# multilineal lección: re-engineering vector calculus

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
• jacobians
• 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
• the de Rham’s complex
• integration on forms
• Stokes’s theorem

PDF draft in spanish: https://juanmarqz.files.wordpress.com/2009/10/lecciontres.pdf

### One response to “multilineal lección: re-engineering vector calculus”

1. c-qit

This particular (gauge theory) lecture touches upon a topic of great practical interest in engineering, namely what the lecture calls “the challenge [of selecting] the gauge that maximizes the effectiveness of analytic methods.”

Very often in engineering, one has a global algebraic invariance that one wishes to promote to a local geometry invariance … and then link to conservation laws … would it be too much to hope for a few remarks on this topic?

Within the context of quantum simulation science, specifically within the (common) problem of dynamically simulating open quantum systems, there is a concrete example of this kind of mathematical challenge