MA 363 Course Description

MA363 ANALYSIS I (3 Cr.)

COURSE DESCRIPTION

Prerequisite: MA 211 (Introduction to Matrix Theory and Linear Algebra) and MA 265 (Calculus III)

Offered: On Demand

Course Description

Set and functions, topological ideas, sequences. Continuity and uniform continuity. Properties of continuous functions and mean value theorems. Integration theory in one and two variables. Evaluation of double and improper integrals.

Objectives of the Course

Upon completion of MA 363, a student must be able to:

  • understand the properties of the real numbers and n-space.
  • understand and recognize legitimate proofs of theorems.
  • initiate proofs, and clearly communicate ideas.
  • understand the concepts found in the study of calculus such as continuity, uniform continuity, limits, differentiation, integration, and other listed in the outline below.
  • view the key theorems in a logical sequence, knowing where they apply and do not apply (use of counter examples).
  • have a general overview of calculus.

Course Content

  1. The System of Real Numbers
    • The properties of an ordered field
    • Natural numbers and induction
    • Sequences of real numbers
    • Completeness
      • Dedikind cut property
      • least upper bound property
      • Cauchy sequences converge
      • Nested interval theorem
    • Functions
  2. The n-space
    • Some topology
    • Functions
    • Sequences and limit points
  3. Continuity of Functions
    • Continuous and uniformly continuous functions
    • Extrema
    • Intermediate Value Theorem
    • Limits of functions
    • Points of discontinuity
  4. Differentiation
    • Law of the Mean and cauchy's Generalized Law of the Mean
    • L'Hospital's Rule
    • Partial differentiation
    • Chain rule
    • Extrema
    • Taylor's Theorem
  5. Integration
    • Upper and lower Darboux sums
    • Riemann sums
    • Upper and lower Darboux integrals and the Riemann integral
    • Existence of integrals of functions which have points of discontinuity
    • Sets with Jordan Content zero and Legesgue measure zero
    • Necessary and sufficient conditions for a function to be Riemann-integrable