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Unformatted text preview: F * α = 1 p ! α μ 1 ...μ p ( y ( x )) dy μ 1 ∧ ··· ∧ dy μ p = 1 p ! ∂ y μ 1 ∂ x i 1 ··· ∂ y μ p ∂ x i p α μ 1 ...μ p ( y ( x )) dx i 1 ∧ ··· ∧ dx i p • In components ( F * α ) i 1 ... i p = ∂ y μ 1 ∂ x i 1 ··· ∂ y μ p ∂ x i p α μ 1 ...μ p ( y ( x )) • Remark. The pullback is welldeﬁned only for covariant tensors and the pushforward is well deﬁned only for contravariant tensors. • Theorem 3.4.1 Let F : M → W. The pullback F * : Λ p W → Λ p M has the properties: 1. F * is linear. 2. For any two forms α and β F * ( α ∧ β ) = ( F * α ) ∧ ( F * β ) 3. F * commutes with exterior derivative. That is, for any pform α F * ( d α ) = d ( F * α ) . Proof : Direct calculation. ± di ﬀ geom.tex; April 12, 2006; 17:59; p. 86...
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 Spring '10
 Wong
 Geometry, Derivative, Continuous function, Manifold, Differential form, Smooth function

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