law for magnetism

curl c tµ ∂

= −∂HE Faraday’s law

4curl c t cε πσ∂

= − +∂EH E Maxwell-Ampere law

where ( ),t=E E r is the electric field and ( ),t=H H r is the magnetic field. Show that the following wave equation can be derived from Maxwell’s equations:

2

2 2 2

4tc t c

µε πµσ∆ ∂ ∂= +

∂∂E EE

Hint: apply operator curl to the last two equation.

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 263 IV.2.10. Line Integral Consider a vector-valued function , where vectors belong to the space curve

C : ( )tr [ ]t a,b∈ with ( )a =r A , ( )b =r B (69) We will consider a line integral which symbolically is written as:

(70)

Let us see how this integral is defined in its physical sense. Set up a partition nP of the curve C into a discrete set of n points:

( ) ( ){ }n 0 1 2 nP a , , ,..., b= = =r r r r r r and define the increment Define the norm of partition as the biggest increment in the partition: n kk

P max ∆= r

Denote the values of function at the points of partition

Form a dot product k k∆⋅a r which has a physical sense of the work performed by the force ka over path k∆r. Then the line integral is defined similarly to definition of the definite integral as a limit of the Riemann sum:

(71)

which physically expresses the work done by the force along the space curve C . Calculation of the line integral. Let the vector function have the following specification:

( ) ( ) ( )1 2 3 1 2 3 1 2 3P x ,x ,x Q x ,x ,x R x ,x ,x= + +i j k

( )i j iP x= i (72) Let the parameterization of the curve C be defined by ( )tr ( ) ( ) ( )1 1 2 2 3 3x t x t x t= + +i i i

(73) differentiation of this equation yields:

( )d tdt

r ( )i ix t′= i or i idx= i (74)

from which follows that k d⋅i r kdx= (75)

( )a r

C

d⋅∫a r

k k k 1∆ −= −r r r

( )a r

( )k k=a a r

P 0n

n

k kn k 1C

d lim ∆→

→∞ =

⋅ = ⋅∑∫a r a r

( )a r

( )a r

( )a r

( ) ( ) ( )1 1 2 3 1 2 1 2 3 2 3 1 2 3 3P x ,x ,x P x ,x ,x P x ,x ,x= + +i i i

( )i ix t= i

( )d tr ( )i ix t dt′= i

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 264

Then the line integral (51) can be transformed to

C

d⋅∫a r

( ) ( ) ( )1 1 2 3 1 2 1 2 3 2 3 1 2 3 3C

P x ,x ,x dx P x ,x ,x dx P x ,x ,x dx = + + ∫

this is the traditional form without parentheses. Then applying i idx x dt′= ,

( ) ( ) ( )b

1 1 2 3 1 2 1 2 3 2 3 1 2 3 3a

P x ,x ,x x P x ,x ,x x P x ,x ,x x dt′ ′ ′ = + + ∫ (76)

Example Find for vector function

3p,q,x=a p,q∈ along the space curve C : ( )tr ( ) ( )1 2 3cos t sin t t= + +i i i t from 0 to π Identify: ( )1x t cos t= ( )1x t sin t′ = −

( )2x t sin t= ( )2x t cos t′ =

( )3x t t= ( )3x t 1′ = Then, using equation (56), one ends up with

[ ]0

p sin t q cos t t dtπ

= − + +∫

2

0

tp cos t q sin t2

π

= + +

2

p p2π

= − + −

2

2 p2π

= − ■

If the curve is defined with the natural parameterization , then

( )d d ds dssdt ds dt dt

= =rr T where T is a unit vector tangent to the curve C .

Then d ds=r T Recall also

ds 2 2 21 2 3dx dx dx= + + 2 2 2

1 2 3x x x dt′ ′ ′= + +

( ) ( ) ( )1 1 2 3 1 2 1 2 3 2 3 1 2 3 3C

P x ,x ,x P x ,x ,x P x ,x ,x d = + + ⋅ ∫ i i i r

( ) ( ) ( )1 1 2 3 1 2 1 2 3 2 3 1 2 3 3C

P x ,x ,x d P x ,x ,x d P x ,x ,x d = ⋅ + ⋅ + ⋅ ∫ i r i r i r

( ) ( ) ( )1 1 2 3 1 2 1 2 3 2 3 1 2 3 3C

P x ,x ,x dx P x ,x ,x dx P x ,x ,x dx= + +∫

C

d⋅∫a r

C

d⋅∫a r

( )sr

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 265

The line integral then is calculated according to

C

d⋅∫a r

( ) ( ) ( )b

2 2 21 1 2 3 1 2 1 2 3 2 3 1 2 3 3 1 2 3

a

P x ,x ,x P x ,x ,x P x ,x ,x x x x dt′ ′ ′ = ⋅ + ⋅ + ⋅ + + ∫ i T i T i T

b2 2 2

1 2 3a

x x x dt′ ′ ′= ⋅ + +∫a T (77)

Therefore, the work is performed only by the tangential component of the force.

Conservative vector fields If ( )a r is a gradient field of some scalar field

( ) ϕ= ∇a r (78)

(in this case function ( )ϕ r is called a potential function for the gradient field

( )a r ), then a linear integral along the curve connecting two points 1r and 2r is equal to the difference between values of the scalar function at these end points:

( ) ( )1 2C C C

d d dϕ ϕ ϕ ϕ⋅ = ∇ ⋅ = = −∫ ∫ ∫a r r r r (79)

It means that the same result will occur for any curve connecting points 1r and

2r , and the line integral is said to be independent of path. Of particular interest are the linear integrals along the closed curves denoted by

C

d⋅∫ a r

It is obvious that for the gradient field

C C

d d 0ϕ⋅ = ∇ ⋅ =∫ ∫a r r

That is why if a vector field is a gradient field of some scalar field it is said to be conservative. We have for a conservative field that d dϕ ϕ= ∇ ⋅ r d= ⋅a r ( )1 1 2 2 3 3P P P d= + + ⋅i i i r

( )1 1 2 2 3 3P d P d P d= ⋅ + ⋅ + ⋅i r i r i r

1 1 2 2 3 3P dx P dx P dx= + + (80) Therefore, 1 1 2 2 3 3P dx P dx P dx+ + is an exact differential. So, the line integral

1 1 2 2 3 3C

P dx P dx P dx+ +∫

is independent of path, if 1 1 2 2 3 3P dx P dx P dx dϕ+ + = is an exact differential. Test for path independence Recall from calculus that the differential form 1 1 2 2 3 3P dx P dx P dx+ + is an exact differential if and only if

i k

k i

P Px x∂ ∂

=∂ ∂

(81)

It is called the test for path independence of a linear integral in space.

( ) ( ) ( )1 1 2 3 1 2 1 2 3 2 3 1 2 3 3C

P x ,x ,x P x ,x ,x P x ,x ,x d = + + ⋅ ∫ i i i r

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 266 IV.2.11. Volume integral Consider a scalar field ( )ϕ r and let V be a volume.

Subdivide the volume V into subvolumes kk

V V∆= ∑ and

construct an integral of the function ( )ϕ r over the volume V as a limit

( )V

dVϕ∫ r ( )k

k kV 0 k lim V∆

ϕ ∆→

= ∑ r (82)

where kr is an arbitrary point in the subvolume kV∆ . IV.2.12. The Divergence Theorem (the Gauss-Ostrogradsky Theorem or the Divergence Theorem) Let V be a volume bounded by a closed surface S . Then flux of the vector field ( )a r through the surface S is equal to the integral

of the divergence of the vector field over the volume V :

S V

d = div dV⋅∫ ∫a S a

(83)

Proof: We will show that equation (83) is approximately valid with any degree

of accuracy, i.e. that

S V

d - div dV ε⋅ <∫ ∫a S a

for any 0ε > .

Subdivide volume V into kk

V V∆= ∑ such that

( )k

kk S

1 d - divV

δ⋅ <∫ a S a r

that is possible according to the definition of the divergence as a limit (65). Multiply this inequality by kV

( )k

k k kS

d - V div Vδ⋅ <∫ a S a r

then summation over all k yields

( )k kkS

d - V div Vδ⋅ <∑∫ a S a r

In this result, the surface integral only over the exterior surface S is left. All interior surfaces kS have to be the boundaries of some adjacent volumes mV and

kV . Having the opposite normal vectors, k m= −n n , the surface integrals over

kS cancel each other in the summation:

k mS S

d d ⋅ + ⋅∫ ∫a S a S

k m

k mS S

dS dS = ⋅ + ⋅∫ ∫a n a n

k k

k kS S

dS dS = ⋅ − ⋅∫ ∫a n a n

0=

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 267

Let k →∞ with kV 0→ , then according to the volume integral definition (82),

( )k

k kV 0 klim V div∆ →

∑ a rV

div dV= ∫ a

and

S V

d - div dV Vδ⋅ <∫ ∫a S a

.

Choose Vεδ = , then

S V

d - div dV ε⋅ <∫ ∫a S a

for any specified 0ε > . ■

Recall

diva 31 2 xx x

1 2 3

aa a=

x x x∂∂ ∂

+ +∂ ∂ ∂

( )ix i na cos , a⋅ = =a n n i

Then the other forms of the Divergence Theorem (Gauss-Ostrogradsky theorem)

can be written as

S V

d = div dV⋅∫ ∫a S a

S V

dS = div dV⋅∫ ∫a n a

nS V

a dS = div dV∫ ∫ a

( ) ( ) ( ) 31 2

1 2 3

xx xx 1 x 2 x 3

1 2 3S V

aa aa cos , a cos , a cos , dS = dV

x x x∂∂ ∂

+ + + + ∂ ∂ ∂ ∫ ∫n i n i n i

Application: The Divergence Theorem has a great importance in mathematical modeling in engineering. In derivation of the governing equations for physical processes in the continuous media, the conservation laws are applied to the control volume yielding an equation which contains both surface integrals and volume integrals. Application of the Divergence Theorem reduces all integrals to the volume integral which allows combination of all terms in one volume integral and concludes with the partial differential equation which governs the physical process under consideration (see example of derivation of the Heat Equation in the Section VIII.1.12).

Chapter IV Vector and Tensor Analysis IV.2 Vector and Tensor Analysis September 23, 2017 268