Real analysis is the branch of mathematics concerned with the analytic properties of real numbers and real-valued functions. Mathematics 370: Applied Real Analysis is intended as a first course in real analysis. It includes units on real numbers, series, topology, functions, differentiation and integration, and norms and inner products. In addition, you will study the application of real analysis to dynamical systems.

Outline

Unit 1: Introduction and Review

Unit 2: The Real Numbers

Unit 3: Series

Unit 4: The Topology of ℝ^{n }

Unit 5: Real-Valued Functions

Unit 6: Differentiation and Integration

Unit 7: Norms and Inner Products

Course Project: Dynamical Systems

Learning outcomes

Upon successful completion of this course, you should be able to

demonstrate a foundational understanding of real analysis through the statement, proof, and application of key theorems.

define the completeness property of the real numbers in terms of limits of sequences, subsequences, and the Cauchy sequence.

define the concepts of convergence, completeness, and compactness in the context of ℝ^{n}.

differentiate between continuous and discontinuous real-valued functions, as well as between uniform continuity and continuity.

distinguish between compactness, the existence of extreme values, and the intermediate value theorem, and list their implications.

demonstrate a foundational understanding of differentiation and integration, culminating with the fundamental theorem of calculus.

define and identify normed vector spaces and demonstrate a basic understanding of the topology of normed vector spaces, paying particular attention to inner product spaces.

apply the principles of real analysis in the field of dynamical systems.

communicate mathematical ideas and analyses in a clear and organized manner.

Evaluation

To receive credit for MATH 370, you must achieve a course composite grade of at least D (50 percent), submit all six course assignments, and achieve a grade of at least 50 percent on each of the two assessments. The weighting of the composite grade is as follows:

Activity

Weight

Assignments 1–6 (5% each)

30%

Course project (dynamical systems)

25%

Midterm assessment

20%

Final assessment

25%

Total

100%

To learn more about assignments and examinations, please refer to Athabasca University’s online Calendar.

Materials

Davidson, K. R., & Donsig, A. P. (2010). Real analysis and applications: Theory in practice. Springer. (Print)

Challenge for credit

Overview

The challenge for credit process allows you to demonstrate that you have acquired a command of the general subject matter, knowledge, intellectual and/or other skills that would normally be found in a university-level course.

Full information about challenge for credit can be found in the Undergraduate Calendar.

Evaluation

To receive credit for the MATH 370 challenge registration, you must complete the two parts of the challenge exam—written on the same day or within two consecutive days—and achieve a grade of at least D (50 percent) on each part.

Athabasca University reserves the right to amend course outlines occasionally and without notice. Courses offered by other delivery methods may vary from their individualized study counterparts.