Today we wrap up section
15.8: Cylindrical coordinates represents one generalization of polar coordinates:
\[
x=r\cos{\theta} \hspace{1in}
y=r\sin{\theta} \hspace{1in}
r^2=x^2+y^2 \hspace{1in}
z=z
\]
So it represents a polar coordinate transformation for two coordinates, and
leaves $z$ untransformed.
The volume element ($dV$) is given by $rdrd{\theta}dz$, as one can see in Figure 7, p. 1054. This is a simple
generalization of the polar formula of the area element $dA$.
Examples:
- #1, p. 1055
- #3
- #13
- let's revisit the problem, #24, p. 1049: we actually did this
one as a cylindrical transformation. If you recall, we left $z$ alone,
and transformed $x$ and $y$.
- Then we'll try #31 -- Mt. Fuji!