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Igor Zelenko, Fall 2009 1 Homework Assignment 9 in Topology I, MATH636 due to Nov 16, 2009 1. Let det : Mn×n (R) → R denote the determinant function on (n×n)-matrices with 2 real entries. (Note: Mn×n (R) ∼ = Rn ). Show that for A ∈ Mn×n (R) satisfying detA 6= 0, the differential det∗ : TA Mn×n (R) → R of the function det at A is the linear map defined by det∗ (M ) = (detA) · tr(A−1 M ), where tr(B) denotes the trace of the matrix B. 2. Show that in a neighborhood of the point (x, y, z) = (1, 1, 1) the equation x2 + y 2 + z 2 + ln(xyz) = 3 defines a twice continuously differentiable function x = φ(y, z) and find grad φ(1, 1) and ∂2φ (1, 1). ∂y∂z 3. Let f : M → N be a diffeomorphism of smooth manifolds. Show that for each p ∈ M , the differential f∗ : Tp M → Tf (p) N is an isomorphism. 4. Solve problem 2, p. 76 in the text. 5. Solve problem 4, p. 80 in the text. 6. Let U be an open subset of a manifold M . Show how one can naturally identify Tp U with Tp M for all p ∈ M .