Let $f: \mathbb{R}^3\to \mathbb{R}^3$ be given by $f(\rho, \phi, \theta) = (\rho\cos\theta \sin \phi, \rho \sin \theta \sin \phi, \rho \cos \phi).$

Question

Let $f: \mathbb{R}^3\to \mathbb{R}^3$ be given by $f(\rho, \phi, \theta) = (\rho\cos\theta \sin \phi, \rho \sin \theta \sin \phi, \rho \cos \phi).$ It is called the spherical coordinate transformation. Take $S$ to be the set $S = [1, 2] \times (0, \pi /2] \times [0, \pi /2]$.

(a) Calculate $D_f$ and $\det D_f$.

(b) Sketch the image under $f$ of the set $S$.

For (a), $D_f(\rho, \phi, \theta)=\begin{bmatrix}\cos \theta \sin \phi &\rho \cos \theta \cos \phi &-\rho\sin\theta\sin\phi \\ \sin\theta\sin\phi &\rho\sin\theta\cos\phi &\rho\cos\theta\sin\phi \\ \cos\phi & -\rho\sin\phi & 0\end{bmatrix}$ and so $\det D_f=\rho^2\cos^2\theta\sin^3\phi+\rho^2\cos^2\theta\cos^2\phi\sin\phi+\rho^2\sin^2\theta\sin^2\phi+\rho^2\sin^2\theta\sin\phi\cos^2\phi$

I do not know what to do in (b), could someone help me please? Thank you very much.

Answer 1

Using properties of the determinant, calculating over the last row, and then using some very basic trigonometry, \begin{align} \det D_f &=\begin{vmatrix}\cos \theta \sin \phi &\rho \cos \theta \cos \phi &-\rho\sin\theta\sin\phi \\ \sin\theta\sin\phi &\rho\sin\theta\cos\phi &\rho\cos\theta\sin\phi \\ \cos\phi & -\rho\sin\phi & 0\end{vmatrix}\\ \ \\ &=\rho^2\sin\phi\,\begin{vmatrix}\cos \theta \sin \phi & \cos \theta \cos \phi &- \sin\theta \\ \sin\theta\sin\phi & \sin\theta\cos\phi & \cos\theta \\ \cos\phi & - \sin\phi & 0\end{vmatrix}\\ \ \\ &=\rho^2\sin\phi\,[\cos\phi(\cos^2\theta\cos\phi+\sin^2\theta\cos\phi)+\sin\phi(\cos^2\theta\sin\phi+\sin^2\theta\sin\phi)]\\ \ \\ &=\rho^2\sin\phi\,[\cos^2\phi+\sin^2\phi]\\ \ \\ &=\rho^2\sin\phi. \end{align}

For part (b), start first with the set $[0,2]\times [0,2\pi]\times[0,\pi]$. Then move to $[1,2]\times [0,2\pi]\times[0,\pi]$, then $[1,2]\times [0,\pi/2]\times[0,\pi]$, and then $[1,2]\times [0,\pi/2]\times[0,\pi/2]$ and $[1,2]\times (0,\pi/2]\times[0,\pi/2]$. If you struggle with the first one, go refresh your spherical coordinates.

Answer 2

For the image of $f$ think of the following:

It is a part of the sphere where $\theta\in [0,\pi/2]$ and $\phi\in (0,\pi/2]$. Here $\rho = 1$.

The following answer "Limits of integration spherical coordinates" shows a nice picture as well.