Courses
This is the site for the courses I am currently teaching. If you experience problems viewing certain symbols, please note that the encoding of this page is UTF-8. Also, some MS Windows fonts do not have a complete set of characters. You should see the page like that (screenshot of an older version of this page). All this stuff is written in spanish, I hope to translate it into english someday...
Update: All the info for current courses will be available at http://www.fc.uaslp.mx/i/moodle/ (and no longer in this page...)
Old Courses
Calculus I
This is an absolutely standard first year calculus course. Maybe the only diversions from the usual path
are the study of numerical series (right after numerical sequences) and a brief excursion into the
integral calculus (we will compute the definite integral of some simple functions using Riemann's sums
viewed as series). Also, we will make use of the CAS Maxima, with its interface wxMaxima. Grab the
software here: Maxima and wxMaxima.
Info for the course here.
Calculus III
I will give an introduction to the study of the notion of differential for real functions defined on open subsets of the normed spaces ℝᵏ. It is a fairly standard course, which begins with the basics of topology we will need, and then moves on to study continuity and differentiability (partial derivatives and the differential). Later on, as applications, we will see some methods for computing extrema of functions of several variables, Taylor developments and the inverse and implicit function theorems. We will be very careful in the course when stating definitions and proving propositions, that is: although elementary in its contents, the course will have a formal character.
The historical roots of Calculus are intimately related to some basic ideas in Physics. Along the course we will try to establish some connections with the knowledge of Mechanics gained in previous semesters. Also, we will make use of modern technology: part of the course will consist in computer sessions in which we will learn to use Maxima, a free CAS (Computer Algebra System) very similar in functions and performance to Mathematica, but with the advantage of being free software!. The interface to Maxima that we will use is wxMaxima, developed by Andrej Vodopivec, a mathematician at the Ljubljana University, in Slovenia.
The materials for the course can be found here (in spanish).
A brief tutorial (in spanish) on how to use Maxima to study Calculus can be downloaded from here.
Selected topics in Applied Mathematics
We will be studying Riemannian and symplectic geometries, with a brief introduction to mechanics and geometric calculus of variations. In the last part, we will see how to generalize some previous constructions to the category of supermanifolds. Info for the course here.
Topology Seminar
I plan to give an introduction to the vast topic of algebraic topology. The idea is to cover in certain detail the fundamental group, touch briefly upon the higher order homotopy groups and then to dedicate the last part to the classification of surfaces and the theory of covering spaces, relating them to the subgroups of the fundamental group. More info here.
Mathematical Analysis I
This course is a natural follow-up of Calculus III and IV. Here we will study the local geometry of germs of differentiable functions, encoded in the algebra of germs ℰ(U), and we will do so through algebraic techniques. To be more specific, we will learn how the group of differentiable automorphisms Aut(U) acts on the different algebraic structures obtained from ℰ(U) such as derivations, 1-forms, etc. Our basic tool for doing that will be the Lie derivative with respect to a vector field.
We will also study 1-parameter groups of autmorphisms, viewing them as the local flow of a vector field, and using them to find symmetries of systems of ordinary differential equations on U ⊂ ℝᵏ. Finally, we will see an introduction to the basic objects of global geometry: differential manifolds.
The materials for the course can be found here (in spanish).
Complex Analysis I
This was a course on the "elementary" part of complex analysis. We studied complex series, holomorphic functions, Cauchy's theorem and other classical results (Liouville, isolated zeros, Identity principle, etc). Complex Analysis II deals with residues and more.
Here is the syllabus, and the problems sheets: 1, 2, 3, 4, 5, 6.
Some notes on line integrals I wrote for another course (Mathematical Methods in Engineering) may be of some use.
