By Loring W. Tu

ISBN-10: 1441973990

ISBN-13: 9781441973993

ISBN-10: 1441974008

ISBN-13: 9781441974006

Manifolds, the higher-dimensional analogues of soft curves and surfaces, are basic items in glossy arithmetic. Combining points of algebra, topology, and research, manifolds have additionally been utilized to classical mechanics, common relativity, and quantum box thought. during this streamlined advent to the topic, the idea of manifolds is gifted with the purpose of assisting the reader in attaining a fast mastery of the basic themes. via the top of the e-book the reader might be capable of compute, not less than for easy areas, probably the most easy topological invariants of a manifold, its de Rham cohomology. alongside the best way the reader acquires the data and abilities worthy for extra examine of geometry and topology. the second one version comprises fifty pages of recent fabric. Many passages were rewritten, proofs simplified, and new examples and workouts additional. This paintings can be utilized as a textbook for a one-semester graduate or complex undergraduate direction, in addition to by way of scholars engaged in self-study. The considered necessary point-set topology is integrated in an appendix of twenty-five pages; different appendices overview evidence from actual research and linear algebra. tricks and strategies are supplied to a number of the workouts and difficulties. Requiring simply minimum undergraduate must haves, "An creation to Manifolds" is usually an exceptional beginning for the author's e-book with Raoul Bott, "Differential varieties in Algebraic Topology."

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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 .

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