Section13.2Changing Coordinate Systems: The Jacobian
After finishing this section, you should...
Just as we did with polar coordinates in two dimensions, we can compute a Jacobian for any change of coordinates in three dimensions. We will focus on cylindrical and spherical coordinate systems.
Remember that the Jacobian of a transformation is found by first taking the derivative of the transformation, then finding the determinant, and finally computing the absolute value.
Exercise13.2.1
The cylindrical change of coordinates is:
\begin{align*}
x\amp =r\cos\theta, y=r\sin\theta, z=z\\
\text{ or in vector form } \amp \\\
\vec C(r,\theta,z) \amp = (r\cos\theta,r\sin\theta, z)
\end{align*}
The spherical change of coordinates is:
\begin{align*}
x\amp =\rho\sin\phi\cos\theta,\ y=\rho\sin\phi\sin\theta,\ z=\rho\cos\phi\\
\text{ or in vector form } \amp \\\
\vec S(\rho,\phi,\theta) \amp = (\rho\sin\phi\cos\theta,\rho\sin\phi\sin\theta,\rho\cos\phi).
\end{align*}
(a)
Verify that the Jacobian of the cylindrical transformation is \(\ds\frac{\partial(x,y,z)}{\partial(r,\theta,z)} = |r|\text{.}\)
(b)
Verify that the Jacobian of the spherical transformation is \(\ds\frac{\partial(x,y,z)}{\partial(\rho,\phi,\theta)} = |\rho^2\sin\phi|\text{.}\)
The previous exercise shows us that, provided we require \(r\geq0\) and \(0\leq \phi\leq \pi\text{,}\) we can write:
\begin{equation*}
dV=dxdydz = rdrd\theta dz = \rho^2\sin\phi d\rho d\phi d\theta,
\end{equation*}
Cylindrical coordinates are extremely useful for problems which involve:
cylinders
paraboloids
cones
Spherical coordinates are extremely useful for problems which involve:
Exercise13.2.2
See Sage.
The double cone \(z^2=x^2+y^2\) has two halves. Each half is called a nappe. Set up an integral in the coordinate system of your choice that would give the volume of the region that is between the \(xy\) plane and the upper nappe of the double cone \(z^2=x^2+y^2\text{,}\) and between the cylinders \(x^2+y^2=4\) and \(x^2+y^2=16\text{.}\) Then evaluate the integral.
Exercise13.2.3
Set up an integral in the coordinate system of your choice that would give the volume of the solid ball that is inside the sphere \(a^2=x^2+y^2+z^2\text{.}\) Compute the integral to give a formula for the volume of a sphere of radius \(a\text{.}\)
Then set up (don't evaluate) an iterated integral that would give the moment of inertia \(I_x\) about the \(x\)-axis, if the density is a constant, so \(\delta =c\text{.}\)
Exercise13.2.4
For the next several exercises be sure to check that you've correctly swapped bounds by having Sage or WolframAlpha actually compute all of the integrals.
Exercise13.2.5
Consider the region \(D\) in space that is inside both the sphere \(x^2+y^2+z^2=9\) and the cylinder \(x^2+y^2=4\text{.}\)
HintStart by drawing the region.
(a)
Set up an iterated integral in Cartesian (rectangular) coordinates that would give the volume of \(D\text{.}\)
(b)
Set up an iterated integral in cylindrical coordinates that would give the volume of \(D\text{.}\)
Exercise13.2.6
Consider the region \(D\) in space that is both inside the sphere \(x^2+y^2+z^2=9\) and yet outside the cylinder \(x^2+y^2=4\text{.}\)
HintStart by drawing the region.
(a)
(i)
For the first integral use the order \(dzdrd\theta\text{.}\)
(ii)
For the second, use the order \(d\theta dr dz\text{.}\)
(b)
Set up an iterated integral in spherical coordinates that would give the volume of \(D\text{.}\)
Exercise13.2.7
The integral \(\ds\int_{0}^{\pi}\int_{0}^{1}\int_{\sqrt{3}r}^{\sqrt{4-r^2}}rdzdrd\theta\) represents the volume of solid domain \(D\) in space. Set up integrals in both rectangular coordinates and spherical coordinates that would give the volume of the exact same region.
Exercise13.2.8
The temperature at each point in space of a solid occupying the region {\(D\)}, which is the upper portion of the ball of radius 4 centered at the origin, is given by \(T(x,y,z) = \sin(xy+z)\text{.}\) Set up an iterated integral formula that would give the average temperature.