# Gauss theorem

## Main Question or Discussion Point

How would you go about confirming the Gauss theorem using cylindrical co-ordinates? Could it be just like Cartesian co-ordinates, or what is the transformation?

Pengwuino
Gold Member
The idea is the same but your integration variables and limits and infinitesimals have to change appropriately (that is, for example, dxdydz does not just become say, $$drd\theta dz$$)

Gauss's theorem (if you mean the divergence theorem) is a special case of the Stokes theorem, of which there is a simple coordinate-independent proof.
However, you can prove it using the basic Gibbs formulation of vector calculus as well. The gist is the same as the poster states above; the volume element (dxdydz for 3-dimensional Euclidean space with the standard Cartesian coordinate system) changes when you change coordinate systems by the Jacobian of the coordinate transformation. Have you learned the Jacobian, or more generally, derivatives of multivariable vector-valued functions (which your coordinate transformation to cylidrical coordinates is) yet?
In case you have not, or do not see why you need to change the volume element, consider the basic 1 cubic unit of volume in 3-dimensional Euclidean space, which is just 1 x unit by 1 y unit by 1 z unit, analogous to the dxdydz. In analytic terms, the dx dy dz comes from limiting form of a cube with sides delta-x, delta-y and delta-z.
Consider that you want to find the volume of the same cube but you are using cylindrical coordinates. Now your fundamental unit cannot be simply delta-r*delta-theta*delta-z, as that forms an extruded sector when integrated over 1 unit of r, 1 unit of theta, and 1 unit of z, not a unit cube. You must correct this by a factor relating the change in volume between the coordinate systems, which is where the derivative comes in, and more fundamentally the determinant (determinants are fundamentally related to volume).
Your text should have an exact derivation.

SpY];2833397 said:
How would you go about confirming the Gauss theorem using cylindrical co-ordinates? Could it be just like Cartesian co-ordinates, or what is the transformation?
Do you have a specific problem where you have to apply this? In that case we could be more concrete...