8. Analytical Mathematics (Calculus)

Short description, basics of logic and set theory. Real numbers numerical sequences and series,cartesian coordinates. Differential and integral calculus for real functions of several variables, limits, continuity, partial derivatives, gradient, differential, implicit functions and Parametric curves, curvilinear integrals of scalar fields. Multiple integrals, parametric surfaces and surface integrals, differential forms and conservative fields. Gauss, Green, and Stokes theorems. Sequences and series of functions, pointwise and uniform convergences. Fourier series of periodic functions. Euler formula. Real functions, derivatives and differentiation rules, properties of differentiable functions on intervals, searching maxima and minima, differential. Taylor polynomial and Taylor formula. Convex functions, qualitative study of the graph of a function. Riemann proper integral. Fundamental Theorem of Calculus. Integration methods. Cauchy problem, existence and uniqueness, continuous dependence on data, regularity and extension of the solution. Qualitative study of solutions to first order differential equations. phase space, stability analysis in the linear case and in the nonlinear case, Discrete dynamical systems, graphical and qualitative analysis, equilibria and periodic orbits, stability criteria, bifurcation, logistic growth, chaotic phenomena. Analytic functions. Functional analysis. Distributions. Fourier transform, Laplace transform and Z transform.

9. C-Programming Language

Short description , basis concepts , overview of computers and programming , algorithm, data type, expression type, parameter passing among functions, local and global variables and the recursive programming.

10. Ordinary Differential Equations

First order equations, explicit first order equations. The linear differential equation. Implicit first order differential equations, complex differential equations, the higher differential equation power series expansions, upper and lower solutions. Maximal and minimal integrals.

## Tuesday, March 25, 2008

### Course description 3

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