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 Real Analysis I 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 Requirements:
Have successfully completed the following Saylor courses, or their equivalents: Single-Variable Calculus I, Single-Variable Calculus II, Multivariable Calculus, Linear Algebra, and Differential Equations from The Core Program in Saylor’s Mathematics discipline, as well as Saylor’s Real Analysis I.
