Mathematics of Physics and Engineering

Functions, Sets, Gradient 123

known as a Jordan curve) divides the plane into two domains, and the
domain enclosed by the curve is simply connected, but the rigorous proof
of this statement is surprisingly hard. The French mathematician MARIE
ENNEMOND CAMILLE JORDAN (1838-1922) was the first to realize the im-
portance and difficulty of the issue. Even though Jordan's proof later turned
out to be wrong, the result is known as the Jordan curve theorem. It was
the American mathematician OSWALD VEBLEN (1880-1960), who, in 1905,
produced the first correct proof. We will take the Jordan curve theorem
for granted. The analog of this result in space, that a closed continuous
simple (that is, without self-intersections) surface in R^3 separates R^3 into
interior and exterior domains, is even harder to prove, and we take it for
granted as well.

We will now discuss continuity of functions on Rn, n = 2,3. A scalar
field / is called continuous at the point A if / is defined in some neigh-
borhood of A and

lim \f(A)-f(B)\=0. (3.1.1)

Since the point A in the above definition is fixed, the condition \AB\ —» 0
means that B is approaching A: B —> A. Then (3.1.1) can be written in
the form that is completely analogous to the definition of continuity for
functions of a real variable:

lim f(B) = f(A).

Similarly, a vector field F is called continuous at a point A if F is defined

in some neighborhood of A and

lim ||F(J4)-F(B)||=0. (3.1.2)

|AB|->0

Next, we discuss the question of differentiability for scalar fields. Let

/ : A H R be a scalar field, where A is either R^2 or R^3. Let u

be a unit vector and, for a real t, define the point Bt so that ABt =

tu. A function / is called differentiable at the point A in the

direction of u if the function g(t) = f(Bt) is differentiable for t = 0.

We also call ^'(0) the (directional) derivative of / at A in the

direction of u and denote this directional derivative by Dnf(A).