Mathematics Optional Calculus & Real Analysis

Real numbers, functions of a real variable, limits, continuity, differentiability, mean value theorem, Taylor's theorem with remainders, indeterminate forms, maxima and minima, asymptotes; curve tracing; functions of two or three variables, partial derivatives, maxima and minima, Lagrange's method, Jacobian; definite and improper integrals, double and triple integrals, areas and volumes.

Scalar and vector fields, gradient, divergence and curl in Cartesian and cylindrical coordinates; vector identities; application to curves in space, curvature, torsion; Serret-Frenet formulae; Gauss, Stokes' and Green's theorems.

Real number system, sequences, Cauchy sequences, convergence of series, continuity, uniform continuity, Riemann integrals, improper integrals, fundamental theorems of calculus, uniform convergence of sequences and series of functions, partial derivatives of functions of several variables, maxima and minima.