Differential Forms, Integration on Manifolds, andStokes’ Theorem

Matthew D. Brown

School of Mathematical and Statistical SciencesArizona State UniversityTempe, Arizona 85287

matthewdbrown@asu.edu

March 30, 2012

Introduction: Theorems of Classical Calculus

The Fundamental Theorem of Calculus∫ b

af (x) dx = f (b)− f (a)

The Fundamental Theorem for Line Integrals∫C∇f · dr = f (r(b))− f (r(a))

Introduction: Theorems of Classical Calculus

Green’s Theorem∫∫A

(∂Q

∂x− ∂P

∂y

)dA =

∮∂A

(P dx + Q dy)

Stokes’ Theorem ∫∫S

(∇× F) · dS =

∮∂S

F · dr

Introduction: Theorems of Classical Calculus

The Divergence Theorem (Gauss’ Theorem)∫∫∫V

(∇ · F) dV =

∫∫∂V

F · dS

Vectors

Definition

A linear map X : C∞(M)→ R is called a derivation at p ∈ M if,for all f , g ∈ C∞(M),

X (fg) = f (p)X (g) + g(p)X (f )

The set of all derivations of C∞(M) at p is a vector space calledthe tangent space to M at p, denoted by TpM.

Covectors

Definition

Let p ∈ M. The cotangent space T ∗p M of M at p, is defined to bethe dual of the tangent space at p,

T ∗p M = (TpM)∗

An element ω ∈ T ∗p M, called a covector, is a linear mapω : TpM → R.

The Differential of a Function

Definition

Let f : M → R be smooth. We define a covector field df , calledthe differential of f , by

dfp(Xp) = Xpf

for all Xp ∈ TpM.

In coordinates, we can write

df =∂f

∂x idx i

Properties of the Differential

The differential satisfies many the properties we would expect it to:

Lemma

Let f , g : M → R be smooth. Then:

For any constants a, b, d(af + bg) = adf + b dg.

d(fg) = f dg + g df .

d(f /g) = (g df − f dg)/g2 on the set where g 6= 0.

If J ⊆ R is an interval containing f (M) and h : J → R issmooth, then d(h ◦ f ) = (h′ ◦ f )df .

If f is constant, then df = 0.

Tensors

Definition

Let V be vector space. A covariant k-tensor on V is a real-valuedmultilinear function of k elements of V :

T : V × · · · × V︸ ︷︷ ︸k copies

→ R

Examples:

The metric g of a Riemannian manifold is a covariant2-tensor.

In classical electrodynamics, the electromagnetic field tensorF is given (in coordinates) by

Fµν =

0 Ex Ey Ez

−Ex 0 −Bz By

−Ey Bz 0 −Bx

−Ez −By Bx 0

Tensor Product

Definition

Let V be a finite-dimensional real vector space and letS ∈ T k(V ),T ∈ T l(V ). Define a map

S ⊗ T : V × · · · × V︸ ︷︷ ︸k+l copies

→ R

by

S ⊗ T (X1, ...,Xk+l) = S(X1, ...,Xk)T (Xk+1, ...,Xk+l)

Some Technical Stuff

Definition

A covariant k-tensor T on a finite-dimensional vector space V issaid to be alternating if

T (X1, ...,Xi , ...,Xj , ...,Xk) = −T (X1, ...,Xj , ...,Xi , ...,Xk)

Alternating Tensors

Definition

Given a covariant k-tensor T , we define the alternating projectionof T to be the covariant k-tensor

AltT =1

k!

∑σ∈Sk

(sgnσ)(σT )

Example - Alternating Projection

1 If T is a 1-tensor, then

AltT = T

2 If T is a 2-tensor, then

AltT (X ,Y ) =1

2(T (X ,Y )− T (Y ,X ))

3 If T is a 3-tensor, then

(AltT )ijk =1

6(Tijk + Tjki + Tkij − Tjik − Tikj − Tkji )

Differential Forms

Definition

A differential k-form is a continuous tensor field whose value ateach point is an alternating tensor.

The Wedge Product

We want a way to produce new differential forms from old ones:

Definition

Given a k-form ω and an l-form η, we define the wedge product orexterior product of ω and η to be the (k + l)-form

ω ∧ η =(k + l)!

k!l!Alt(ω ⊗ η)

Some Properties of the Wedge Product

Bilinearity

Associativity

Anticommutativity:

ω ∧ η = (−1)klη ∧ ω

The Exterior Derivative

Theorem

For every smooth manifold M, there are unique linear mapsd : Ak(M)→ Ak+1(M) defined for each integer k ≥ 0 andsatisfying the following three conditions:

If f is a smooth real-valued function (a 0-form), then df isthe differential of f .

If ω ∈ Ak and η ∈ Al , then

d(ω ∧ η) = dω ∧ η + (−1)kω ∧ dη

d2 = d ◦ d = 0.

The Exterior Derivative

In coordinates,

d

(∑′

J

ωJdxJ

)=∑′

J

dωJ ∧ dxJ

, where dωJ is just the differential of the function ωJ .

Orientations of Vector Spaces

Definition

Any two bases (E1, ...,En) and (E1, ..., En) of a finite-dimensionalvector space V are related by a transition matrix B = (B j

i ),

Ei = B ji Ej

We say that (E1, ...,En) and (E1, ..., En) are consistently ordered ifdet(B) > 0.

”Consistently ordered” defines an equivalence relation on the set ofall (ordered) bases of V . There are exactly two equivalence classes,which we refer to as orientations of V . A vector space along witha choice of orientation is called an oriented vector space.The standard orientation of Rn is [e1, ..., en].

Orientations of Arbitrary Manifolds

Definition

A pointwise orientation of a manifold M is just a choice oforientation of each tangent space. An orientation of M is acontinuous pointwise orientation. M is said to be orientable ifthere exists an orientation for it.

Manifolds with Boundary

Informally, an n-dimensional manifold with boundary is a spacewhich is ”like” Rn except at certain boundary points. Formally,

Definition

An n-dimensional topological manifold with boundary is asecond-countable Hausdorff space M in which every point has aneighborhood homeomorphic to an open subset U of Hn, whereHn is the upper half-space,

Hn ={(

x1, ..., xn)∈ Rn | xn ≥ 0

}.

Examples:

The unit interval, [0, 1].

Closed balls Br (x) in Rn.

Integration of Forms in Rn

First, a little terminology:

Definition

The support of an n-form is ω is the closure of the set{p ∈ M |ω(p) 6= 0}. ω is said to be compactly supported if itssupport is a compact set.

Definition

Let ω = fdx1 · · · dxn be an n-form on Rn and D ⊆ Rn be compact.We define ∫

Dω =

∫D

f dx1 · · · dxn

Integration on Arbitrary Manifolds

Definition

Let M be a smooth, oriented n-manifold, and let ω be an n-form onM. If ω is compactly supported in the domain of a single orientedcoordinate chart (U, φ), we define the integral of ω over M to be∫

Mω =

∫φ(U)

(φ−1)∗ω

If we require more than a single chart to cover the support of ω,then, informally speaking,

∫M ω is the sum of the integrals over

each chart, minus overlap.

Stokes’ Theorem

Theorem

Let M be a smooth, oriented n-dimensional manifold withboundary, and let ω be a compactly supported smooth(n − 1)-form on M. Then∫

Mdω =

∫∂M

ω

The Fundamental Theorem for Line Integrals

Theorem

Let C : [a, b]→ Rn be a smooth curve in Rn such thatM = C ([a, b]) is a 1-dimensional submanifold with boundary ofRn. Let f : Rn → R be smooth. Then∫

C∇f · dr = f (r(b))− f (r(a))