At the end of the course, each student should be able to…
- C09: PolCylSph. Apply polar, cylindrical, and spherical transformations of variables.
C09. Polar, Cylindrical, and Spherical Coordinates
- The polar coordinate transformation
\(\vect p(r,\theta)=\<r\cos\theta,r\sin\theta\>\) has Jacobian
\(\frac{\partial\vect p}{\partial\<r,\theta\>}=r\).
- Thus if the region \(R\) in the \(xy\) plane is described by polar coordinates \(G\), then \(\iint_R f(x,y)\,dA=\iint_G f(\vect p(r,\theta))|r|\,dA\).
- Assuming \(r\geq 0\), and assuming the region is described by the outside polar curve \(r=g_2(\theta)\), inside polar curve \(r=g_1(\theta)\), and angles \(\alpha\leq\theta\leq\beta\), then \(\iint_R f(x,y)\,dA=\int_\alpha^\beta\int_{g_1(\theta)}^{g_2(\theta)} f(\vect p(r,\theta)) r\,dr\,d\theta\).
- The cylindrical coordinate transformation
\(\vect c(r,\theta,z)=\<r\cos\theta,r\sin\theta,z\>\) has Jacobian
\(\frac{\partial\vect p}{\partial\<r,\theta,z\>}=r\).
- Since \(z\) is preserved by this transformation, this is equivalent to reinterpretting the shadow \(R\) in the \(xy\) plane with polar coordinates \(G\).
- Assuming \(r\geq 0\), this yields \(\iiint_D f(x,y,z)\,dV = \iint_R\int_{g_1(x,y)}^{g_2(x,y)}f(x,y,z)\,dz\,dA = \iint_G\int_{g_1(\vect p(r,\theta))}^{g_2(\vect p(r,\theta))} f(\vect c(r,\theta,z))r\,dz\,dr\,d\theta\).
- The spherical coordinate transformation
\(\vect s(\rho,\phi,\theta)=\<r\cos\theta,r\sin\theta,\rho\cos\phi\>\)
expands to \(\vect s(\rho,\phi,\theta)=
\<\rho\sin\phi\cos\theta,\rho\sin\phi\sin\theta,\rho\cos\phi\>\)
and has Jacobian
\(\frac{\partial\vect s}{\partial\<\rho,\phi,\theta\>}=\rho^2\sin\phi\).
- Thus if the solid \(D\) in \(xyz\) space is described by polar coordinates \(H\), then \(\iiint_D f(x,y,z)\,dA=\iiint_H f(\vect s(\rho,\phi,\theta))\rho^2|\sin\phi|\,dA\).
- Assuming \(0\leq\phi\leq\pi\), and assuming the solid is described by the outside spherical surface \(\rho=g_2(\phi,\theta)\), inside spherical surface \(\rho=g_1(\phi,\theta)\), and is between the angles \(\gamma\leq\phi\leq\delta\) and \(\alpha\leq\theta\leq\beta\), then \(\iiint_D f(x,y,z)\,dV=\int_\alpha^\beta\int_\gamma^\delta \int_{g_1(\phi,\theta)}^{g_2(\phi,\theta)} f(\vect s(\rho,\phi,\theta)) \rho^2\sin\phi\,d\rho\,d\phi\,d\theta\).
Textbook References
- University Calculus: Early Transcendentals (3rd Ed)
- 14.4 (exercises 1-22)
- 14.7 (exercises 1-6, 15-26, 33-38, 43-62)