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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{z} = \frac{3R}{8}}

Example

Calculate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iiint \left( x^2+y^2 \right) \,\mathrm{d}V} over the ice-cream cone Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2+y^2 \le z^2 \le 1-x^2-y^2}

• Integrand: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2+y^2 = \rho^2 \sin^2\phi}
• Differential: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{d}V = \rho^2\sin\phi \,\mathrm{d}\rho\,\mathrm{d}\phi\,\mathrm{d}\theta}
• Limits of Integration: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le \rho \le 1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le \theta \le 2\pi} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le \phi \le \tfrac{\pi}{4}}

Set up and solve.