MATHS 472: Real Analysis 2 (3)
Syllabus
1. Prerequisite: MATHS 471.
2. Course Description: The Riemann-Stieltjes integral and Fundamental Theorem of Calculus. Sequences and series of functions. Differential calculus of functions of several variables. Inverse and Implicit Function Theorems. Extremum problems. Lebesgue integration and comparison with the Riemann integral.
3. Course Objectives: This course, a continuation of MATHS 471, continues the rigorous treatment of calculus. Students completing the course will see most of the fundamental ideas of theory of integration and multivariate differential calculus treated in depth. Among these are the Riemann integral and the Fundamental Theorem of Calculus. The notion of nonzero derivative is generalized to functions of several variables (Inverse/Implicit Function Theorems). The behavior of sequences of functions under differentiation and integration is analyzed. Students completing the sequence MATHS 471-472 will gain a deep understanding of most aspects of calculus. Students will also be introduced to Lebesgue theory, an indispensable tool of integration in modern analysis. They will gain experience in writing proofs and solving problems related to the content of the course.
4. Course Rationale: This course primarily serves Mathematics majors and graduate students (since the course is taught with MATHS 572). Students who take this course will gain a better understanding of the theoretical underpinnings of calculus. Since the course provides essential knowledge required for further study in analysis, geometry, and applied mathematics, it is an essential course for students contemplating earning a doctorate in the mathematical sciences.
5. Course Content: The Riemann-Stieltjes integral and the Fundamental Theorem of Calculus;.sequences and series of functions; differentiability of functions of several variables, including partial derivatives, directional derivatives, the total derivative, the Jacobian matrix, Inverse Function theorem, Implicit Function Theorem, and extremum problems; and Lebesgue integration, including comparison with the Riemann integral and L^2 theory.
6. Course Format: lecture/discussion. The amount of material to be covered does not allow for complete proofs to be given in class of all the results which are important. The instructor is responsible for choosing a representative sample of proofs that illustrate the most common techniques. Students are expected to understand other proofs through careful reading of the text.
7. Methods of Evaluating Student Performance: Typical components are homework assignments, projects, and take-home exams (possibly involving group work), in-class exams, student presentations, and oral exams. The evaluation and weight of these components are at the discretion of the instructor. Students may be evaluated on their understanding of the content material of the course, their ability to solve problems related to the course material, and on their ability to communicate mathematically to others orally and/or in writing.
8. Evaluation of the Course: The instruction of the course is evaluated by departmental student evaluations and peer evaluation. The course is reviewed and revised periodically by the Departmental Undergraduate Programs Committee.
Ahmed Mohammed, Michael Karls, Spring 2007