Vector Calculus

Abstract

In the previous chapters, we have studied properties of functions defined on $\mathbb {R}$ (the line), $\mathbb {R}^2$ (the plane), or $\mathbb {R}^3$ (the space) with values in $\mathbb {R}$, which are called real-valued or scalar functions, or scalar fields. Here, we would like to study the calculus of functions taking values in $\mathbb {R}^2$ or $\mathbb {R}^3$, instead of $\mathbb {R}$. Those functions are called vector-valued functions or vector fields. In Sect. 9.2, the concept of a vector will be introduced along with its basic algebraic properties. Vector fields and their continuity and differentiation properties are discussed in Sect. 9.3 along with the notions of gradient, divergence and curl. Moreover, we explain how curves and surfaces are described by such functions. Integrals of vector fields are introduced in Sect. 9.4, first the line integral for scalar and vector fields, and then the surface integral for scalar fields. In Sect. 9.5, we present three fundamental theorems of vector calculus, namely, the Green–Ostrogradski theorem, the Gauss divergence theorem, and the theorem of Stokes. Section 9.6 is devoted to certain applications of the vector calculus to science and engineering, with an emphasis on problems from various parts of mechanics.