Advanced Calculus (M.S., Applied M.S., Ed.D.)

Preparatory Courses: Math 4143, 4153 Advanced Calculus involves a rigorous development of the calculus of one and several variables which prepares the student for further study in statistics, probability, differential equations, analysis, etc. The student should be familiar with the computational aspects of calculus and linear algebra and have some experience with rigorous arguments.

1.

Real numbers: Algebraic properties, completeness, countable and uncountable sets, sequences and series of constants, tests for convergence, and partial summation

2.

Metric spaces: Topology of $Rn$, Cauchy-Schwarz Inequality, completeness, compactness, continuity, and uniform continuity

3.

Calculus of functions of one variable: differentiation, Riemann integration, monotone functions, Riemann-Stieltjes integration, and improper integrals

4.

Power series: Uniform Convergence, operations on power series, Taylor's Theorem

5.

Sequences and series of functions: Weierstrass M-Test, integration and differentiation of sequences nd series of functions, Weierstrass approximation theorem

6.

Calculus of functions on $Rn$: Partial derivatives, differentials, tangent planes, chain rule, Jacobians, multiple integration, Fubini theorem, functions defined by integrals, and the inverse and implicit function theorems

7.

Curves and surfaces: Parameterization of curves and surfaces, rectifiable curves, arc length, line and surface integrals, Green's theorem

REFERENCES: T.M. Apostol, Mathematical Analysis; R.C. Buck, Advanced Calculus; Walter Rudin, Principles of Mathematical Analysis