4. Differential forms A. The Algebra And Integral Calculus Of Forms 4.1 Definition Of Volume – The Geometrical Role Of Differential Forms 4.2 Notation.

4. Differential forms

A. The Algebra And Integral Calculus Of Forms

4.1 Definition Of Volume – The Geometrical Role Of Differential Forms

4.2 Notation And Definitions For Antisymmetric Tensors4.3 Differential Forms4.4 Manipulating Differential Forms4.5 Restriction Of Forms4.6 Fields Of Forms4.7 Handedness And Orientability 4.8 Volumes And Integration On Oriented Manifolds 4.9 N-vectors, Duals, And The Symbol Ij…k

4.10 Tensor Densities 4.11 Generalized Kronecker Deltas 4.12 Determinants And Ij…k

4.13 Metric Volume Elements.

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 Harmonics4.30 Bibliography

Concepts that are unified and simplified by forms • Integration on manifolds• Cross-product, divergence & curl of 3-D euclidean geometry• Determinants of matrices• Orientability of manifolds• Integrability conditions for systems of pdes• Stokes' theorem• Gauss' theorem• …

E. Cartan

4.1. Definition Of Volume – The Geometrical Role Of Differential Forms

2 vectors define an area (no metric required).

Different pairs of vectors can have same area.

3. , , ,area a b area a c area a b c

:area V V R , ,a b area a b

area( , ) is a (02) skew-tensor

such that

2. , 0area a a 1. , ,area a b area b a →

Ex. 4.1

,x y

x y

V Varea V W

W WFor vectors in the x-y plane:

4.2. Notation And Definitions For Antisymmetric Tensors

A (0p) tensor is completely antisymmetric if

, , , , , ,U V V U ,U V

Totally antisymmetric part of a (0p) tensor:

1, , ,

2!A U V U V V U

1, , , , , , , ,

3!A U V W U V W V W U W U V

, , , , , ,V U W W V U U W V

Index-notation:

1

2!A i j j ii j

i j

1

3!A i jk j k i k i j j i k k ji j k ii j k i j k

A skew (0p) tensor on an n-D space has at most

!

! !np

nC

p n p

independent components

4.3 Differential Forms

p-form = completely antisymmetric (0p) tensor ( p = degree of form).

0-form = scalar function. 1-form = covariant vector.

Wedge (exterior) product :

p q p q q p

Let , ,p q r be 1-forms. Then

0p p →

be the vector basis & 1-form basis, resp.

ieLet & ie

Then is a basis for 2-forms.

; 1, ,i je e i j n

1

ni j

i ji j

e e

1

2!i j

i j e e

,i j i je e

j i

(antisymmetry)

1,

2!k l l k

k l i je e e e e e 1

2!k l l k

k l i j i je e e e e e e e

1

2!k l l k

k l i j i j 1

2! i j j i i j

2dim nC

= (vector) space of all p-forms at x M px M dim n

pC

2x M

p q r p q r p q r

Grassmann algebra = { all p-forms , +, }

Ex. 4.8:

Show that

(associativity)

Dim = 0

1 1 2n

nn np

p

C

[ ]p q

p i j k li jk lp q C p q

i j k j k i k i ji j kp q p q p q p q [ ]3 i j kp q if

1 2&p q

4.4 Manipulating Differential Forms

Attention: signs

Let ,p q be p- & q-forms, resp. Then pqp q q p

Proof: Let

1 pp

be 1-forms such thatj

1p p qq

Then1 1p p p qp q

1 1 2p p p p p q

2 1 2 1 3p p p p p p q

1 1q p p p q p

pqq p

Proof using basis:

1

1

1

!p

p

iii ip p e e

p 1

1

1

!q

q

iii iq q e e

q

11

1 1

1

! !p p p q

p p p q

i i iii i i ip q p q e e e e

p q

1 21

1 1

1

! !p p p p q

p p p q

p i i i iii i i ip q e e e e e

p q

1 2 31

1 1

2 1

! !p p p p p q

p p p q

p i i i i iii i i ip q e e e e e e

p q

1 1

1 1

1

! !p p q p

p p p q

pq i i iii i i ip q e e e e

p q

pqq p

Contraction:

1

,p empty slots

Let be a vector & a p-form.

i.e., ii j kj k

Define

Example: p q where ,p q are 1-forms

p q p q q p p q q p p q q p

1 1 2 2 1p p pi i ii i i i ie e e e e e e e

1 2][! pii ip e e

1 2 2 1p pi ii i i ie e e e e e

1 2 2 1p pi ii i i ie e e e [ p! terms]

[ ]! i j kp e e

1

1

1

!p

p

iii i e e

p 1 2

1 2

]p

p

ii ii i i e e →

1 1 2][!p pi ii i ie e p e e

1 1p pi i i i

1 2

1 2

p

p

ii i

i i ie e

1 2

1 2

1

1 !p

p

ii ii i i e e

p

pp p p In general

1

1 !i j k

i j k e ep

= (p–1)-form with components 1

1 22

pp

ii i ii i

4.5. Restriction of Forms

A p-form is a (0p) tensor → its domain is

, , , ,W

X Y X Y

Wof

, ,X Y W

dimW p

p

p factors

V V V

The restriction (section) to a subspace W of V is

0W

→

dimW p → W is 1-D

(annulled by W)

4.6. Fields of Forms

A field Ωp(M) of p-forms on a manifold M

= a rule that gives a p-form at each point of M.

Ditto vector field.

A submanifold S of M picks a subspace VP of TP PS.

→ Restriction of p-form field to S

= restriction of p-form at P to VP PS.

4.7. Handedness and Orientability

1n nnDim M C Dim M n →

Let nP M

If ie is a basis for TP(M), then 1, , 0ne e iff 0 at P.

1

1

, , 0

, , 0n

in

e eright handede is if

e eleft handed

Relative handedness is independent of choice of

M is orientable if it is possible to define handedness continuously over it,

i.e., a continuous basis with the same handedness everywhere on M.

i.e., 0P P M E.g. En is orientable.

The Mobius band is not.

Absolute handedness is fixed by the choice of the coordinate chart.

4.8. Volumes and Integration on Oriented Manifolds

Integration of a Function

11

, , nn

x xx x

1 1 11

, ,n n nn

dV x x dx dx x xx x

( parallelepiped / cell )

(volume of cell)

:f M R (function)

1 1, , n n n

cell

f x x d x f x x

Integration of f over cell :

1 11

, ,n nn

f dx dx x xx x

11

, , nn

x xx x

cell

1 nf dx dx ( n-form )

Integration of f over U M :

n

cellU

f d x cell U

1 n

U

f dx dx

Change of Variables

is independent of coordinates up to an overall sign.

,f d d ,f d d

d dx dyx y

, ,x y

dx dyx y y x

E.g., M is 2-D :

Changing coordinates

d dx dyx y

d d dx dy dx dyx y x y

→

,

,

x yJ

x y

x y

,

,dx dy

x y

= Jacobian

J dx dy J dy dx

d d J dxdy

Riemannian integration:

11, , n

nx e x e 11

, , nn

cell x xx x

Orientability

1 nf dx dx

1 , , ne e

nf d x cell

cell

Let be another basis which differs from

1 11 , ,n n

ncell f dx dx x e x e

11 , , n

ncell x e x e

1, , ne e only in handedness.

'cell cell

cell cell

cell

Let the entire region of integration be orientable, then

By convention, a right-handed basis is always assumed in

nf d x cell

cell

Integration on Submanifold

1, , n pn n

1, , n pn n

is defined only for n-form on an n-D manifold M,

or p-form over a p-D submanifold S.

Relation between the orientabilities of M and S ?

( Domain must be internally orientable )

Let M be orientable and a right-handed n-form at PS.

the p-form is a right- handed restriction of to S

not tangent to S at P,

Given n–p independent normal vectors

1, , n pn n determines an external orientation for S at P.

S is externally orientable if it is possible to define an external orientation continuously over it.

If U M is orientable, then S U is either both internally and externally orientable, or it is neither.

Otherwise, S may be one but not both.

Mobius strip embedded in R3.

M is not externally orientable in R3.

A curve is always internally orientable

→ it can't be externally orientable inside a nonorientable submanifold

C1 is not orientable in M

But C2 is both internally & externally orientable in M

4.9. N-vectors, Duals, and the Symbol

Dual Maps

n-form

p vector n p forms

g g

p forms n p vectors

* T T T

* T T

1 1

1 1

1

!q n

n q

j j iii i j jT e e e e

n T

g = metric tensor

: dual map

Dual of a q-vector T

*: *q n q n q V V

1

1

q

q

i i

i iT e e T

1

1

1

!n

n

iii i e e

n

1

1

1

!q

q

i i

i iT e eq

1

1

q

q

i i

i iT e e

1 1

* *n q n qi i i i

T T

1 1

1 1

1

!q q n

q q n

j j i inq j j i iC T e e

n

T

1 1

1 1

1

!q n

n q

j j iii i j jT e e e e

n T

1 1 11 1

1 1 1 1

q q q qn n

n q n q

j j j j i ii ii ii i j j i i j jT e e e e T e e

1 1

1 1

q q n

q q n

j j i ij j i i T e e

11

1 1

1

! !q n q

q n q

ij j ij j i i T e e

q n q

*T is an (n-q)-form with components

1 n qi i

T

1

1 1

1

!q

q n q

j jj j i i T

q

,U V

123 1

U V

i j j iU U U U

i ji jU V e e

2 3 32 1dim * 3 dimC C V V

Example: Cross Products in E3

Then

be vectors &Let ,U V the associated 1-forms.

12!

2!i j

i jU V e e

2!i j i jU V

2 3 3 1 1 22 3 3 2 3 1 1 3 1 2 2 1U U U U e e U U U U e e U U U U e e

W U V i j kj k iU V e * i j k

j k iW U V e Let →

Setting gives *W U V * U V

The cross product exists only in E3 , where

cell cell

1 1

1 1

1

! !q n q

q n q

j j i i

j j i iT S e e e eq n q

ST

cell cell cell

1 1

1 1

1* ,

!q n q

q n q

j j i ij j i i T S

q

T S

1 1

1

1

! !q q n

n

j j j jj jT S e e

q n q

1 1

1

1

!q q n

n

j j j jnq j jC T S e e

n

= n-vector with components 1 1 1n q q nj j j j j jnqC T S T S

11! n qqn q T S if

12 1n

cell cell cell cell cell cell

cell cell cell cell cell cell