MATH 251 Lecture 26

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Spherical Coordinates

Recall that

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

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

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

$V=\mathrm {d} \rho \mathrm {d} \phi \mathrm {d} \theta$, so $V'=\rho ^{2}\sin \phi \mathrm {d} \rho \mathrm {d} \phi \mathrm {d} \theta$

Example

Find the center of mass of a hemisphere of radius $R$ with a constant density of $\sigma _{0}$.

$M=\iiint _{R}\sigma _{0}\mathrm {d} V=\sigma _{0}{\frac {2}{3}}\pi R^{3}$

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

$z={\frac {1}{M}}\iiint _{R}z\sigma _{0}\,\mathrm {d} V=\int _{0}^{2\pi }\int _{0}^{\tfrac {\pi }{2}}\int _{0}^{R}\sigma _{0}\rho \cos \phi \rho ^{2}\sin \phi \,\mathrm {d} \rho \,\mathrm {d} \phi \,\mathrm {d} \theta$

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

Example

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

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

Set up and solve.