In Calculus I we had the Fundamental
Theorem of Calculus that told us how to evaluate definite integrals. This told us,
It turns out that there is a version of this for line
integrals over certain kinds of vector fields.
Here it is.
Theorem
Note that represents the initial point on C while represents the final point on C.
Also, we did not specify the number of variables for the function since
it is really immaterial to the theorem.
The theorem will hold regardless of the number of variables in the
function.
Proof
This is a
fairly straight forward proof.
For the
purposes of the proof we’ll assume that we’re working in three dimensions,
but it can be done in any dimension.
Let’s start by
just computing the line integral.
Now, at this
point we can use the Chain Rule
to simplify the integrand as follows,
To finish this
off we just need to use the Fundamental Theorem of Calculus for single integrals.

Let’s take a quick look at an example of using this theorem.
The most important idea to get from this example is not how
to do the integral as that’s pretty simple, all we do is plug the final point
and initial point into the function and subtract the two results. The important idea from this example (and
hence about the Fundamental Theorem of Calculus) is that, for these kinds of
line integrals, we didn’t really need to know the path to get the answer. In other words, we could use any path we want
and we’ll always get the same results.
In the first section on line integrals (even though we
weren’t looking at vector fields) we saw that often when we change the path we
will change the value of the line integral.
We now have a type of line integral for which we know that changing the
path will NOT change the value of the line integral.
Let’s formalize this idea up a little. Here are some definitions. The first one we’ve already seen before, but
it’s been a while and it’s important in this section so we’ll give it
again. The remaining definitions are
new.
Definitions
With these definitions we can now give some nice facts.
Facts
These are some nice facts to remember as we work with line
integrals over vector fields. Also notice
that 2 & 3 and 4 & 5 are converses of each other.