# 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

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