Smoothness of Functions

Most of the functions we encounter have continuous partial derivatives. To keep our theory simple we shall concentrate on such functions in this chapter.

The definition for three or more variables is similar.

The Increment Theorem for a differentiable function of one variable shows that the increment Δz is very close to the differential dz, and leads to the notion of a tangent line. In this section we introduce the increment and total differential for a function of two variables. Then we state an Increment Theorem for a smooth function of two variables, which leads to the notion of a tangent plane.

Let z depend on the two independent variables x and y,

z = f(x, y).

Let Δx and Δy be two new independent variables, called the increments of x and y. Usually Δx and Δy are taken to be infinitesimals.