1. Homework 7 Let X; Y; Z be normed vector spaces over R: (1) Let bil(X ×Y; Z) be the space of bounded bilinear maps from X ×Y into Z: (For the definition of bounded bilinear maps, see class notes.) For each T 2 bil(X × Y; Z); we define kT k = sup kT (x; y)kZ : kxkX =kykY =1 Prove that (bil(X × Y; Z); k · k) forms a normed vector space. (2) A linear map φ : X ! Y is called an isomorphism of normed vector spaces over R if φ is an isomorphism of vector spaces such that kφ(x)kY = kxkX for any x 2 X: Prove that the map b b ' : L (X; L (Y; Z)) ! bil(X × Y; Z);T 7! 'T defined by 'T (x; y) = T (x)(y) is an isomorphism of normed vector spaces. (3) Compute Df(0; 0);D2f(0; 0) and D3f(0; 0) for the following given functions f: (a) f(x; y) = x4 + y4 − x2 − y2 + 1: (b) f(x; y) = cos(x + 2y): (c) f(x; y) = ex+y: (4) Let f :[a; b] ! R be continuous. Suppose that Z b f(x)h(x)dx = 0 a for any continuous function h :[a; b] ! R with h(a) = h(b) = 0: Show that f is the zero function on [a; b]: (5) Let C1[a; b] be the space of all real valued continuously differentiable functions on [a; b]: On C1[a; b]; we define 0 1 kfkC1 = kfk1 + kf k1 for any f 2 C [a; b]: 1 (a) Show that (C [a; b]; k · kC1 ) is a Banach space over R: 1 (b) Let X be the subset of C [a; b] consisting of functions f :[a; b] ! R so that f(a) = f(b) = 0: Prove that X forms a closed vector subspace of C1[a; b]: Hence X is also a Banach space. 3 3 (6) Let L : U ⊆ R ! R be a smooth function defined on an open subset of R and 1 Xα,β = ff 2 C [a; b]: f(a) = α; f(b) = βg where α; β 2 R: We consider the functional S : Xα,β ! R defined by Z b S(f) = L(f(t); f 0(t); t)dt: a Compute the Euler-Lagrange equation for S when L is given below. (a) L(x; y; z) = sin y (b) L(x; y; z) = x2 − y2 (c) L(x; y; z) = 2zy − y2 + 3x2y: (7) Let f :[a; b] ! R be a nonnegative continuously differentiable function on [a; b]: The area of the surface of revolution generated by rotating the curve y = f(x) about x-axis is Z b p S(f) = 2π f(x) 1 + (f 0(x))2dx: a 1 2 Find the C1-curve y = f(x) with f(a) = f(b) = 0 and f ≥ 0 on [a; b] minimizing S : X ! R: 1 2 (8) A C -curve on R is a map 2 γ : [0; 1] ! R ; t 7! (x(t); y(t)) such that x(t); y(t) are C1-functions on [0; 1]: If γ(0) = A and γ(1) = B; we say that 2 γ is a curve from A to B: Here A; B are points of R : The arclength of γ is defined to be Z 1 p L(γ) = (x0(t))2 + (y0(t))2dt: 0 Then L defines a real valued function on the space 1 2 PA;B = fγ 2 C ([a; b]; R ): γ(0) = A; γ(1) = Bg 1 2 of C -curve from A to B in R : Find the Euler Lagrange equation for L and prove that γ is the solution to the Euler-Lagrange equation for L if and only if γ the straight line connecting A and B:.