I 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?

Related Differential Geometry News on Phys.org

Orodruin

Staff Emeritus
Homework Helper
Gold Member
2018 Award
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 $dr\wedge dr = 0$.

Homework Helper

quickAndLucky

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

Last edited:

"Deriving the spherical volume element"

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving