Calculus Small Changes, Big Ideas

Section 18.4 Cylindrical and spherical coordinates

• 3D Jacobians are set up the same way, but might need to show how determinant works (expansion by minors or maybe Rule of Sarrus would be enough)

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• Cylindrical coordinates:

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  \begin{align*} x \amp = r\cos\theta\\ y \amp = r\sin\theta\\ z \amp = z \end{align*}
  Jacobian: \(\DS\abs{\pdv{(x,y,z)}{(r,\theta,z)}}=r\)

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• Spherical coordinates:

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  \begin{align*} x \amp = \rho\sin\phi\cos\theta\\ y \amp = \rho\sin\phi\sin\theta\\ z \amp = \rho\cos\phi \end{align*}
  Jacobian: \(\DS\abs{\pdv{(x,y,z)}{(\rho,\theta,\phi)}}=\rho^{2}\sin\phi\)

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  This is assuming \(\rho\) is spherical radius, \(\theta\) is longitude (azimuthal angle), \(\phi\) is co-latitude (polar angle).

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• Show how to use them to convert triple integrals, applied problems (most likely center of mass, moment of inertia)

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