Let be the cylindrical coordinates in . Derive the Laplacian in cylindrical coordinates.
So what I know is that , so that the last term in the Laplacian doesn't change.
I know where I want to go (I have the expression right in front of me), but I'm not seeing how to derive the other two.
Here is a pdf for spherical and polar as well.
In cylindrical coordinates the basis of 1-forms is
and the Laplacian is given by
Where * is the Hodge dual
taking the Hodge dual gives
taking the exterior derivative gives
and taking the Hodge dual again gives
You can do the same this with spherical coordinates ( or any curvilinear coordinate system) but the basis of 1-forms is