An introduction to proof techniques, with a focus on topics relevant to computer science. Topics include: fundamental proof techniques, boolean logic, sequences and summations, set theory, algorithm analysis, recursion, mathematical induction, recurrence relations, an introduction to number theory, combinatorics, discrete probability, and graph theory. The class will be adequate preparation for students choosing to continue on the pure math track (Real Analysis, Abstract Algebra, etc.) or the theoretical computer science track (Analysis of Algorithms, Theory of Computation, etc.).
A study of solution techniques and models in ordinary differential equations including first order equations, linear differential equations, series solutions, Laplace transform methods, and concepts of numerical and graphical techniques applied to equations and systems.
A study of the differential and integral calculus of functions of several variables together with an introduction to vector calculus. Topics include partial derivatives, directional derivatives, gradients, tangent planes to surfaces, double and triple integrals, change of variables in multiple integrals, vector fields, line integrals, Green's Theorem, and surface integrals.
The algebra and geometry of complex numbers, sequences and series of complex numbers, derivatives, and integrals of functions of a complex variable. The Cauchy-Goursat Theorem, the Cauchy Integral Formula and its consequences, Taylor series, classification of singularities, the Residue Theorem, Laurent series, harmonic functions, conformal mappings, and, if time permits, miscellaneous applications.