# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 127C: Real Analysis

**Approved:**2015-11-06, John Hunter and Janko Gravner

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

James R. Munkres, Analysis On Manifolds, Westview Press, 1997. (Chapters 1--4 provide an introduction to multi-variable calculus on R^n)

**Course Description:**

Metric spaces and multi-variable calculus.

**Suggested Schedule:**

Suggested number of lectures are indicated in parentheses.

- (1) Metric spaces. Norms with examples from R^n.
- (1) Sequences, limits, and completeness.
- (1) Open, closed, and connected sets.
- (1) Compact sets. Heine-Borel theorem.
- (2) Continuous functions on metric spaces and R^n.
- (1) Derivatives of functions of several variables. Partial derivatives.
- (1) Continuously differentiable functions.
- (1) Higher order partial derivatives.
- (3) Chain rule.
- (2) Inverse function theorem.
- (1) Definition of the Riemann integral of functions of several variables over a rectangle.
- (1) Statement of the Lebesgue condition.
- (2) Iterated integrals and the Fubini theorem.
- (1) The Riemann integral over a bounded set.
- (2) Properties of the integral.
- (2) Improper integrals.
- (3) Change of variables.

Total: 26 lecture