8—Multivariable Calculus 2088.16 Compute the volume of a sphere using spherical coordinates. Also do it using rectangular coor-
dinates. Also do it in cylindrical coordinates.
8.17 Finish both integrals Eq. (8.21). Draw sketches to demonstrate that the limits stated there are
correct.
8.18 Find the volume under the plane 2 x+ 2y+z= 8aand over the triangle bounded by the lines
x= 0,y= 2a, andx=yin thex-yplane. Ans: 8 a^3
8.19 Find the volume enclosed by the doughnut-shaped surface (spherical coordinates)r =asinθ.
Ans:π^2 a^3 / 4
8.20 In plane polar coordinates, compute∂ˆr/∂φ, also∂φ/∂φˆ. This means thatris fixed and you’re
finding the change in these vectors as you move around a circle. In both cases express the answer in
terms of theˆr-φˆvectors. Draw pictures that will demonstrate that your answers are at least in the
right direction. Ans:∂φ/∂φˆ =−rˆ
8.21 Compute the gradient of the distance from the origin (in three dimensions) in three coordinate
systems and verify that they agree.
8.22 Taylor’s power series expansion of a function of several variables was discussed in section2.5.
The Taylor series in one variable was expressed in terms of an exponential in problem2.30. Show that
the series in three variables can be written as
e
~h.∇f(x,y,z)
8.23 The wave equation is (a) below. Change variables toz=x−vtandw=x+vtand show that
in these coordinates this equation is (b) (except for a constant factor). Did youexplicitlynote which
variables are kept fixed at each stage of the calculation? See also problem8.53.
(a)
∂^2 u
∂x^2
−
1
v^2
∂^2 u
∂t^2
= 0 (b)
∂^2 u
∂z∂w
= 0
8.24 The equation (8.23) comes from taking the gradient of the Earth’s gravitational potential in an