Here a summary of calculus in several variables, also known as vector’s calculus or vector analysis. Here also will re-engineer it, incorporating the basic elements of the multilinear algebra already devised.
Pointing to the most important tool i would say that the chain’s rule is the one. The chain’s rules is about the derivative of a composition of functions and where it is absolutely essential when applied to the geometry of surfaces, by the way, the ideal arena where the elementary concepts of curvature are showed.
We will jump in it as we finished the multilineal algebra of euclidean vector spaces.
- scalar fields
- vector fields
- map composition
- parameterizations of curves and surfaces
- partial derivatives, derivations
- Leibnitz’ rule
- chain’s rule
- regular points and regular values
- critical points and critical sets
- surfaces as a pre-images of regular values
- total differential
- euclidean differential forms
- exterior derivative
- wedge product
- gradient, divergence and rotational
- the de Rham’s complex
- integration on forms
- Stokes’s theorem
PDF draft in spanish: https://juanmarqz.files.wordpress.com/2009/10/lecciontres.pdf