# Covariant derivatives in Wolfram Math

1. Dec 27, 2009

### textbooks

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?

2. Dec 28, 2009

### kakaz

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:

$A_r = CQ/r^2$
$A_{\phi} = 0$
$A_{\theta} = 0$

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/ [Broken]

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

Last edited by a moderator: May 4, 2017