A Comprehensive Introduction to Differential Geometry, VOL. by Michael Spivak

By Michael Spivak

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Also to be thought of as the velocity of a curve through that point. A vector field is a (smooth) assignment of a tangent vector to each point on a manifold and might, for example, model the velocity field of a fluid 37 38 CHAPTER 2. APPETIZERS flow. Just as a vector is the velocity of a curve through a point, so a vector field is the “velocity” of a smooth family of deformations of the manifold. Tangent vectors and vector fields are introduced in chapter 4 and studied in further depth in chapter 7.

Hint: Find the Jacobian Jt := det[DF lt (x)] and then convert the integral above to one just over D(0) = D. 6. Let GL(n, R) be the nonsingular n × n matrices and show that GL(n, R) is an open subset of the vector space of all matrices Mn×n (R) and then find the derivative of the determinant map: det : GL(n, R) → R (for each A this should end up being a linear map D det|A : Mn×n (R) → R). What is ∂ ∂xij det X where X = (xij ) ? 7. Let A : U ⊂ E → L(F, F) be a C r map and define F : U × F → F by F (u, f ) := A(u)f .

1 There are at least two issues that remain even if we restrict ourselves to Banach spaces. First, the existence of smooth cut-off functions and smooth partitions of unity (to be defined below) are not guaranteed. The existence of smooth cut-off functions and smooth partitions of unity for infinite dimensional spaces is a case-by-case issue while in the finite dimensional case their existence is guaranteed. Second, there is the fact that a subspace of a Banach space is not a Banach space unless it is also a closed subspace.

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