Good resource on covariant/contravariant, magnetic coordinates, and jacobians?

[chrysaetos13]

First post here in PF, so forgive me if this question is in the wrong place.

I'm a student in computational plasma physics. The code I work with utilizes magnetic field aligned coordinates, and as a necessity, it is sometimes useful to convert between spatially regular coordinates (cartesian, cylindrical, toroidal, etc) to our field aligned coordinates. Although I have a system of equations which defines our magnetic coordinates, and a general idea that "magnetic field lines are straight in these coordinates" I find the coordinate system largely unintuitive.

Does anyone know of a good resource for developing intuition on complicated coordinate systems? I have always felt a little fuzzy even on the relatively simple description of covariant and contravariant coordinates provided in Jackson's E&M. In one sentence, what is the salient benefit of using these coordinates.

I also see descriptions of a scalar Jacobian, which I gather/guess is the determinant of the Jacobian matrix. Is this correct? I believe the matrix can be thought of as a way to convert between coordinate systems. Is there an intuitive, spatial way to think of what this scalar Jacobian value represents?

 

[Muphrid]

The determinant of a linear operator describes how the volume element is dilated or shrunk under the action of the operator. Thus it's common to see in volume integrals, as well as easy to work with because unlike the transformations of vectors, this factor is just a number.

I wouldn't describe the distinction between contravariant and covariant as one that is useful so much as necessary. You can decompose a vector in terms of vectors tangent to coordinate axes (these are contravariant components), which is what's usually done, but on a 2d plane, you can use a different basis, one that is perpendicular to all other coordinate axes. These are the covariant components, and as you change coordinate systems, the components change differently, just as a consequence of their differing definitions.

Basically, though, the Jacobian is key, and as you go back and forth between coordinate systems, individual vectors will transform by the Jacobian (or more exactly, but either its inverse or its transpose or both, depending on which coordinate system you start with).