MA 281 Honors Mathematical Analysis

MA 281 - Honors Mathematical Analysis III

Covers the same material as that dealt with in MA 221, but with more breadth and depth.

Prerequisites:    MA 182

Course Objectives

This is the second installment of a 3-course sequence in mathematical analysis for beginners. The course is intended for math majors and students with a serious mathematical interest. The topics covered will be approximately those of a standard Calculus II course (MA116). The style and format, though, will be rather different from the standard Calculus sequence. A higher level of mathematical rigor, and a deeper understanding of the few fundamental concepts involved, will be preferred to a more extensive syllabus.

Learning Outcomes

  1. Uniform continuity: understand the concept of uniform continuity and what role it plays for integrability of continuous functions.
  2. Taylor polynomials and error estimates: Know how to approximate functions by Taylor polynomials, how to evaluate the error of approximation, be familiar with Landau’s symbols o and O and be able to apply Taylor polynomials to calculating limits of indeterminate form.
  3. Hyperbolic functions and inverse trigonometric functions: Demonstrate a working knowledge of hyperbolic and inverse trigonometric functions and be able to derive derivatives for trigonometric functions using the implicit differentiation technique.
  4. Improper integrals and tests for convergence: Understand the concept of improper integrals and know how to test improper integrals for convergence.
  5. Series, convergence: Distinguish between the necessary and sufficient conditions for series convergence, recognize special types of series (geometric and alternating) and understand the concepts of absolute and conditional convergence.
  6. Test for series convergence: Apply convergence tests such as the comparison test, integral test, ratio test, root test and Leibnitz’s rule for corresponding types of series.
  7. Sequences of functions: Distinguish between pointwise and uniform convergence and apply Weierstrass Test to test uniform convergence of series.
  8. Power series: calculate the radius of convergence for power series and determine the radius of convergence of integrated and differentiated power series.
  9. Taylor series: Know sufficient conditions for a function to be represented in the form of Taylor series, determine radius of convergence of Taylor series, be able to give an example of a function, which cannot be represented by a Taylor series and know Taylor series of elementary functions (sine, cosine, logarithm, exponential).
  10. Basic operations of vector algebra: Demonstrate a working knowledge of basic operations with vectors (addition, subtraction and dot and cross products) and a working knowledge of basic concept of vectors (vector norm, orthogonal vectors and angle between two vectors).
  11. Scalar and vector fields: Understand limits of functions of many variables and demonstrate a working knowledge of directional and partial derivatives, total differential and gradient of a scalar field.

Textbooks

Apostol, T.M., Calculus, Volumes I and II, second edition, Wiley