Differential calculus for vector-valued functions

Abstract

In resuming the study of vector-valued functions started in Chapter 4, we begin by the various definitions concerning differentiability, and introduce the Jacobian matrix, which gathers the gradients of the function’s components, and the basic differential operators of order one and two. Then we will present the tools of differential calculus: among them, the so-called chain rule for differentiating composite maps has a prominent role, for it lies at the core of the idea of coordinate-system changes. After discussing the general theory, we examine in detail the special, but of the foremost importance, frame systems of polar, cylindrical, and spherical coordinates.