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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.
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:
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.
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)
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.
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Saylor Academy’s mission is to open education to all.
Saylor Academy’s mission is sustained by the continued evolution of an open educational ecosystem, and we are dedicated partners in this movement. Saylor’s commitment to the open education ecosystem is founded not just on open educational resources and open source learning technologies, but also on open access to credentials, and ongoing open learning opportunities.
Guided by these beliefs, Saylor Academy is currently focused on the following projects:
Open Courses: Maintenance and Learner-Centered Improvements
- A commitment to the OER community means that we’ll continue to replace open access materials with openly licensed ones in an effort to make Saylor courseware as reusable and remixable as possible.
- Open courses require more instructional supports for learners, so our current improvements focus on ensuring better and more frequent opportunities for Saylor students to practice what they’re learning.
Open Credentials: Adding New Opportunities and Bolstering Existing Ones
- We’re working on expanding our suite of Saylor Direct credit recommended exams, and we’re also keen on working with university partners to develop innovative and flexible partner degree launching and completion programs. ...
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