Mathematical Tools for Physics

8—Multivariable Calculus 220

What is the total kinetic energy because of this oscillation? It is

∫

udA. To evaluate it, use polar coordinates
and integrate over the area of the drumhead.

∫

udA=

∫R

0

r dr

∫ 2 π

0

dθ

σ

2

z^20

(

(1−r^2 /R^2 )ωsinωt

) 2

=

σ

2

2 πz^20 ω^2 sin^2 ωt

∫R

0

dr r

(

1 −r^2 /R^2

) 2

=σπz 02 ω^2 sin^2 ωt

1

2

∫r=R

r=0

d(r^2 )

(

1 −r^2 /R^2

) 2

=σπz 02 ω^2 sin^2 ωt

1

2

R^2

1

3

(

1 −r^2 /R^2

) 3

(−1)

∣

∣

∣

∣

r=R

0

=

1

6

σR^2 πz^20 ω^2 sin^2 ωt

(14)

See problem 10 and following for more on this.

8.8 Cylindrical, Spherical Coordinates
The three common coordinate systems used in three dimensions are rectangular, cylindrical, and spherical coor-
dinates, and these are the ones you have to master. When you need to use prolate spheroidal coordinates you
can look them up.

x

z

y

θ

r

z

φ

θ r

−∞< x <∞ 0 < r <∞ 0 < r <∞

−∞< y <∞ 0 < θ < 2 π 0 < θ < π

−∞< z <∞ −∞< z <∞ 0 < φ < 2 π