Integrals in polar coordinates

Question

Polar and spherical coordinates seem very useful for areas, however I don't understand why I can't seem to keep a direction after a spherical integral. In Cartesian coordinates, it's very easy to integrate component by component and end up with a vector, but is it possible to do the same in polar/spherical?

Answer 1

Can "keep a direction" if you understand the underlying idea. See the formulas at http://plaza.obu.edu/corneliusk/mp/suv.pdf.