Differential Geometry: Lecture 4 part 2: Jacobian and push-forward

I define the Jacobian matrix for a mapping from Rn to Rm. We see how it contains many directional derivatives at once. Then, we change focus to study the transport of curves by such a mapping and the corresponding push-forward of tangent vectors. This, ideally, motivates the definition of the push-forward which I offer. Incidentally, there is a different definition in Oneill which is just based on curves, but, I'm trying to stick with derivations, even in inelegant situations. Finally, I mention briefly the content of the implicit and inverse function theorems. everything in this lecture is treated in greater depth in my usual advanced calculus course. Lecture 5 is where we begin the real content of this course.