B. The Differential Calculus of Forms and Its Applications 4.14 The Exterior Derivative4.15 Notation for Derivatives4.16 Familiar Examples of Exterior Differentiation4.17 Integrability Conditions for Partial Differential Equations4.18 Exact Forms4.19 Proof of the Local Exactness of Closed Forms4.20 Lie Derivatives of Forms4.21 Lie Derivatives and Exterior Derivatives Commute4.22 Stokes' Theorem4.23 Gauss' Theorem and the Definition of Divergence 4.24 A Glance at Cohomology Theory4.25 Differential Forms and Differential Equations4.26 Frobenius' Theorem (Differential Forms Version)4.27 Proof of the Equivalence of the Two Versions of Frobenius' Theorem4.28 Conservation Laws4.29 Vector Spherical Harmonics

4.14. The Exterior Derivative

The exterior derivative is a map

1: p pd M M by p d p

s.t.

(i) d d d

, &p qM M

( distributive )

(ii) pd d d ( anti- Leibniz / derivation )

(iii) 2 0d d d ( Poincare lemma )

The Poincare lemma is a consequence of the irrelevance of the order of partial derivatives.

Useful properties (see Ex.4.14) : d f d g d f d g

1

1

1

!p

p

iii id d dx dx

p

1

1 ,

1

!p

p

iiji i j dx dx dx

p

11

,1

ppi i jj i i

d p

1 p

p

i i jd

4.15. Notation for Derivatives

2

, j k k j

ff

x x

k j f

Partial derivation is not a tensor operation:

Let'

' 'i

i i i ii i

xdx dx dx

x

' ' '

ii

i i i ii

x

x

Then

' ', ' ',

i i i jj i jj

V V

' '' '

i j i ij j j iV V ' '

' 'j i i j i i

j j i j i jV V

' ', ' , '

i i j i i ji j j i j jV V

''

j i ij i jV

or'

, 'i i j

i j jV

Exception: , i if f are the components of the 1-form d f

4.16. Familiar Examples of Exterior Differentiation

Details of vector analysis (in 3) can be found in Frankel, §§ 2.5e & 2.9c.

Let iia a be a vector field in 3 described by Cartesian coordinates.

Its covector is the 1-form field iia a dx with

iia a

iid a d a dx→ ,j i

i ja dx dx , ,

1

2i j

j i i ja a dx dx

iib a i j kk j a i j k

j ka (valid only for Cartesian 3)

i i j ki l m i l m j kb a j k k j

l m l m j ka l m m la a

1

2k i j

k i jd a b dx dx

, ,m l l ma a

kk i ji j

d a b ki j k b ki j k a

1

2k i j

k i jd a b dx dx å å

1

2i j i jdx dx dx dx å

ki j k l m i j

l mdx dx dx dx

k l m i j j il m l m

k l m i j j il m l m

k i j k j i 2 k i j

i j k i jkdx dx å →

1

2k m i j

k i j mb 1

2k m i i m

k i k i mb

13

2m i

m ib b iib i ia a

1 2 1 21 2 3

1

2dx dx dx dx å

3

etc…Treating the forms as tensors, we have

1 21 2 3

1

2dx dx

→ j

i j ib a

Using the formula

i j k

1

1 11

1*

!p

p n pn p

j j

j j i ii ip p

p

1*

2!l j k

j k i lid a b

we have

1

2!j k lj k i lb

12!

2!li lb ib

* iid a b dx

Alternatively:

iid a b a å

d a=å

1*

2!i j l m

l mk i jk

dx dx

* i j ki j kdx dx dx

c.f.

1

2k i j

k i jd a b dx dx

(Cartesian Coord)

Divergence:

→

iia a å å

1

2!j k

i i j k dx dx å * idx mi m jk ij k

åi j k

1

2!i j k

i j ka dx dx *a a

,

1

2!i m j k

m i j kd a a dx dx dx å

,

1 3!

3! 2!l mn p

i j k l pmni j kd a a å ,

1

2!p l mn

l i j k pmna ,

1

2!p l mn

l i j k pmna

,p l

l i j k pa ,l

l i j ka i j ka

1

3!i j k

i j kd a a dx dx dx å a *d a

Caution: For curvilinear coordinates, 1

3!i j k

i j k dx dx dg x x

See Frankel §2.9

4.17. Integrability Conditions for Partial Differential Equations

Consider the PDEsf

gx

, i if a

fh

y

i.e., with 1 2, ,x x x y

→ d f a

1 2, ,a a g h

(coordinate independent)

0d d f d a ,j i

i ja dx dx

→ ,,

10

2!k li j k li ja a ( Integrability conditions )

1, 2 2 ,1 0a a g h

y x

2 2f f

y x x y

General case: Frobenius’ theorem

4.18. Exact Forms

A form α is exact if β s.t. d

d d

A form α is closed if 0d

exact → closed

But a closed form is guaranteed exact only locally.

i.e., a closed form is exact in the neighborhood of a non-singular point that can be described in a single coordinate patch.

Proof: See §4.19.

Note that β is not unique since d

Example: 2 2

xdy ydx

x y

2 2

2 22 2 2 22 2 2 2

1 2 1 2x yd dx dy dy dx

x y x yx y x y

0

For (x, y) (0,0)

Let (0,0) be inside C1 , then α is closed in D2.

1tany

fx

2 2

2

1 1

1

yd f d dy dx

y x xx

θ is multi-valued in D → α is not (globally) exact in D.

α is (locally) exact in any region not enclosing C1.

α is globally exact in D.

(x, y) = (0,0) is a singular point

• Homology: study of topological invariants by means of chains.• Chains are linear combinations of simplexes.

– 0-simplex = point. – 1-simplex = closed interval.– 2-simplex = triangle.– 3-simplex = tetrahedron.– …

• Chains have simple boundaries.• Chains form a vector space.• Cohomology: homology of co-chains (duals of chains).• Integration of forms makes use of cohomology theory.

For an introduction, see Aldrovandi, Chap 2.

For a more detailed treatment, see C.Nash & S.Sen, “Topology & Geometry for Physics”, Chaps 4 & 6.

4.19. Proof of the Local Exactness of Closed Forms

Let U be a n-D region diffeomorphic to an open n-ball in n.

Then, with suitable scaling, the coordinates of a point in U can be written as

1

1

1

!p

p

iii i dx dx

p

a p-form closed in U.

1

1 ,

10

!p

p

iiji i jd dx dx dx

p i.e.

1, , nx x x with 21

1n

i

i

r x

Let

x U

Let iir x then

r

2

2

1

1 !p

p

iijj i ix dx dx

p

1 1 1 1p p

ji i j i i

x

1 ,0

pi i j

1 1p

jj i ix

Let

1 1

1 1

1

1 !p

p

i i

i i dx dxp

with

1 1 1 1

11

0p p

p ji i j i ix d t t x t x

→

1 1

1 1 ,

1

1 !p

p

i iji i jd dx dx dx

p

1 11 1

,ppi i jj i i

d p

1 1p

j i ip

1 1 1 1 1 1

11

, ,

0p p p

p k ki i j j k i i k i i jx d t t t x t x t x

1 1 1 1

11

,

0p p

p p kj i i k i i jd t t t x t x t x

Since 1 1p pi i i i

we have

1 11 1 1 1

11

, ,0

pp p

p p kj i ii i j k i i j

x d t t t x t x t x

1 1 1 1 1 2 1 1 1 1 1 1 1

1

, , , , ,2

1 ! !p p p m m m p p

p

k i i j k i i j i k i i j i i i k i i j j i i km

p p

1 1 ,0

pk i i j

0d →

1 1 1 11 1

11

,0

p pp

p p kj i i k i i jj i i

d d t pt t x pt x t x

we have

where k means that the position of index k is fixed.

1 1 1 1 1 1 1 1, ,m m m p m m m pi i i k i i j k i i i i i j

Since1 1 ,pk i i j

1 1 1 1 1 1, , ,

1 ! !p p pk i i j k i i j j i i k

p p p

0

1 1 1 1, ,p pk i i j j i i kp

1 1 1 1

11

,

0p p

p p kj i i j i i kd t pt t x t x t x

1 1 ,pj i i k

1 1

1

0p

pj i i

dd t t t x

d t

1 1pj i i x

i.e. d

4.20. Lie Derivatives of Forms

Vd V d V L

Proof: Reminder:

1. is a derivation on Ωp :

2. d is an anti-derivation:

V V V L L L

d d d

3.

,i i j

jVdx V dxL

Let 1

1

1

!p

p

iii i dx dx

p

Then 1

1

1 11

1

,

,1

1

!

1

!

p

p

pk k k

p

iiji i jV

pii i ii j

i i jk

V dx dxp

V dx dx dx dx dxp

L

V Vd i i d see Choquet,

pp. 206, 552

,j

i i jVV L

,i

iVf f VL

1 11 2

1 2 1 1 ,pk k k

k k k p

ii i ii i ji i i i i i jV dx dx dx dx dx dx

1 11 2

1 2 1 1 ,pk k k

k k p k

ii i ii iji i i j i i iV dx dx dx dx dx dx

1

2 1,p

p

iijj i i iV dx dx

1 12 1

2 1 1 1,pk k k

k k k p

ii i ii iji i i j i i iV dx dx dx dx dx dx

( all j ik )

( all i1 ik )

1 12 1

2 1 1 1,pk k k

k k k p

ii i ii ijj i i i i i iV dx dx dx dx dx dx

( i1 ik in dx )

( j ik in ω )

1 11 2

2 1 1 1,pk k k

k k k p

ii i ii ijj i i i i i iV dx dx dx dx dx dx

1

1 2 1, ,

1

!p

p p

iij ji i j j i i iV

V p V dx dxp

L →

1

1

1 11

1

,

,1

1

!

1

!

p

p

pk k k

p

iiji i jV

pii i ii j

i i jk

V dx dxp

V dx dx dx dx dxp

L

1

1 2 1, ,

1

!p

p p

iij ji i j j i i iV

V p V dx dxp

L

2

2

1

1 !p

p

iijji iV

i V V dx dxp

1 2

2 1 2 1, ,

1

1 !p

p p

ii ij jji i i ji i iV

d i d V V V dx dx dxp

1

1 ,

1

!p

p

iiji i jd dx dx dx

p

1

1 ,

11

!p

p

iij

V i i ji d d V p V dx dx

p

1

1 ,

11

1 !p

p

iiji i jp dx dx dx

p

1 2

2 1 2 1, ,

1

!p

p p

ii ij jji i i ji i ip V p V dx dx dx

p

1 2

12 1 2,,

1

!p

p p

ii ij jij i i i j i i

p V p V dx dx dxp

1 1 2 1, , ,

1 ! !p p pi i j i i j j i i i

p p p

→ Vd V d V L

2nd term represents p terms resulted from interchanging j with ik .

For a 1-form ω :

,i i j

jVdx V dxL

,i

iVf f VL

Alternative Derivation

d f V idV

i i ii i iV V V Vdx dx dx L L L L

i ii id V dx dV

i i i ii i i id dx V dx V d d V V d

d V d V

4.21. Lie Derivatives and Exterior Derivatives Commute

Vd V d V L

Vd d d V dd V L → d d V

Vd d V L V

d dd V L

Vd L ω

V Vd dL L

In general, d commutes with all derivations.

4.22. Stokes' Theorem

U Ud

Called by Arnold the Newton-Leibniz-Gauss-Ostrogradski-Green-Stokes-Poincare theorem.

Ampere & Kelvin also contributed.

One of the most important theorems of all Mathematics:

For proof, see Frankel, §3.3.

For an introduction to homology, see Aldrovandi, Chap 3.

Consider an n-form ω over an open region U of an n-D manifold M.

The boundary U of U is an (n-1)-D submanifold that divides M into 2 disjoint sets U & CU.

→ Any smooth curve joining U & CU must contain a point in U.

Example:

1 and 2 are not boundaries,

but 1 2 and 3 are.

1 2 is disconnected.

Let U be smooth, orientable, and, for convenience, connected.

An Indirect Proof

Let ξ be any vector field on M and Lie drag U along ξby ε.

0U U U

Let V be a patch of U covered by coordinates { x2, … , xn }.

By Lie dragging V along ξ, one creates a region of coordinates { x1 =ε , x2, … , xn }.

(ξ not tangent to U )

1 1, , n nf x x dx dx Let

1 2

00

n

V Vf dx dx dx

2 2 2

00, , , n n

Vf x x dx dx O

2

V UO

1x

The formula is also applicable a region where ξ is tangent to U.

In which case, δV = 0 and ω(ξ) = 0.

Hence 00

1lim

U U U

d

d

U U

0

1lim

VV

V

V Uwhere

Let

*

U U : M N

then See Frankel §3.1.f

Lie dragging U along ξ by ε means 0U U

where σε is the local transformation along the integral curves of ξ by ε.

*

0U U

00

1lim

U U U

d

d

*

00

1lim

U

0U L

U

0Ud d

If U is a p-D submanifold, then restricting the p+1 form dω to U gives 0.

→

U U

dd

d

E.g. :C MR

b ii

i iC t Ma

d xdx t d t

d t

then

* * *: p pT N T M

U U U

d

ξ

U Ud

α = any (n1)-form→ Stokes theorem :

Example: M = 2

iidx ,

j ii jd dx dx ,

12

2!j i

i j dx dx

,2 j ii jd

, ,j i i j

Let l = d/dλ be a vector tangent to U.

Restricting α to U means confining the domain of α to l.

→ 2 11, 2 2 ,1U U

d dx dx

ib

ia

dxd

d

iiUdx

U

~S

d α S

~C

dα l

4.23. Gauss' Theorem and the Definition of Divergence

Let 1 ng dx dx

Then 2

2

1

1 !n

n

iijj i ii g dx dx

n

1

1

1

!n

n

iii ig dx dx

n

1 2

21,

1

1 !n

n

ii ijj i i

id i d g dx dx dx

n

1 2 1 2

1 2 21,

1 1

1 ! !n n

n n

k k k ii iji i i j k k

kg dx dx dx

n n

1 2 1 2

1 2 2 1 2 2

n n

n n n n

k k k k k ki i i j k k i i i j k k

1 1 2

1 21,

1

!n

n

ik i iji i i j

kd g dx dx dx

n

1 2

1 2 ,

1

!n

n

ii iji i i

jg dx dx dx

n

,

1 j

jg

g

volume form

1 2

1 2 2

n

n n

k k ki i i j k k 1

1 21 !

n

ki i i j n

*

*d

å

d å

,

1 j

jd i d g

g

div

(divergence of vector fieldξ)

Stokes theorem → U U

d vd i

Let n be a 1-form normal to U , i.e.

n Let

0n T U

then U U

n

U UU

div n

U

→

or ( Green’s theorem )U U

dV n dS

Ex. 4.20-4

*d d å

,

1 j

jdiv g

g

4.24. A Glance at Cohomology Theory

Let Z p (M) be the set of all closed p-forms on M, i.e.,

0p pZ M M d

Let B p (M) be the set of all exact p-forms on M, i.e.,

p pB M M d

2 closed forms are equivalent if they differ by an exact form, i.e.,

1 2 1 2 d

Let H p (M) be the set of all equivalent classes of closed p-forms on M, i.e.,

/p p pH M Z M B M ( Quotient space )

= pth de Rham cohomology vector space of M

dim H p (M) = b p = pth Betti number of M.

Local exactness: In any U diffeomorphic to an open n-ball in n,

0d d

→ All closed forms are equivalent (& hence to the zero form) in U.

i.e., 0 1pH U p

If M is connected, then

0 0Z M f d f = space of constant functions =

0 0B M Since there is no (1)-form, where 0 is the zero function.

→ f g f g

0 0H M Z M R

In general, 0 mH M R for M with m connected components.

( Poincare lemma )

Some interesting results concerning S n :

[ Proof: see M.Spivak, “A Comprehensive Introduction to Differential Geometry”,Vol 1, PoP (70) ]

n nH S R

0 0p nH S p n

0 nH S R

Comment:

Homology & cohomology are theories on topological invariants.

The study of cohomology in terms of differential forms was initiated by de Rham, hence the name de Rham cohomology.

c.f. Ex 4.24

Ex. 4.25-7

4.25. Differential Forms and Differential Equations

,dy

f x ydx

0a f f

→

so that

,

,x yP x yV a f V Sx Ty

0dy f dx

A solution y = g(x) defines a curve ( 1-D solution manifold S ) on state space M = 2.

i.e., a

In general, a particular solution to a set of DEs is a submanifold S whose tangent vectors at each point annul the set of differential forms representing the DEs.

V dy f dx V ( V annuls ω )

Locally, the space Vp of annulling vectors at p is a vector subspace of TP (M).

Globally, whether these VP s can be meshed to give a tangent bundle T(S) is determined by the Frobenius theorem (existence of solutions).

Example: Harmonic Oscillator2

22

0d x

xdt

Equivalent 1st order eqs:dx

ydt 2dy

xdt

Equivalent set of forms: dx yd t 2dy xd t

State space M is 3-D with (global) coordinates ( x, y, t ).

Solution space S is 1-D, i.e., a spiral whose projection on the x-y plane is an ellipse.

Ex. 4.32

4.26. Frobenius' Theorem (Differential Forms Version)

Let B = {βi } be a set of forms of arbitrary degrees.

• The annihilator XP of B at pM is the set of all vectors that annuls every βi in B.

• XP is a subspace of TP .

• The complete ideal (Bp) of B at p is the set of all forms at p whose restriction to XP vanishes.

• Reminder: restriction of a form to V means confining its domain to V.

See Choquet, §§ IV.3-4

A left ideal of a ring X is a subring of X s.t. xi Xi x I I

A right ideal of a ring X is a subring of X s.t. xx Xi i I I

An ideal is a left ideal that is also a right ideal.

deg deg i

i i pB I

Let XP be spanned by basis {e1 , …, em },

which is augmented by {em+1 , …, en } to span TP.

Let { ωm+1 , … , ωn } be the dual basis to {em+1 , …, en }, i.e.,

0 1, ,for

1, ,i

ijj

j me

j m n

and i = m+1, … , n.

Hence, the annihilator of { ωm+1 , … , ωn } is also Xp .

If β (Bp), then

Any q-form βon M can be written as 1

1

1

!q

q

i ii iq

10

qi i

1, ,ji n

unless ij { m+1, … , n }

for some {γi }.Hence, β (Bp) → 1

ni i

i m

Ex 4.30: Let { αj , j = 1, … , m } be a set of independent 1-forms.

Then any form γ is in their complete ideal iff 1 0m

Let B = {βi } be a set of fields of forms of arbitrary degrees.

• The complete ideal (B) of B is the set of all fields of forms whose restriction to the annihilator XP vanishes pM .

• An ideal is a differential ideal if γ → dγ .

• A differential system { αj = 0 } is closed if dαj ( { αj } ).

i.e. dαj α1 … αm = 0 j = 1, …, m

( Ex. 4.30 )

Ex.4.31:

a) The 1-forms of a closed system generate a differential ideal.

b) On an n-D manifold, any lin. indep. set of n, or n1, 1-form system is closed.

Frobenius theorem (on differential forms):

Let {αi ; i = 1, … , m } be a set of lin. indep. 1-form fields in U M (dimM = n).

Then 0 is closed where , are functions & , 1, ,ji i i jP dQ P Q i j m

Proof: see §4.27

Interpretation: Consider the set of differential system 0i

Its solution is the submanifold S whose tangent space at each point is the annihilator of {αi }.

Let be the complete ideal generated by {αi }.

If a solution exists, then dαi = 0, is a differential ideal and {αi = 0 } is closed.

Conversely, {αi } is not closed → no solution exists.

Frobenius’ theorem: αi = 0 → dQj = 0 ( S is then defined by Qj = const ) .

The {αi } are then said to be surface forming.

Requirement of closedness is dual to that of the annihilator being a Lie algebra.

Frobenius theorem → f exists iff a is closed, i.e. da = 0.

Example: PDE discussed in §4.17x

y

f g

f h

i if a or d f a or

Another example: Ex. 4.32

Example: Harmonic Oscillator

0dx yd t 2 0dy xd t y x

By Ex 4.31, a set of two 1-forms in a 3-D manifold is closed.

Frobenius theorem : a,b,c,e,f,g s.t. a df b dg c df e dg

Solution submanifold is 1-D on which f and g are constant.

DEs described by higher forms may not be equivalent to those described by the 1-forms that generate the same ideal (see Choquet)

Alternative Formulation

See Aldrovandi, §7.3.14.

; 1, ,i i m A set of independent 1-forms, is cotangent to a submanifold

iff 1 0i md 1, ,i m

i.e., { θi = 0 } is closed.

i.e., where , are functions & , 1, ,ji i jP dQ P Q i j m

E.g., S2 can be specified by d r = 0.

4.27. Proof of the Equivalence of the Two Versions of Frobenius' Theorem

Vector field version: A set of vector fields { V(i) , i = 1, … , q }, which at every point form a p-D vector space, will mesh to form a p-D submanifold iff they are closed under the Lie bracket.

1-form field version: A set of 1-form fields { α(i) , i = 1, … , p } can be written as

iff they are closed. j

i ji P dQ

whereupon, { Qj = const } defines a (np)-D submanifold on which α(i) = 0.

Reminder:

An r-D subspace XP of the n-D TP (M) defines an (nr)-D subspace of T*P (M)

that is annulled by XP .

An q-D subspace ΩP of the n-D T*P (M) defines an (nq)-D subspace of TP (M)

that annulls ΩP .

→ An r-D submanifold of M can be defined by either an r-D XP

or an (nr)-D ΩP annulled by XP .

Let S be an p-D submanifold of an n-D manifold M.

→ S is defined by np equations { Qk = const } so that np coordinates of a point on S can be set to 0.

, 0k PdQ V V T S → { dQk } are lin. indep. and

TP(S) defines an (np)-D subspace ΩP of T*P(M) that is annulled by TP(S), i.e.,

, 0 &P PV V T S

k

i ki P dQ

Obviously, { dQk } is a basis of ΩP .

Any other basis { α(i) } of ΩP can be written as

Let { V(i) , i = 1, … , p } be a basis for TP(S).

→ , 0 1, , & 1,i jV i n p j p

0 ,k i jV

VL , ,k ki j i jV V

V V L L

, , 0k ki j i jV V

V V L L

, , ,k i j i k i k jV

V d V d V V L ,i k jd V V

,i k jd V V

, , ,ki j i k jVV V V L

If V(j) are closed under Lie bracket

so that , 0i k jd V V

i.e., dα(i) are in the complete ideal → { α(i) = 0 } is closed.

The converse is proved by retracing the argument . QED.

→ , , , 0i k j i k jd V V V V

, , , 0i k j i mmm

V cV V

4.28. Conservation Laws

Let the solution of a set of equations be equivalent to finding surfaces that annul a set of forms { αi }.

Let i ii

A s.t.

0d

Then, for some suitable region U of a solution surface H, σ s.t.

d and 0H H

d

Stokes theorem →U U

d

0U H

(conservation law)

Example: Harmonic Oscillator, M = { (x,y,t) }

dx yd t 2dy x d t

2x y

H is 1-D curve → γ = dσ are 1-forms on H → σ is a 0-form (function)

Eliminating dt, we have

On H,

0d

y x

2 2x dx y d t y dy x d t 2x dx y dy

2 2 21 1

2 2d x y

0 → 0 2 2 21 1

2 2d x y

2 2 21 1

2 2x y const = energy since

0U U U

d

d

2

1

2 2 21 1

2 2

p

p

x y

4.29. Vector Spherical Harmonics

Finite dimensional representations of SO(3) on L2(S2) :

basis = { Ylm ; m = l, … , l } = Spherical harmonics

Representations of SO(3) on L21,0

(S2) :

basis = Vector spherical harmonics

21 , 0 , sing g g g Let the metric tensor on S2 be

Then L21,0

(S2) is the set of all vector fields V on S2 s.t.

2

2, exists

SV V V g ω = volume 2-form

The vector spherical harmonics are eigenvectors of lz and L2.

Given Ylm , one gets the 1-form dYlm and its dual vector

A ABl m l m B

Y g dY ,AB

lm Bg Y

2

11 , 0 ,

sing g g g

where A, B = {θ ,φ } and

z z

AB

l lmBl lmY g d Y L L AB

lm Bim g dY

lmim Y

22 ABlm lm B

Y g d LL Y 1 lml l Y

Since g & d commutes with 2&

zlLL

→ lm lmY Y is a basis vector of T (S2) for given l, m.

The other basis vector can be chosen as lm lmY dY å

Other tensor spherical harmonics can be similarly constructed once covariant derivatives on S2 are defined.

Let S2 be a submanifold of M, say, E3.

Then l m l ml m

f f r Yx

TV V V

T l m l m l m l ml m

V u r Y u r Y x

with r l m l ml m

V e v r Y x