Mathematical Analysis II

Real functions of several variables. Euclidean spaces. Topological properties of subsets of Euclidean metric space. Limits and continuity. Partial derivative, the concept of directional derivatives. Total differential and the gradient function. Applications. Geometric interpretation gradient, outline methods steepest descent method. Discussion relationships between the fundamental concepts of calculus. Differentials of higher orders, Taylor polynomials, Taylor's theorem. Theorem of implicit function. Weierstrass theorem on the global extrema, local extrema. Criteria existence of local extreme. Constrained local extrema, Lagrange multipliers method. Search global extremes - practices. Riemann double integral, basic properties. Fubini phrases in double integrals. Substitution theorem for double integrals, applications of double integrals Riemann triple integrals, basic properties. Fubini theorems for integrals. Substitution theorem for integrals. Applications. First order differential equations, the theorem on the existence and uniqueness of the Cauchy problem. Linear differential equation 1 order, the equation with separated variables.