MATHS 472: Real Analysis 2 (3)

1. Prerequisite: MATHS 471 or 571.
MATHS 572: Real Analysis 2 (3)
Syllabus
Note: Not open to students who have credit in MATHS 472.
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 571, 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 571-572 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 472). 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.
Although this course may be taught with an undergraduate course, students earning graduate credit for this
course will complete work and be evaluated on that work appropriate to graduate study. Specifically, these
students will be expected to (i) demonstrate comprehension and proper use of terminology; (ii) attain
proficiency in formal logic, foundational theory, and proof writing; (iii) master core graduate level
computation skills.

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 Graduate
Programs Committee.
9. Addendum: Suitable textbooks for this course include the following.
Rudin, W., Principles of Mathematical Analysis (3 rd Edition)
R. Stankewitz, 10/10/09 (Adapted from Maths 472 syllabus of Ahmed Mohammed, Michael Karls, Spring
2007); R. Stankewitz, 2/15/12