Math 452 - Advanced Calculus II Homework # 4. Due Mar. 20, 2015, noon Q 1. For each positive integer n, the Bessel function Jn (x) may be defined by: xn Jn (x) = 1 · 3 · 5 · · · (2n − 1)π Z 1 (1 − t2 )n−1/2 cos xt dt. −1 Prove that Jn (x) satisfies Bessel’s differential equation 1 0 n2 00 Jn + Jn + 1 − 2 Jn = 0. x x Q 2. Let A ⊂ Rm and B ⊂ Rn be contented sets, and f : Rm → R and g : Rn → R integrable functions. Define h : Rm+n → R by h(x, y) = f (x)g(y), and show that Z Z Z g . f h= A×B A B Conclude as a corollary that v(A × B) = v(A)v(B). Q 3. Consider the n-dimensional solid ellipsoid ( ) n 2 X x i E = x ∈ Rn : ≤1 . 2 a i=1 i Note that E is the image of the unit ball B1 (measured in the euclidean norm) in Rn under the map T : Rn → Rn defined by: T (x1 , . . . , xn ) = (a1 x1 , . . . , an xn ). Show that v(E) = a1 a2 · · · an v(B1 ). 1 Q 4. Let R be the solid torus in R3 obtained by revolving the circle (y − a)2 + z 2 ≤ b2 , in the yz-plane, about the z-axis. Note that the mapping T : R3 → R3 defined by x = (a + w cos v) cos u, y = (a + w cos v) sin u, z = w sin v, maps the interval Q = {(u, v, w) : u, v ∈ [0, 2π] and w ∈ [0, b]} onto R. Apply the change of variables formula to calculate the volume of this torus. Q 5. Find the volume of Br , the ball of radius r (measured in the euclidean norm) in Rn . HINT: See exercise 5.17 in chapter 4 of the textbook. 2

© Copyright 2020