Mathematics A

obligatory courses

Basic applications of integrals. Differentiation of functions of several variables, local maxima and minima, extreme values of differentiable functions of several variables on compact sets. Separable and linear differential equations. Second order linear differential equations with constant coefficients.

Definite integral. Riemann sums. Basic applications of integrals: area of plane sets bounded by graphs of two functions, volume of a solid as an integral of area of plane sections, length of a curve, area of surface of revolution, center of mass. Improper integrals. Linear spaces, including some functional spaces like

polynomials or polynomials of bounded degree, differentiable functions. Complex numbers: absolute value, argument, de Moivre's formula, exponential with complex exponent, Euler's formulae. Several-variable functions: continuity and partial derivatives, Jacobian matrix, gradient, vectors tangent to a level surface of a

differentiable function. Taylor's formula (second order). Locally extreme values and saddles. Separable differential equations. Linear differential equations of first and second order. Quasi-polynomials. Solving second order ODE's with constant coefficients and quasi-polynomial right-hand side. Some iterated integrals. More general areas, volumes and mass centers.