textbooks said:
In the Wolfram Mathworld section on spherical coordinates there's given a list of nine covariant derivatives. The derivatives are given with respect to radius, azmuth, and zenith using the usual symbols r, theta and phi. The question is: what would be examples of the vectors whose derivatives are taken, sybolized by A(subscript)r, A(subscript)theta, and A(subscript)phi. These vectors appear explicitly on the right side of the equations. I would have expected the covariant derivatives to be of the position vector parameterized by r, theta, and phi, but not so. Anyone?
You may compute covariant derivative for any covarinat tensor, and in this case for A_i. As expressions are in spherical coordinate system then subscript i must agree with names of coordinates, so then i \in {r,\theta, \phi}. You may treat it as usual as with {x,y,z}. The proper use of Christoffel symbols, and covariant derivatives is exactly for this - for computing with this coefficients as close as in Cartesian system.
So You ask for example of vector You may put into this formulas. Here You are ( please make some picture): simple Culomb force, notice angular parts are vanish so, it is easy to compute with it just exactly in spherical coordinates:
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A_r = CQ/r^2<br />
<br />
A_{\phi} = 0<br />
<br />
A_{\theta} = 0<br />
Here is picture of something similar:
Of course You may substitute anything You want for A_{r},A_{\theta},A_{\phi}, then You may obtain interesting vector fields. If You have Sage computing environment You may create some pictures with it: http://www.sagenb.org/home/pub/216/
Any vector (A_x,A_y,A_z) may be described in spherical coordinate system by taking simple and well known transformation between coordinate systems, but when there is no spherical symmetry this may be a paint to compute with it.
Best regards ;-)
Kazek