Covariant derivatives in Wolfram Math

In summary: SołtysiakIn summary, the Wolfram Mathworld section on spherical coordinates lists nine covariant derivatives with respect to radius, azimuth, and zenith. The question is for examples of vectors represented by A(subscript)r, A(subscript)theta, and A(subscript)phi. These vectors appear on the right side of the equations, and can be computed with the use of Christoffel symbols and covariant derivatives. One example of a vector that can be used is the Coulomb force, where the angular parts are zero and can be easily computed in spherical coordinates. Any vector can be described in spherical coordinates using simple transformations, but it can be challenging when there is no spherical symmetry.
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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?
 
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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 [itex]A_i[/itex]. As expressions are in spherical coordinate system then subscript i must agree with names of coordinates, so then [itex] i \in {r,\theta, \phi}[/itex]. You may treat it as usual as with [itex] {x,y,z} [/itex]. 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:

[itex]
A_r = CQ/r^2
[/itex]
[itex]
A_{\phi} = 0
[/itex]
[itex]
A_{\theta} = 0
[/itex]

Here is picture of something similar:
index.jpeg

Of course You may substitute anything You want for [itex]A_{r},A_{\theta},A_{\phi}[/itex], 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 [itex](A_x,A_y,A_z)[/itex] 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
 
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FAQ: Covariant derivatives in Wolfram Math

What is a covariant derivative in Wolfram Math?

A covariant derivative is a mathematical tool used to describe how a geometric object changes along a given direction. In Wolfram Math, it is commonly used to describe the change of a vector field on a manifold.

How is a covariant derivative calculated in Wolfram Math?

In Wolfram Math, the covariant derivative is calculated using the built-in function "CovariantD". This function takes in the vector field and the direction in which the derivative is to be calculated and returns the result.

What is the difference between a covariant derivative and a partial derivative in Wolfram Math?

A partial derivative only considers the change of a function in one direction, while a covariant derivative takes into account how the function changes in all directions on a curved surface. In other words, a covariant derivative is a more general form of a partial derivative.

Can a covariant derivative be applied to any type of mathematical object in Wolfram Math?

Yes, a covariant derivative can be applied to any type of mathematical object, such as tensors, vector fields, and differential forms, in Wolfram Math. However, the calculation process may vary slightly depending on the type of object.

What are some practical applications of covariant derivatives in Wolfram Math?

Covariant derivatives are commonly used in differential geometry, general relativity, and other areas of mathematical physics. They can also be used in computer graphics and image processing to calculate the change of a vector field on a curved surface.

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