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.
http://banach.millersville.edu/~bob/.../Laplacian.pdf
Just as a side not with the use of differential forms you can derive the Laplacian and many other operators from vector calculus.
In cylindrical coordinates the basis of 1-forms is
and the Laplacian is given by
Where * is the Hodge dual
This gives
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