SCV0007 - MATHEMATICAL ANALYSIS B

The complex field: Gauss' plane, operations, algebraic and trigonometric form of complex numbers, De Moire formula, theorem on roots of complex numbers, the fundamental theorem of Algebra, subsets of the Gauss plane. Linear spaces, basis and dimension: scale product, triangular inequality, Cauchy-Schwarz inequality, orthonormal basis and Gram-Schmidt procedure, projections theorem. Linearity: approximation and the superposition principle, the vector space Mat(m,n), row-column product, transpost, diagonal, triangular, and symmetric matrices, rotations in the plane, representation theorem, linear maps between finite dimensional vector spaces, composition of linear maps. Determinants, Laplace theorem and properties. Binet formula, rank of a matrix, kernel and rank of linear maps, nullstellensatz theorem. Inverse of a matrix and applications to linear systems of equations, Touche-Capelli theorem. Eigenvalues and eigenvectors, diagonalization. Real functions of several variables, level curves, limits and continuity, properties. Directional derivatives, linear approximation and the tangent plane, differentiability and the gradient formula. Optimization and the method of restrictions, the increment method. Fermat's theorem, higher order derivatives, Schwarz theorem, the chain rule, taylor expansion. The Hessian matrix. Constrained optimization, the Lagrange multiplier theorem. Surfaces, vector fields and the Jacobian matrix, coordinates transformations in the plane and in the space. Curve integrals and vector fields, conservative and local conservatives fields. Divergence, Curl, Laplacian and differential identities. Multiple integrals, Riemann sums and integrability of continuous functions, simple domains and iterated integrals. Properties of the Riemann integral, change of variables in double and triple integrals, polar coordinates, spherical and elliptic coordinates. Surface integrals, the divergence theorem, the Stoke theorem.