Chapter 3
Vector Analysis and Classical
Electromagnetic Theory
3.1 Functions of Several Variables
3.1.1 Functions, Sets, and the Gradient
In the previous chapter, we saw how the tools of vector analysis work in
modelling the mechanics of point masses and rigid bodies. In this chapter
we will see how the tools of vector analysis work in continuum mechanics,
that is in the study of continuous media such as fluids, heat flow, and
electromagnetic fields.
Mathematically, a continuous medium is described using certain func-
tions, called scalar and vector fields. Recall that a function is a corre-
spondence between two sets A and B so that to every element A & K there
corresponds at most one element f(A) G B. Despite its simplicity, it was
only in 1837 that the German mathematician JOHANN PETER GUSTAV
LEJEUNE DIRICHLET (1805-1859) introduced this definition of a function.
We write / : A -> B.
For most functions in this chapter, the set A will be either R^2 or
R^3 , and the set B, R, R^2 , or R^3. When B = R, the collection of pairs
{(P, f(P)), P G A} is called a scalar field; the function / is also called
a scalar field. When B = A, that is, when f(P) is a vector, the collection of
pairs {(P, f(P)), P € A} is called a vector field; the function / is also
called a vector field. Following our convention from the previous chapters,
we will denote vector fields with bold-face letters.
Let us review the main definitions related to the sets in R^2 and R^3.
Recall that the distance between two points A, B in R™, n = 2,3, is denoted
by \AB\. A neighborhood of a point A is the set {B : \AB\ < b] for some
b > 0, that is, an open disk in R^2 and an open ball in R^3 , with center121