Topics treated in this course are: Calculus for functions of several variables, sequences and series of functions, ordinary differential equations, Lebesgue measure and integral. Emphasis will be given to examples and applications.
i) Calculus in several variables. Neighborhoods in several variables, continuity in several variables, directional derivatives, differential of functions in several variables, Theorem of Total Differential, gradient of scalar functions, Jacobian matrix for vector-valued functions, level curves of scalar functions. Parametrized surfaces, tangent and normal vectors, changes of coordinates. Higher order derivatives and differentials, Hessian matrix, Schwarz's Theorem, Taylor's Series.
(ii) Optimization problems for functions in several variables. Critical points, free optimization, constrained optimization, Lagrange's Multiplier Theorem, Implicit and inverse function theorem, Contraction Principle.
(iii) Integral of functions in several variables. Fubini and Tonelli theorems, integral on curves, change of variables formula.
(iv) Integral of scalar function on surfaces, vector fields, conservatice vector fields, scalar potentials, curl and divergence of a vector fields, introduction to differential forms, closed and exact forms, Poincare lemma, Gauss-Green formulas.
(v) Flux through surfaces, Stokes' Theorem, Divergence Theorem
(vi) Introduction to metric spaces and normed spaces, spaces of functions, sequence of functions, uniform convergence, function series, total convergence, derivation and integration of a series of functions.
(vii) Introduction to Lebesgue's Measure Theory. Measurable sets and functions, stability of measurable functions, simple functions, approximation results, Lebesgue integral. Monotone Convergence Theorem, Fatou's Lemma, Dominated convergence Theorem and their consequences.
(viii) Ordinary differential equation, existence and uniqueness results, Cauchy-Lipschitz's Theorem. Extension of a solution, maximal solution, existence and uniqueness results for systems of ODE, linear ODE of order n, Variation of the constants method,
other resolutive formulas.
(ix) Fourier's series for periodic functions, convergence results, application to solutions of some PDE.