Complements of Mathematics for Mathematicians and Physicists

Set theory. Cardinality. Axiom of choice.Construction of integer, rational and real numbers. Completeness of R. Limits of sequences. Limit points, upper and lower limits. Cauchy criterion, infinite limits. Numerical series and convergence criteria.Topology of R^n.Metric spaces, completeness and the contraction theorem.Sequences and series of functions, pointwise and uniform convergence.Power series. Limits of derivatives and integrals. Functions of several real variables. Partial and directional derivatives. Differential. Totaldifferential theorem. Hessian matrix.A few hints on functions from R^n to R^m, Jacobian matrix, differentiation of composite functions.Implicit functions. Regular curves. Conservative vector fields and potentials.Differential equations and Cauchy problems. Local existence and uniqueness theorem. Linear differential equations and systems.