Divergence in spherical coordinate system

[PrakashPhy]

I have been studying gradience; divergence and curl. I think i understand them in cartesian coordinate system; But i don't understand how do they get such complex stuffs out of nowhere in calculating divergence in spherical and cylindrical coordinate system. Any helps; links or suggestion referring del operator in these coordinate system would be appreciable.

 

[Muphrid]

The simplest answer is that in spherical coordinates, the basis vectors change based on position, so they get differentiated too. The exact form of del can be computed using the Jacobian matrix and the expression for del in Cartesian.

 

[algebrat]

Ooh playing with rates (differentials and partial derivatives, aka covectors and vectors) is really cool in changes of coordinates.

In Boas' "Mathematical Methods of the Physical Sciences", there is a pretty quick treatment (two or three sections) of curvilinear coordinates and these infinitesimals (although I've never gotten around to figuring out how to marry Boas' notation with Jack Lee's in Introduction to Smooth Manifolds, which is harder to dig through quickly, but seems to have a more consistent development).

I conjecture that the main idea is finding the relation between dx and dθ, and ∂/∂x and ∂/∂θ. I'm not sure without trying it out this moment, but you might be able to find these relations yourself.

And the rest follows -maybe- if you're careful. ∇=(∂_x,∂_y). Hmm, I've got to work on some other stuff, so I have to leave it there.