## Course Syllabus for "MA242: Real Analysis II"

Real Analysis II is the sequel to Saylor’s Real Analysis I, and together these two courses constitute the foundations of real analysis in mathematics. In this course, you will build on key concepts presented in Real Analysis I, particularly the study of the real number system and real-valued functions defined on all or part (usually intervals) of the real number line. The main objective of MA241 was to introduce you to the concept and theory of differential and integral calculus as well as the mathematical analysis techniques that allow us to understand and solve various problems at the heart of science—namely, questions in the fields of physics, economics, chemistry, biology, and engineering. In this course, you will build on these techniques with the goal of applying them to the solution of more complex mathematical problems. As long as a problem can be modeled as a functional relation between two quantities, each of which can be expressed as a set of real numbers, the techniques used for real-valued functions of one variable should suffice. However, most practical problems cannot be modeled via functions of a single real variable. For instance, modeling a moving particle in space requires three real variables in the three-dimensional coordinate system of real numbers. In another example from physics, the altitude a projectile will reach—a quantity described by one real variable—depends on two factors: the weight of the projectile as well as the initial velocity it has acquired from some external force. Sometimes, depending on the answer desired, a problem may be modeled as a single-variable or a multivariable function. For example, a particle moving in three-dimensional space through a force field (think of a dust particle floating in the air as it is blown by gusts of wind) may be modeled either as a function of time (a single-variable function) to describe the coordinates of the particle at each instance of time; or, if one is interested in the final resting place of the particle as a function of its initial position, the same problem may be modeled as a multivariable function that requires three inputs (the coordinates of the initial position) in order to produce three outputs (the coordinates of the resting place). In this course, you will learn about some of the intricacies of the geometry of higher-dimensional spaces, how the theory of multivariable functions is developed, and how to apply the advanced techniques of differentiation and integration to such functions. Finally, you will explore applications of these advanced techniques to the solution of complex mathematical problems.

### Learning Outcomes

Upon successful completion of this course, you will be able to:

- define and compute directional and partial derivatives;

- prove properties of the directional and partial derivatives;

- summarize the role of matrices in the theory of differentiation;

- define differentiable functions;

- prove properties of the differential;

- state, use, and prove the chain rule and the Cauchy invariant rule;

- compute and prove properties related to repeated differentiation;

- use and prove Taylor’s Theorem in higher dimensions;

- define, compute, and prove properties related to the Jacobian;

- state, prove, and use the Inverse Function Theorem;

- define and use the technique of a Baire category;

- locate local and global maxima and minima;

- apply the method of Lagrange multipliers to locate conditional maxima and minima;

- define the Riemann integral of a function ;

- define and recognize sets of zero content;

- state and prove the basic properties of the Riemann integral;

- identify Riemann integrable functions;

- relate multiple and iterated integrals;

- compute Riemann integrals using iterated integrals;

- define and recognize Jordan measurable sets;

- summarize the role of Jordan measurable sets in the theory of the Riemann integral;

- formulate and prove the Change of Variables formula;

- compute integrals with the Change of Variables formula;

- use polar and spherical coordinates;

- define differential forms, their product, the exterior derivative, and the Hodge star operation;

- compute with differential forms;

- define n-chains and identify boundaries and cycles;

- define, compute, and use the winding number of a curve;

- define the integral of k-forms on n-chains;

- state and prove Stokes’ Theorem; and

- obtain the formulas of Green, Gauss, and Stokes as special cases.

### Course Requirements

In order to take this course, you must:

√ have access to a computer;

√ have continuous broadband Internet access;

√ have the ability and permission to install plug-ins and/or
software (e.g., Adobe Reader or Flash).

√ have the ability to download and save files and documents to a
computer;

√ have the ability to open Microsoft files and documents (.doc,
.docx, .ppt, .xls, etc.);

√ have competency in the English language;

√ have read the Saylor Student
Handbook;
and

√ have successfully completed the following Saylor courses, or their
equivalents: MA101, MA102, MA103, MA211,
and MA221 from The Core
Program in Saylor’s
Mathematics discipline, as well as Saylor’s
MA241 (Real Analysis I).

### Course Information

Welcome to **MA242**. General information on this course and its
requirements can be found below.

**Course Designer:** Ittay Weiss, Ph.D.

**Primary Resources:** This course draws on a range of different free,
online educational materials, with primary use of the following online
textbooks:

- Professor Elias Zakon’s
*Mathematical Analysis: Volume II*(PDF) - Trinity University: Dr. William Trench’s
*Introduction to Real Analysis*(PDF) - Cornell University: Dr. Reyer Sjamaar’s
*Manifolds and Differential Forms*(PDF)

**Requirements for Completion:** In order to complete this course, you
will need to work through each unit and all of its assigned readings and
materials. Be sure to follow carefully the instructions for each unit
and assignment, as these guidelines are designed to lead you in an
efficient study of the material. As instructed, you also will need to
complete all the assigned problem sets within specific units and
subunits of this course. Finally, you must successfully complete and
pass the course’s Final Exam.

Please note that you will receive an official grade only on your Final
Exam. In order to pass this course, you will need to earn a 70% or
higher on the Final Exam. Your score on the exam will be tabulated as
soon as you complete it. If you do not pass the exam, you may take it
again.

**Time Commitment:** This course should take you a total of **137
hours** to complete. Each unit includes time advisories that list the
amount of time you are expected to spend on each subunit and assignment.
These time advisories should help you plan your coursework accordingly.
It may be useful for you to take a look at these time advisories and to
determine how much time you have over the next few weeks to complete
each unit, and then to set goals for yourself. For example, Unit 1
should take you approximately 66 hours to complete. Perhaps you can sit
down with your calendar and decide to complete Subunit1.1 (a total of 4
hours) on Monday and Tuesday nights; Subunit 1.2 (a total of 6 hours) on
Tuesday and Wednesday nights; etc.

**Tips/Suggestions:** The course is a continuation of Saylor’s
MA241, Real Analysis I, and thus
assumes a level of mathematical maturity that would be expected upon
mastery of MA241. Please keep in mind that the indicated time advisories
for each unit, subunit, and assignment of this course are not absolute,
and heavily depend on your own personal mastery of the material in
MA241. For example, you may find yourself spending more time on certain
topics than is suggested in the time advisory for each unit; or you may
need to spend more time on reviewing the background material presented
in MA241. If this happens, just keep in mind that it is not necessarily
a problem, but rather another opportunity to improve, enhance, and hone
your previous knowledge. Please note that you do not need a calculator
for this course, but be sure to have plenty of scrap paper and a writing
utensil available to you at all times, since you will be performing
calculations, stating proofs, and working through problem sets
throughout each unit of this course and during the Final Exam.

Please note that understanding the exercises presented in the below
course materials is an absolutely essential part of internalizing the
new subject matter. Although it is not always necessary that you
complete each and every exercise for a given assignment, it is
imperative that you take the exercises supplementing each reading
seriously. To aid you in the learning process, the instructions for each
unit and assignment may indicate which exercises are more or less
crucial than others. In particular, the time advisory for each
assignment, unless stated otherwise, is not meant to indicate how long
it should take you to solve all the exercises for a particular reading,
but rather how much time you should spend on both the reading and your
attempts to solve the exercises. When certain exercises are imperative,
or when solving many exercises is required to understand fully the
material, this requirement (and additional allotted time) will be
indicated clearly in the time advisory.