====== Lecture 17 ======
We follow Pugh 5.2.
  * A function $f: \R^n \to \R^m$ is differentiable at $p \in \R^n$ if $f$ can be approximable by a constant plus a linear term plus a remainder. 
  * If approximation exists, then the differential is unique (and has a formula)
  * If the partial derivative exists, and is continuous, then the total derivative exists. (Proof: show the remainder is sublinear, by examine componentwise)
  * (key) total derivative satisfies all the nice properties. 
  * A function is differentiable at a point $p$, iff all its components are differentiable. 
  * Two mean value theorems: a crude one on length; a more precise one, using averaging. 
  * Differentiation commute with integration.