Recall that in single variable calculus,
The Second Fundamental Theorem of Calculus tells us that given a constant
and a continuous function
there is a unique function
for which
and
In particular,
is this function. We are about to investigate an analog of this result for path-independent vector fields, but first we require two additional definitions.
If
is a subset of
or
we say that
is
open provided that for every point in
there is a disc (in
) or ball (in
) centered at that point such that every point of the disc/ball is contained in
For example, the set of points
in
for which
is open, since we can always surround any point in this set by a tiny disc contained in the set (as illustrated by point
in
Figure 12.4.5). However, if we change the inequality to
then the set is not open, as any point on the circle
cannot be surrounded by a disc contained in the set; any disc surrounding a point on that circle will contain points outside the set, that is with
(as illustrated by the point
in
Figure 12.4.5). We will also say that a region
is
path-connected provided that for every pair of points in
there is a path from one to the other contained in
We summarize the result of
Activity 12.4.7 below. Much like the Second Fundamental Theorem of Calculus, which tells us that a function is an antiderivative for another function, but leaves the antiderivative in terms of a definite integral, this theorem tells us that a function is a potential function for a vector field, but the definnition of the potential function is in terms of a line integral.