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 ∈ 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 ∈ X. Prove that the map b b ϕ : L (X, L (Y,Z)) → bil(X × Y,Z),T 7→ ϕT

defined by ϕT (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 = kfk∞ + kf k∞ for any f ∈ 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 = {f ∈ C [a, b]: f(a) = α, f(b) = β}

where α, β ∈ 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 = {γ ∈ C ([a, b], R ): γ(0) = A, γ(1) = B} 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.