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Transcript of Differential Forms, Integration on Manifolds, and Stokes ...paupert/BrownStokes.pdf · Di erential...
Differential Forms, Integration on Manifolds, andStokes’ Theorem
Matthew D. Brown
School of Mathematical and Statistical SciencesArizona State UniversityTempe, Arizona 85287
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