Differential Forms, Integration on Manifolds, and Stokes Theorem

Transcription

1 Differential Forms, Integration on Manifolds, and Stokes Theorem Matthew D. Brown School of Mathematical and Statistical Sciences Arizona State University Tempe, Arizona March 30, 2012

2 Introduction: Theorems of Classical Calculus The Fundamental Theorem of Calculus b a f (x) dx = f (b) f (a) The Fundamental Theorem for Line Integrals f dr = f (r(b)) f (r(a)) C

3 Introduction: Theorems of Classical Calculus Green s Theorem ( Q x P ) da = y A A (P dx + Q dy) Stokes Theorem ( F) ds = S S F dr

4 Introduction: Theorems of Classical Calculus The Divergence Theorem (Gauss Theorem) ( F) dv = F ds V V

5 Vectors 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 called the tangent space to M at p, denoted by T p M.

6 Covectors Let p M. The cotangent space T p M of M at p, is defined to be the dual of the tangent space at p, T p M = (T p M) An element ω T p M, called a covector, is a linear map ω : T p M R.

7 The Differential of a Function Let f : M R be smooth. We define a covector field df, called the differential of f, by for all X p T p M. In coordinates, we can write df p (X p ) = X p f df = f x i dx i

8 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 + bdg. d(fg) = f dg + g df. d(f /g) = (g df f dg)/g 2 on the set where g 0. If J R is an interval containing f (M) and h : J R is smooth, then d(h f ) = (h f )df. If f is constant, then df = 0.

9 Tensors Let V be vector space. A covariant k-tensor on V is a real-valued multilinear function of k elements of V : Examples: T : V } {{ V } R k copies The metric g of a Riemannian manifold is a covariant 2-tensor. In classical electrodynamics, the electromagnetic field tensor F is given (in coordinates) by 0 E x E y E z F µν = E x 0 B z B y E y B z 0 B x E z B y B x 0

10 Tensor Product Let V be a finite-dimensional real vector space and let S T k (V ), T T l (V ). Define a map by S T : V } {{ V } R k+l copies S T (X 1,..., X k+l ) = S(X 1,..., X k )T (X k+1,..., X k+l )

11 Some Technical Stuff A covariant k-tensor T on a finite-dimensional vector space V is said to be alternating if T (X 1,..., X i,..., X j,..., X k ) = T (X 1,..., X j,..., X i,..., X k )

12 Alternating Tensors Given a covariant k-tensor T, we define the alternating projection of T to be the covariant k-tensor AltT = 1 (sgnσ)( σ T ) k! σ S k

13 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 (T (X, Y ) T (Y, X )) 2 3 If T is a 3-tensor, then (AltT ) ijk = 1 6 (T ijk + T jki + T kij T jik T ikj T kji )

14 Differential Forms A differential k-form is a continuous tensor field whose value at each point is an alternating tensor.

15 The Wedge Product We want a way to produce new differential forms from old ones: Given a k-form ω and an l-form η, we define the wedge product or exterior product of ω and η to be the (k + l)-form ω η = (k + l)! Alt(ω η) k!l! Some Properties of the Wedge Product Bilinearity Associativity Anticommutativity: ω η = ( 1) kl η ω

16 The Exterior Derivative Theorem For every smooth manifold M, there are unique linear maps d : A k (M) A k+1 (M) defined for each integer k 0 and satisfying the following three conditions: If f is a smooth real-valued function (a 0-form), then df is the differential of f. If ω A k and η A l, then d(ω η) = dω η + ( 1) k ω dη d 2 = d d = 0.

17 The Exterior Derivative In coordinates, d ( ) ω J dx J = J J dω J dx J, where dω J is just the differential of the function ω J.

18 Orientations of Vector Spaces Any two bases (E 1,..., E n ) and (Ẽ 1,..., Ẽ n ) of a finite-dimensional vector space V are related by a transition matrix B = (B j i ), E i = B j i Ẽ j We say that (E 1,..., E n ) and (Ẽ 1,..., Ẽ n ) are consistently ordered if det(b) > 0. Consistently ordered defines an equivalence relation on the set of all (ordered) bases of V. There are exactly two equivalence classes, which we refer to as orientations of V. A vector space along with a choice of orientation is called an oriented vector space. The standard orientation of R n is [e 1,..., e n ].

19 Orientations of Arbitrary Manifolds A pointwise orientation of a manifold M is just a choice of orientation of each tangent space. An orientation of M is a continuous pointwise orientation. M is said to be orientable if there exists an orientation for it.

20 Manifolds with Boundary Informally, an n-dimensional manifold with boundary is a space which is like R n except at certain boundary points. Formally, An n-dimensional topological manifold with boundary is a second-countable Hausdorff space M in which every point has a neighborhood homeomorphic to an open subset U of H n, where H n is the upper half-space, Examples: H n = {( x 1,..., x n) R n x n 0 }. The unit interval, [0, 1]. Closed balls B r (x) in R n.

21 Integration of Forms in R n First, a little terminology: The support of an n-form is ω is the closure of the set {p M ω(p) 0}. ω is said to be compactly supported if its support is a compact set. Let ω = fdx 1 dx n be an n-form on R n and D R n be compact. We define ω = f dx 1 dx n D D

22 Integration on Arbitrary Manifolds Let M be a smooth, oriented n-manifold, and let ω be an n-form on M. If ω is compactly supported in the domain of a single oriented coordinate chart (U, φ), we define the integral of ω over M to be ω = (φ 1 ) ω M φ(u) 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.

23 Stokes Theorem Theorem Let M be a smooth, oriented n-dimensional manifold with boundary, and let ω be a compactly supported smooth (n 1)-form on M. Then dω = ω M M

24 The Fundamental Theorem for Line Integrals Theorem Let C : [a, b] R n be a smooth curve in R n such that M = C([a, b]) is a 1-dimensional submanifold with boundary of R n. Let f : R n R be smooth. Then f dr = f (r(b)) f (r(a)) C