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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