# An Introduction to Manifolds by Loring W. Tu

By Loring W. Tu

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Extra resources for An Introduction to Manifolds

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X1 ∂ x2 ∂ x1 ∂ x2 Therefore, d 2 ( f dxI ) = 0. (iii) This is simply the definition of the exterior derivative of a function as the differential of the function. 8 (Characterization of the exterior derivative). 7 uniquely characterize exterior differentiation on an open set U in Rn ; that is, if D : Ω∗ (U) → Ω∗ (U) is (i) an antiderivation of degree 1 such that (ii) D2 = 0 and (iii) (D f )(X ) = X f for f ∈ C∞ (U) and X ∈ X(U), then D = d. Proof. Since every k-form on U is a sum of terms such as f dxi1 ∧ · · · ∧ dxik , by linearity it suffices to show that D = d on a k-form of this type.

A linear subspace of V of dimension n − 1 is called a hyperplane in V . (b) Show that a nonzero linear functional on a vector space V is determined up to a multiplicative constant by its kernel, a hyperplane in V . In other words, if f and g : V → R are nonzero linear functionals and ker f = ker g, then g = c f for some constant c ∈ R. 3. A basis for k-tensors Let V be a vector space of dimension n with basis e1 , . . , en . Let α 1 , . . , α n be the dual basis for V ∨ . 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 , .

Vn ) = det[v1 · · · vn ], viewed as a function of the n column vectors v1 , . . , vn in Rn , is n-linear. 10. A k-linear function f : V k → R is symmetric if f vσ (1) , . . , vσ (k) = f (v1 , . . , vk ) for all permutations σ ∈ Sk ; it is alternating if f vσ (1) , . . , vσ (k) = (sgn σ ) f (v1 , . . , vk ) for all σ ∈ Sk . Examples. (i) The dot product f (v, w) = v • w on Rn is symmetric. (ii) The determinant f (v1 , . . , vn ) = det[v1 · · · vn ] on Rn is alternating. (iii) The cross product v × w on R3 is alternating.