Derivation of the divergence in the spherical coordinate

[elgen]

Hi,

I am reviewing some vector calculus and have a problem on the derivation of the divergence in the spherical coordinate.

Assume there is a small volume located at $$r_0, \theta_0, \phi_0$$ with a volume of $$r_0^2\sin\theta_0 \Delta r \Delta \theta \Delta \phi$$.

My question is that why there is an $$r^2$$ sitting behind the partial derivative operator? The problem is attached.

My feeling is that as I take the limit of the volume going to 0, I can cannot cancel $$r_0^2$$. However, I cannot give a good justification to that.

Would anyone knowledgeable with vector calculus give an explanation? Thank you.


elgen

 

[fresh_42]

The procedure is described here: https://en.wikipedia.org/wiki/List_of_common_coordinate_transformations#From_spherical_coordinates

If we change coordinates in an integral we have to adjust the volume elements by the determinant of the transformation matrix, and this determinant is $r^2 \sin \vartheta$. It is the transformation theorem for integrals. In words: you must not neglect the radius, as it is part of the formulas for volume. Without it you would set $r=1$ which makes no sense if we want to consider an infinitesimal volume element,