ABSTRACT
In this chapter, the author explains functions of several variables, the concept of differentiability in the framework, and properties of differentiable functions. Although some of the material can be traced all the way to Newton and Leibniz, most of it was rigorously developed at the end of the nineteenth century, and the beginning of the twentieth century. The author shows that the mixed second-order partial derivatives at (0, 0) need not be equal. It is exactly the fact that they are not continuous that is to blame. One of the unifying themes of calculus is the idea of approximating complicated functions by simpler ones. Now that the readers are studying functions of several variables, the author looks for a similar formula. He looks at some basic properties of the derivative of functions of n variables.
