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
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.
- Uniform continuity: understand the concept of uniform continuity and what role it plays for integrability of continuous functions.
- 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.
- 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.
- Improper integrals and tests for convergence: Understand the concept of improper integrals and know how to test improper integrals for convergence.
- 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.
- 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.
- Sequences of functions: Distinguish between pointwise and uniform convergence and apply Weierstrass Test to test uniform convergence of series.
- Power series: calculate the radius of convergence for power series and determine the radius of convergence of integrated and differentiated power series.
- 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).
- 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).
- 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.
Apostol, T.M., Calculus, Volumes I and II, second edition, Wiley