8—Multivariable Calculus 211constraints is on the edge of this allowed region, and that it is at the corner of intersection of the two
inequalities. This is the beginning of the subject called “linear programming.”
Ans: a cube
8.46 Plotθversusbin equation (8.45) or (8.46).
8.47 A disk of radiusRis at a distancecabove thex-yplane and parallel to that plane. What is the
solid angle that this disk subtends from the origin? Ans: 2 π
[
1 −c/
√
c^2 +R^2
]
8.48 Within a sphere of radiusR, what is the volume contained between the planes defined byz=a
andz=b? Ans:π(b−a)
(
R^2 −^13 (b^2 +ab+a^2 )
)
8.49 Find the mean-square distance, V^1
∫
r^2 dV, from a point on the surface of a sphere to points
inside the sphere. Note: Plan ahead and try to make this problem as easy as possible. Ans: 8 R^2 / 5
8.50 Find the mean distance,V^1
∫
rdV, from a point on the surface of a sphere to points inside the
sphere. Unlike the preceding problem, this requires some brute force. Ans: 6 R/ 5
8.51 A volume mass density is specified in spherical coordinates to be
ρ(r,θ,φ) =ρ 0
(
1 +r^2 /R^2
)[
1 +^12 cosθsin^2 φ+^14 cos^2 θsin^3 φ
]
Compute the total mass in the volume 0 < r < R. Ans: 32 πρ 0 R^3 / 15
8.52 The circumference of a circle is some constant times its radius (C 1 r). For the two-dimensional
surface that is a sphere in three dimensions the area is of the formC 2 r^2. Start from the fact that you
know the integral
∫∞
−∞dxe
−x^2 =π 1 / (^2) and write out the following two dimensional integral twice. It
is over the entire plane.
∫
dAe−r
2using dA=dxdy and using dA=C 1 rdr
From this, evaluateC 1. Repeat this fordV andC 2 r^2 in three dimensions, evaluatingC 2.
Now repeat this in arbitrary dimensions to evaluateCn. Do you need to reread chapter one? In
particular, what isC 3? It tells you about the three dimensional hypersphere in four dimensions. From
this, what is the total “hypersolid angle” in four dimensions (like 4 πin three)? Ans: 2 π^2
8.53Do the reverse of problem8.23. Start with the second equation there and change variables to see
that it reverts to a constant times the first equation.
8.54 Carry out the interchange of limits in Eq. (8.22). Does the drawing really represent the integral?
8.55 Isx^2 +xy+y^2 a minimum or maximum or something else at(0,0)? Do the same question for
x^2 + 2xy+y^2 and forx^2 + 3xy+y^2. Sketch the surfacez=f(x,y)in each case.
8.56 Derive the conditions stated after Eq. (8.33), expressing the circumstances under which the
Hessian matrix is positive definite.
8.57 In the spirit of problems8.10et seq.what happens if you have a rectangular drumhead instead
of a circular one? Let 0 < x < aand 0 < y < b. The drumhead is tied down at its edges, so an
appropriate function that satisfies these conditions is