By Loring W. Tu

ISBN-10: 1441973990

ISBN-13: 9781441973993

ISBN-10: 1441974008

ISBN-13: 9781441974006

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Sample text

Div   P curl Q R 43 = 0. Proposition C. On R3 , a vector field F is the gradient of some scalar function f if and only if curl F = 0. Propositions A and B express the property d 2 = 0 of the exterior derivative on open subsets of R3 ; these are easy computations. Proposition C expresses the fact that a 1-form on R3 is exact if and only if it is closed. Proposition C need not be true on a region other than R3 , as the following well-known example from calculus shows. Example. If U = R3 − {z-axis}, and F is the vector field −y x , ,0 x2 + y2 x2 + y2 F= on R3 , then curlF = 0, but F is not the gradient of any C∞ function on U.

Times in total. 7) can be rewritten as A(A( f ) ⊗ g) = k! A( f ⊗ g). The equality in (ii) is proved in the same way. 25 (Associativity of the wedge product). Let V be a real vector space and f , g, h alternating multilinear functions on V of degrees k, ℓ, m, respectively. Then ( f ∧ g) ∧ h = f ∧ (g ∧ h). Proof. m! m! ℓ! (k + ℓ)! ℓ! 1 A(( f ⊗ g) ⊗ h). m! (ℓ + m)! m! 1 A( f ⊗ (g ⊗ h)). m! f ∧ (g ∧ h) = Since the tensor product is associative, we conclude that ( f ∧ g) ∧ h = f ∧ (g ∧ h). ⊔ ⊓ By associativity, we can omit the parentheses in a multiple wedge product such as ( f ∧ g) ∧ h and write simply f ∧ g ∧ h.

Then d(ω ∧ τ ) = d( f g dxI ∧ dxJ ) ∂ ( f g) i dx ∧ dxI ∧ dxJ ∂ xi ∂f ∂g = ∑ i g dxi ∧ dxI ∧ dxJ + ∑ f i dxi ∧ dxI ∧ dxJ . ∂x ∂x =∑ In the second sum, moving the 1-form (∂ g/∂ xi ) dxi across the k-form dxI results in the sign (−1)k by anticommutativity. Hence, ∂f i ∂g dx ∧ dxI ∧ g dxJ + (−1)k ∑ f dxI ∧ i dxi ∧ dxJ i ∂x ∂x = d ω ∧ τ + (−1)k ω ∧ d τ . d(ω ∧ τ ) = ∑ (ii) Again by the R-linearity of d, it suffices to show that d 2 ω = 0 for ω = f dxI . We compute: 40 §4 Differential Forms on Rn d 2 ( f dxI ) = d ∂f ∑ ∂ xi dxi ∧ dxI =∑ ∂2 f dx j ∧ dxi ∧ dxI .