Volume element in Spherical Coordinates $dxdydz$

integrationmultivariable-calculusvolume

So I know in Cartesian coords $dV = dxdydz$. I also know, that in Spherical coordinates, $dV = Jd\phi d\theta dx$ where $J =\frac{\partial(x,y,z)}{\partial(r,\phi,\theta)}$. However, when I find the differentials of x,y,z, as below,

$ π‘₯=π‘Ÿsinπœƒcosπœ™,
𝑦=π‘Ÿsinπœƒsinπœ™,
𝑧=π‘Ÿcosπœƒ$

$𝑑π‘₯=sinπœƒcosπœ™π‘‘π‘Ÿ+π‘Ÿcosπœ™cosπœƒπ‘‘πœƒβ€“π‘Ÿsinπœƒsinπœ™π‘‘πœ™$

$𝑑𝑦=sinπœƒsinπœ™π‘‘π‘Ÿ+π‘Ÿsinπœ™cosπœƒπ‘‘πœƒ+π‘Ÿsinπœƒcosπœ™π‘‘πœ™$

$𝑑𝑧=cosπœƒπ‘‘π‘Ÿβ€“π‘Ÿsinπœƒπ‘‘πœƒ$

If I calculate $dxdydz$ it does not equal what it should equal. There are $dr^3$ and $d\theta^3$ terms included. I don't understand why this is true? Can someone help me understand?

Best Answer

Consult the Wikipedia page, for instance. Coordinate changes change the volume element by the jacobian. Your expressions for $\operatorname dx, \operatorname dy$ and $\operatorname dz$ are correct. But when you multiply them, you actually have an exterior, or wedge, product of differential forms. Instead of $\operatorname dr^3$, you'll have $\operatorname dr\wedge \operatorname dr\wedge\operatorname dr=0$. And so on. I instruct you to do some reading on the subject.

Use the fact that the wedge product is anticommutative and you can get it to work out.

Maybe try Calculus on Manifolds by Spivak, among others.

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