# An Introduction to Manifolds (2nd Edition) (Universitext) by Loring W. Tu

By Loring W. Tu

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Extra resources for An Introduction to Manifolds (2nd Edition) (Universitext)

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Show that a basis for the space Lk (V ) of k-linear functions on V is {α i1 ⊗ · · · ⊗ α ik } for all multi-indices (i1 , . . , ik ) (not just the strictly ascending multi-indices as for Ak (L)). In particular, this shows that dim Lk (V ) = nk . 4. A characterization of alternating k-tensors Let f be a k-tensor on a vector space V . Prove that f is alternating if and only if f changes sign whenever two successive arguments are interchanged: f (. . , vi+1 , vi , . ) = − f (. . , vi , vi+1 , . ) for i = 1, .

N be the dual basis for V ∨ . Introduce the multi-index notation I = (i1 , . . , ik ) and write eI for (ei1 , . . , eik ) and α I for α i1 ∧ · · · ∧ α ik . A k-linear function f on V is completely determined by its values on all k-tuples (ei1 , . . , eik ). If f is alternating, then it is completely determined by its values on (ei1 , . . , eik ) with 1 ≤ i1 < · · · < ik ≤ n; that is, it suffices to consider eI with I in strictly ascending order. 28. Let e1 , . . , en be a basis for a vector space V and let α 1 , .

3) ∂ ∂ xi . p Since Dv is clearly linear in v, the map φ is a linear map of vector spaces. 1. If D is a point-derivation of C∞ p , then D(c) = 0 for any constant function c. Proof. Since we do not know whether every derivation at p is a directional derivative, we need to prove this lemma using only the defining properties of a derivation at p. By R-linearity, D(c) = cD(1). So it suffices to prove that D(1) = 0. 2), D(1) = D(1 · 1) = D(1) · 1 + 1 · D(1) = 2D(1). Subtracting D(1) from both sides gives 0 = D(1).