Deriving the spherical volume element

[quickAndLucky]

I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using

$$dxdydz = \left (\frac{\partial x}{\partial r}dr + \frac{\partial x}{\partial \theta }d\theta + \frac{\partial x}{\partial \phi }d\phi \right )\left ( \frac{\partial y}{\partial r}dr + \frac{\partial y}{\partial \theta }d\theta + \frac{\partial y }{\partial \phi}d\phi \right )\left ( \frac{\partial z}{\partial r}dr + \frac{\partial z}{\partial \theta }d\theta + \frac{\partial z}{\partial \phi}d\phi \right )$$

Unfortunately, I can’t see how I will arrive at the correct expression, $r^{2}sin\theta drd\theta d\phi$.

For one reason, when completely expanded, I get terms with repeated differentials like $dr^{3}$ that don’t cancel.

Why is my method of derivation invalid?

 

[Orodruin]

You are not looking at the volumes actually spanned by the basis vectors corresponding to the spherical coordinates. The volume element spanned by three vectors $\vec a_i$ is the triple product $\vec a_1 \cdot (\vec a_2 \times \vec a_3)$. Hence, if you look at the volume of a small parallelepiped spanned by the coordinate lines with coordinate differences $dy^i$, then this is given by
$$
dV = \frac{\partial \vec x}{\partial y^1} \cdot \left(\frac{\partial \vec x}{\partial y^2} \times \frac{\partial \vec x}{\partial y^3}\right) dy^1 dy^2 dy^3.
$$

Alternatively, the volume element can be seen as the wedge product between the differentials. You must then remember that the wedge product is completely anti-symmetric and therefore anything like $dr\wedge dr = 0$.

 

[BvU]

You seem to think that $dV$ expressed in cartesian coordinates is identical to $dV$ expressed in polar coordinates. Not so. Check out https://en.wikipedia.org/wiki/Spher..._and_differentiation_in_spherical_coordinates

 

[quickAndLucky]

Thanks for the help! Using a wedge between the terms in my original expression got me to the answer!