MATH 223 Problems Problem Set for Quiz 1, Spring 2017

Math 223 - Advanced Calculus, Spring 2017 Review exercises Exercise 1. Verify whether or not the following functions are con- tinuous at the point v = (0, 0): (1) f : R2 → R, f(x, y) = x3y2 x2+y4 if (x, y) ̸= (0, 0), f(0, 0) = 0; (2) f : R2 → R, f(x, y) = x2y2 x4+y4 if (x, y) ̸= (0, 0), f(0, 0) = 0; (3) f : R2 → R, f(x, y) = x2y3+2xy4 (x2+y2)2 if (x, y) ̸= (0, 0), f(0, 0) = 0. Exercise 2. Let V be a real vector space, and let ∥ · ∥ be a norm on V . Recall that the ball of radius r > 0 and center x ∈ V is the set Br(x) = {v ∈ V : ∥v − x∥ < r}. (1) Show that ∥av + bw∥ ≤ |a|∥v∥ + |b|∥w∥ for all a, b ∈ R and v, w ∈ V . (2) A subset U of V is called convex if it satisfies the following property: for all v, w ∈ U and 0 ≤ t ≤ 1 we have tv + (1 − t)w ∈ U. Show that B1(0) is convex. (Hint: use part (1)). (3) Show that any ball Br(x) is convex (Hint: one can use for instance (2)). Exercise 3. Let f : R2 → R be defined as f(x1, x2) = cos(x1x2). Compute the Hessian matrix of f at a general point p = (x0 1, x0 2) ∈ R2. Exercise 4. Let V be a vector space. Recall that two norms ∥ · ∥1 and ∥ · ∥2 on V are called equivalent if there are constants C, D > 0 such that C∥v∥1 ≤ ∥v∥2 ≤ D∥v∥1 for all v ∈ V . (1) Let V = Rn with the standard Euclidean norm ∥·∥, and stan- dard basis e1, . . . , en. Let ∥ · ∥1 be another norm on Rn, and define M = max{∥e1∥1, ∥e2∥1, . . . , ∥en∥1}. Show that ∥v∥1 ≤ nM∥v∥ for all v ∈ Rn. (Hint: consider the case ∥v∥ = 1 first). (2) Let V = Rn, ∥ · ∥, ∥ · ∥1 be as in (1), and consider in Rn the Euclidean metric d(x, y) = ∥x − y∥. Show that the function ∥ · ∥1 : Rn → R is Lipschitz with respect to this metric. (3) Show that ∥ · ∥1 admits a maximum D > 0 and a minimum C > 0 on the set S1(0) = {v ∈ Rn : ∥v∥ = 1}. (Hint: S1(0) is compact (why?) and ∥ · ∥1 is continuous by (2)). (4) Using (3), show that ∥ · ∥ and ∥ · ∥1 are equivalent. Deduce that if V be a finite dimensional vector space, all norms on V are equivalent. Exercise 5. Let f : R2 → R2 be a continuous, bijective function with (continuous) inverse f −1 : R2 → R2 (that is, we have f ◦ f −1(v) = v for all v ∈ R2). Suppose that f is differentiable on R2. Does it follow that f −1 is differentiable on R2? (If yes, give a proof, otherwise find a counterexample). Exercise 6. Let f : R2 → R be a differentiable function, and fix v ∈ R2. We define the directional derivative ∂f ∂v as ∂f ∂v(p) = limt→0 f(p+tv)−f(p) t . (1) Show that the limit exists, so that ∂f ∂v is indeed well defined. (2) Let Jf(p) be the Jacobian of f at p (which is a 1 × 2 row vector). Show that ∂f ∂v(p) = Jf(p) · v. (3) Let p1, p2 ∈ R2 be such that p2 − p1 = v. Let S = {(1 − t)p1 + tp2 : 0 ≤ t ≤ 1} (S is the segment with endpoints p1, p2). Show that there is a point q ∈ S such that ∂f ∂v(q) = f(p2) − f(p1). (Hint: show that (1 − t)p1 + tp2 = p1 + tv, define g : [0, 1] → R as g(t) = f(p1 + tv) and apply the mean value theorem). Suggested problems (for those who look for a harder challenge!): Rudin, Chapter 9, ex. 7,8,11,13,14.