# MA 227 Standard C09

Polar, Cylindrical, and Spherical Coordinates

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)