$\frac{\partial \varphi}{\partial x}$ from the much simpler

$\frac{\partial x}{\partial \varphi}$. Note though that here we

are dealing with functions depending on several variables, so the

*Jacobian* has to be used (see [Wikipedia](https://en.wikipedia.org/wiki/Inverse_function_theorem). A direct calculation is in this particular case more easy.

*Jacobian* has to be used (see [Wikipedia](https://en.wikipedia.org/wiki/Inverse_function_theorem)). A direct calculation is in this particular case more easy.

Note that this procedure also carries over to other coordinate systems,

although the calculations can become quite tedious. In these cases,

...

...

@@ -344,15 +344,15 @@ Problems

3. Find the spherical coordinates of the points

$${\bf r} = (3/2, \sqrt{3}/2, 1).$$

2. [:grinning:] *Geometry and different coordinate systems}*

2.[:grinning:] *Geometry and different coordinate systems}*

What geometric objects do the following boundary conditions describe?

What geometric objects do the following boundary conditions describe?

1. $r<1$ in cylindrical coordinates,

2. $\varphi=0$ in cylindrical coordinates,

3. $r=1$ in spherical coordinates,

4. $\theta = \pi/4$ in spherical coordinates,

5. $r=1$ and $\theta=\pi/4$ in spherical coordinates.

1. $r<1$ in cylindrical coordinates,

2. $\varphi=0$ in cylindrical coordinates,

3. $r=1$ in spherical coordinates,

4. $\theta = \pi/4$ in spherical coordinates,

5. $r=1$ and $\theta=\pi/4$ in spherical coordinates.

3. [:smirk:] From the transformation from polar to Cartesian