Multivariate Calculus: Lecture 7 part 1: surfaces and coordinates

I once more discuss quadratic surfaces. We play the identification game a few minutes. Then we examine several parametrizations. We see how choosing a domain for the parameters can select just part of the surface in question. Then, I turn our attention to cylindrical and spherical coordinates where we find another way to uncover parametrizations. When we can use the cylindrical, spherical or cartesian coordinates as a parameter for a given surface then that allows us to use the formulas which are known for the coordinate systems to simplify our thinking. For arbitrary parametrizations, you just have to do or "see" the algebra. That, is an art.