Connection coefficients entering differential operators

[masudr]

I have worked out how the connection coefficients enter into the expression for the Laplacian, for example, in different coordinate systems.

Does anyone have general expressions for how the coefficients enter into other expressions such as divergence and curl and grad and so on?

Thanks in advance.

 

[Chris Hillman]

Not sure I understand the question...

Hi, masudr,

I have worked out how the connection coefficients enter into the expression for the Laplacian, for example, in different coordinate systems.

Does anyone have general expressions for how the coefficients enter into other expressions such as divergence and curl and grad and so on?

Thanks in advance.

Are you asking about Christoffel coefficients or connection one-forms? What do you mean by "enter into the expression for"? I can't tell if you are trying to work out an expression using index gymnastics which is valid for any coordinate chart (components taken wrt the coordinate basis), or something else entirely. Also, it might be relevant to say whether you are working in special dimension such as three, or what kinds of objects (forms? vector fields?) you are trying to evaluate a "div" or "curl" on. In absence of clarification I assume you are talking about elementary vector calculus in E^3.

If you simply want to know how to conveniently compute grad and curl in popular charts on E^3, such as cylindrical, polar spherical (trig or rational), paraboloidal, oblate spheroidal (trig or rational), prolate spheroidal (trig or rational), harmonic charts such as
ds^2 = dz^2 + \frac{-u+\sqrt{u^2+v^2}}{4 \, (u^2+v^2-u \sqrt{u^2+v^2)}} \; \left( du^2 + dv^2 \right)
(in this example the transformation to a Cartesian chart is given by u = x^2-y^2, \; v = 2xy), etc.--- all of these find application in electromagnetism--- then the most efficient route is probably to use exterior calculus with a frame field. I long ago explained why in a series of posts to sci.physics called something like "The Joy of Forms". Working these computations is a great way to see the power and convenience of exterior calculus! See also the classic and very readable textbook Harley Flanders, Differential Forms, with Applications to the Physical Sciences, Dover reprint.