HW 3: Let F : R → R^{2}, F (x) = (2, 3)t G : R^{2} → R^{3}, G(x, y) = (xyz, x^{2}, x + 2y + 3z + 5) H : R → R^{2}, H(x) = (x^{2}, x^{3}) k : R^{2} → R, k(x, y) = x^{8} + 5xy. 1.) Use the chain rule to calculate D(G ∘ F)_{2} 2.) Use the product rule to calculate D(F H)_{2} 3.) Let a = (3, 4). Let X_{a} = 9E_{1a} − E_{2a}. Then X_{a}^{∗}(k) = F is a C^{r}-diffeomorphism if

(1) F is a homeomorphism

(2) F, F^{−1} ∈ C^{r} F is a diffeomorphism if F is a C^{∞}-diffeomorphism. 4.) Give an example of a homeomorphism which is analytic (ie C^{∞} and near a, f(x) = f(a) + f′(a)(x − a) + \frac{f′′(a)}{2!}(x − a)^{2} + ... , its Taylor series) which is not a diffeomorphism. 5.) Suppose F′(x) = 0 for all x ∈ U^{open} ⊂ R^{n}. Show F cannot be a homeomorphism. What can you say about F (hint: MVT). 6.) Suppose f : R → R, f′(x) ≠ 0 for all x ∈ U. Show that the derivative of f^{−1} exists for all y ∈ f(U) 7.) Suppose F is a C^{1}-diffeomorphism. Show that DF_{x} is nonsingular (ie det(DF_{x}) ≠ 0)∀x ∈ dom(F) 8.) Ex 1: Show F : R^{n} → R^{n}, F (x) = x + a is a diffeomorphism. 9.) Ex 2: Determine when F : R^{n} → R^{m}, F (x) = Ax, where A is an m × n matrix, is a diffeomorphism. DF_{x} = . Note that if F and G are diffeomorphism, then F ∘G is a diffeomorphism (when F ∘G is defined). Thm 6.5 (Contracting mapping theorem): Let M be a complete metric space and let T : M → M. Suppose there exists a constant λ ∈ [0, 1) such that for all x, y ∈ M, d(T (x), T (y)) ≤ d(x, y). Then T has a unique fixed point. Proof: See class notes (Recall T^{n}(x_{0}) is a Cauchy sequence. Since M be a complete metric space, T^{n}(x_{0}) converges, say to a. Then d(T (a), a) = 0). Thm 6.4 (Inverse Function Theorem): Suppose F : W^{open} ⊂ R^{n} → R^{n} ∈ C^{r}. Suppose for a ∈ W, det(DF_{a}) ≠ 0. Then there exists U such that a ∈ U^{open}, V = F (U) is open, and F : U → V is a C^{r}-diffeomorphism. Moreover, for x ∈ U and y = F (x), DF_{y}^{−1} = (DF_{x})^{−1}