# MAT 403 Advanced Analysis II

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### Revised Course Description

MAT 403 covers the theoretical foundations of differential and integral calculus: differentiation, integration of functions of one and several variables, and applications. Additional topics chosen by the instructor.

MAT 403 meets for three hours of lecture per week.

### Prerequisites

Students must have earned at least a "C" in a course equivalent to MAT 401 Advanced Analysis I to be eligible to enroll in MAT 403.

### Objectives

After completing MAT 403 the student should know, understand and be able to use techniques and concepts from elementary topology and real analysis to prove basic facts about differentiation and integration of functions on R^n including:

• sums, products, and quotients of differentiable functions are differentiable where defined
• the chain rule
• the mean value theorem
• definition of Riemann integrability
• continuous functions are integrable on compact sets
• basic rules of integration (sum rule, constant multiple rule)
• integration by substitution
• integration by parts
• approximate integration
• additional topics chosen by the instructor.

The student must be able to solve concrete problems and write clear and coherent mathematical arguments incorporating these techniques and concepts.

### Expected outcomes

Students should be able to demonstrate through written assignments, tests, and/or oral presentations, that they have achieved the objectives of MAT 403.

### Method of Evaluating Outcomes

Evaluations are based on homework, class participation, short tests and scheduled examinations covering students' understanding of the foundations of integration and differentiation and related topics that are covered in MAT 403.

### Text

Advanced Calculus, by R. Creighton Buck. McGraw-Hill, 1978.

#### Chapters from the Text

• 3. Differentiation
• 3.4. Differentiation of Composite Functions
• 3.5. Taylor's Theorem
• 3.6. Extremal Problems
• 4. Integration
• 4.1. Preview
• 4.2. The Definite Integral
• 4.3. Evaluation of Definite Integrals
• 4.4. Substitution in Multiple Integrals
• 4.5. Improper Integrals

The instructor may choose to cover supplementary material or additional chapters:

• 5. Series
• 6. Uniform Convergence [includes power series, improper integrals with parameters, gamma function, Fourier series]
• 7. Differentiation of Transformations [includes implicit and inverse function theorems]
• 8. Applications to Geometry and Analysis
• 9. Differential Geometry and Vector Calculus
• 10. Numerical Methods.

Students' grades are based on homework, class participation, short tests, and scheduled examinations covering students' understanding of the topics covered in MAT 403. The instructor determines the relative weights of these factors.

### Attendance Requirements

Attendance policy is set by the instructor.

### Policy on Due Dates and Make-Up Work

Due dates and policy regarding make-up work are set by the instructor.

### Schedule of Examinations

The instructor sets all test dates except the date of the final exam. The final exam is given at the date and time announced in the Schedule of Classes.