Credit Points: 4
Prerequisites: Differential and Integral Methods; Linear Algebra
The real numbers as an ordered field, limits of infinite sequences, divergence, uniqueness and arithmetics of limits, limits of special sequences. Bolzano-Weierstrass theorem.  Infinite series as limits of partial sums, summation of special series, divergent series, convergence criteria and the ratio test, the nth root test, Leibniz’s theorem, absolute and conditional convergence, reordering theorem. Improper integrals, Euler’s function, the comparison test. Power series: Cauchy-Hadamard theorem, differentiation and integration, multiplication of power series. Convergence of sequences of functions and series of functions, piecewise convergence, Weierstrass “M” test. Exchange of the limit sum integral and derivative. Differentiation and integration with respect to parameters. Functions of two variables: limits and continuity, interactive limits, partial derivatives, the total differential and chain rule, changing the order of derivatives. The Jacobian, implicit differentiation theorem Taylor’s with the remainder, extreme points, Lagrange multipliers for funding the values of the extremas. Double integrals: Existence conditions, changing variables and Jacobians in polar, cylindrical and spherical coordinates. Applications for calculating areas, volumes, moments and center of mass. Surface integral of the first kind orientation on a surface with a continuous normal and integral of the second type. Green’s, Gauss and stokes theorems. Field theory.