(I don't know why this question posted by me has been migrated here from physics stackexchange website; do I need to tell people in physics group that there exists "mathematical physics" tag in physics website? Do I need to tell people that electrodynamics which is nothing but vector field theory is physics stuff ? !!)

Consider spherical co-ordinate system. Let vector field be $ \vec V = k \vec r $.

$k$ is constant.

Consider a sphere of radius $R$.

I want to verify divergence theorem using this sphere and above mentioned vector field.

(a) When centre of sphere is at the origin of co-ordinate system, it is easy to verify divergence theorem.

$ \int_{vol} \vec \nabla.\vec V d \tau = \int_{surf} \vec V . \vec{da} $

We solve LHS, we get $ 4k \pi R^3$. We solve RHS, we get $ 4k \pi R^3$. And theorem is verified for this example of sphere.

(b) Now let us suppose centre of sphere is not at the origin (0,0,0); but at any but different location e.g. at $x=2R, y=2R, z=2R$. Evaluation of surface integral using spherical co-ordinates becomes hard in this case. Can anybody show how to evaluate this surface integral in spherical co-ordinate system?

By the way, I will be grateful if I get answers considering that I am a physicist and not mathematician!


migrated from physics.stackexchange.com Feb 12 '18 at 11:37

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  • $\begingroup$ Why do you expect the choice of the axes' origin to be relevant? $\endgroup$ – valerio Feb 12 '18 at 11:34
  • $\begingroup$ because choice of origin is necessarily relevant to the question itself. $\endgroup$ – atom Feb 12 '18 at 11:36

This problem is equivalent to keeping the origin of coordinates fixed and instead integrate the flux of $\vec{V}'= k \vec{r} + \vec{b}$, with $\vec{b}$ a constant vector that takes care of the shift in position. The divergence of this vector is still $\nabla \cdot \vec{V}'= 3k \, .$ So we should find that $\int_{\mathcal{S}} \vec{b} \cdot \mathrm{d} a = 0$. This is simple to check: the flux of a uniform vector over a sphere vanishes.


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