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.

 

[tacman]

The connection coefficients play a crucial role in differential operators, such as the Laplacian, divergence, curl, and gradient. These coefficients represent the change in the basis vectors as we move from one coordinate system to another. In the expression for the Laplacian, the connection coefficients enter by accounting for the change in the metric tensor as we move from one coordinate system to another.

For other operators, such as divergence and curl, the connection coefficients enter in a similar way by taking into account the change in the basis vectors and metric tensor. However, the exact expressions for how they enter may vary depending on the specific operator and coordinate system being used.

One way to determine the general expressions for how the connection coefficients enter into these operators is by using the Christoffel symbols. These symbols represent the connection coefficients in terms of the metric tensor and its derivatives. By using these symbols, we can derive the general expressions for how the connection coefficients enter into different operators.

Alternatively, specific expressions for the connection coefficients can also be derived for different coordinate systems, such as Cartesian, polar, or spherical coordinates. These expressions can then be used to determine how the coefficients enter into the various differential operators.

In conclusion, the connection coefficients play a crucial role in various differential operators, and their specific expressions for how they enter may vary depending on the operator and coordinate system being used. The use of Christoffel symbols or specific expressions for different coordinate systems can help determine the general expressions for how the connection coefficients enter into these operators.