8—Multivariable Calculus 2438.12 Repeat the preceding problem for the drumhead mode of problem 10. The exact result, calculated in terms
of roots of Bessel functions is 3. 832
√
T/σR^2. Ans: 4√
T/σR^28.13 Sketch the gravitational field of the Earth from Eq. ( 16 ). Is the direction of the field plausible?
8.14 Prove that the unit vectors in polar coordinates are related to those in rectangular coordinates by
ˆr=ˆxcosθ+yˆsinθ, θˆ=−ˆxsinθ+ˆycosθWhat areˆxandˆyin terms ofˆrandθˆ?
8.15 Prove that the unit vectors in spherical coordinates are related to those in rectangular coordinates by
ˆr=xˆsinθcosφ+yˆsinθsinφ+zˆcosθ
ˆθ=xˆcosθcosφ+yˆcosθsinφ−ˆzsinθφˆ=−ˆxsinφ+ˆycosφ8.16 Compute the volume of a sphere using spherical coordinates. Also do it using rectangular coordinates. Also
do it in cylindrical coordinates.
8.17 Finish both integrals Eq. ( 15 ). 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 linesx= 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∂r/∂θˆ , also∂θ/∂θˆ. 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-ˆθvectors.
Draw pictures that will demonstrate that your answers are at least in the right direction. Ans:∂θ/∂θˆ =−ˆr
8.21 Compute the gradient of the distance from the origin (in three dimensions) in three coordinate systems and
verify that they agree.