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 .