MATH 251 Lecture 26

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Spherical Coordinates

Recall that

• $x=\rho \sin \varphi \cos \theta$
• $y=\rho \sin \varphi \sin \theta$
• $x=\rho \cos \varphi$

and the derivative of the transformation (Jacobian) is ${\rho }^2\sin \varphi$

For small changes $\mathrm{d}\rho$, $\mathrm{d}\varphi$, and $\mathrm{d}\theta$, the transformation of a small cube multiplies its volume by the Jacobian:

$V=\mathrm{d}\rho \mathrm{d}\varphi \mathrm{d}\theta$, so $V^{\prime }={\rho }^2\sin \varphi \mathrm{d}\rho \mathrm{d}\varphi \mathrm{d}\theta$

Example

Find the center of mass of a hemisphere of radius $R$ with a constant density of ${\sigma }_0$.

$M=\underset{R}{\iiint }{\sigma }_0\mathrm{d}V={\sigma }_0\frac{2}{3}\pi R^3$

By symmetry, $\overline{x}=\overline{y}=0$, so let's evaluate $\overline{z}$ in polar coordinates:

$z=\frac{1}{M}\underset{R}{\iiint }z{\sigma }_0\mathrm{d}V=\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{\frac{\pi }{2}}{\int }}\underset{0}{\overset{R}{\int }}{\sigma }_0\rho \cos \varphi {\rho }^2\sin \varphi \mathrm{d}\rho \mathrm{d}\varphi \mathrm{d}\theta$

After much evaluation, the answer is $\overline{z}=\frac{3R}{8}$

Example

Calculate $\iiint (x^2+y^2)\mathrm{d}V$ over the ice-cream cone $x^2+y^2\le z^2\le 1-x^2-y^2$

• Integrand: $x^2+y^2={\rho }^2{\sin }^2\varphi$
• Differential: $\mathrm{d}V={\rho }^2\sin \varphi \mathrm{d}\rho \mathrm{d}\varphi \mathrm{d}\theta$
• Limits of Integration: $0\le \rho \le 1$, $0\le \theta \le 2\pi$, $0\le \varphi \le \frac{\pi }{4}$

Set up and solve.