This is a free online course offered by the Saylor Foundation.
'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, which focused on the study of the real number system, including real numbers and real-valued functions defined on all or part (usually intervals) of the real number line. In particular, MA241introduced you to differentiation and integration, powerful analysis techniques that enable the solution of many problems at the heart of science, including questions in the fields of physics, economics, chemistry, biology, and engineering. Real Analysis II will help you extend these techniques to the solution of more complex mathematical and scientific problems.
As long as a problem can be modeled as a functional relation between two quantities, each of which can be expressed as a single real number, the techniques of single-variable real-valued functions should suffice. However, quite often a problem fundamentally involves information requiring more than one real number to describe, or it depends on more than one variable, or both. For instance, a particle moving in a room requires three coordinate real numbers to determine its location. Or, in another example from physics, the altitude a projectile will reach – a quantity measurable by one real number – depends on the weight of the projectile as well as the initial velocity it acquired from some external force.
Sometimes a problem can be modeled as a single-variable or multivariable function depending on the answer desired. For example, a particle in three-dimensional space moving through a force field (think of a dust particle floating in the air as it is blown by strong or minute gusts of wind) can be modeled both as a function of time (a single-variable function) describing 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 problem can be modeled as a multivariable function requiring three inputs and producing three outputs.
In this course, you will learn about some of the intricacies of the geometry of higher-dimensional spaces. You will develop the theory of multivariable functions and apply advanced techniques of differentiation and integration to such functions. Finally, you will explore applications of these advanced techniques in solving scientific problems.'
