MATHEMATICAL ANALYSIS II

The Euclidean space R^. Functions between Euclidean spaces, limit and continuity of functions. Differentiation of vector-valued functions of a single variable, application in mechanics and differential geometry, polar, cylindrical and spherical coordinates. Differentiable functions (partical derivatives, directional derivatives, differential). Vector fields. Gradient-divergence-curl. Fundamental theorems of differentiable functions (mean value theorem, Taylor). Inverse function theorem. Implicit function theorems. Functional dependence. Local and conditional extremes. Double and triple integrals: definitions, integrability criteria, properties. Change of variables, applications. Contour integrals: Contour integral of the first and second kind, contour integrals independent of path, Green’s Theorem, simply and multiply connected domains of R^2 and R^3. Elements of surface theory. Surface integrals of the first and second kind. Fundamental theorems of vector analysis (Stokes and Gauss) applications.

DIFFERENTIAL EQUATIONS

Introduction to differential equations (definitions). First order differential equations (separable variables, total differential and Euler multiplier, linear, Bernoulli, homogeneous Riccati, Clairaut, Lagrange). Qualitative theory of differential equations (general). Higher order linear differential equations (general theory). Linear differential equations with constant coefficients (solution of linear equations, variation of parameter method, method of undetermined coefficients, Euler’s differential equations, applications).