This course develops the calculus of real and complex valued functions in depth. The emphasis throughout is on careful argument and proof. After clearly stating the properties of the real numbers that we accept as given, we develop in detail the basic topics of mathematical analysis, proving all claimed results precisely. These topics include: euclidean space topology, numerical sequences and series, continuity of functions, differentiation of functions, integration theory, uniform convergence, and special functions. This Advanced Calculus sequence serves as a pillar of mathematics at the undergraduate level, preparing one for advanced course work at the graduate level.
Students who intend to take Measure and Integration (and perhaps subsequent courses such as Stochastic Calculus) need to have the knowledge and mathematical maturity developed by a thorough study of the principles of mathematical analysis, at the level of Advanced Calculus I-II.
The purpose of this capstone course is for students to bring together knowledge from previous courses to read current research, formulate specific project ideas, develop computational experiments to support their own conclusions, hone written and oral presentation skills, and practice teamwork to produce a polished final product under time-limited conditions.