Fundamental Theorem for Line Integrals

7/29/2019 Fundamental Theorem for Line Integrals

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Fundamental Theorem for Line Integrals

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 vectorfields. Here it is.

Theorem

Suppose that Cis asmooth curve given by , . Also suppose

that fis a function whose gradient vector, , is continuous on C. Then,

Note that represents the initial point on Cwhile 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 thenumber of variables in the function.

Proof

This is a fairly straight forward proof.

For the purposes of the proof well assume that were working in three dimensions, but itcan be done in any dimension.

Lets start by just computing the line integral.
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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 singleintegrals.

Lets take a quick look at an example of using this theorem.

Example 1 Evaluate where

and Cis any path that starts at and

ends at .

Solution

First lets notice that we didnt specify the path for getting from the first point to the second point. The

reason for this is simple. The theorem above tells us that all we need are the initial and final points on thecurve in order to evaluate this kind of line integral.

So, let be any path that starts at and ends at

. Then,
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The integral is then,

Notice that we also didnt need the gradient vector to actually do this line integral. However, for the

practice of finding gradient vectors here it is,

\

The most important idea to get from this example is not how to do the integral as thatspretty 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 didntreally need to know the path to get the answer. In other words, we could use any path we

want and well always get the same results.

In the first section on line integrals (even though we werent 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 NOTchange the value of the line integral.

Lets formalize this idea up a little. Here are some definitions. The first one weve

already seen before, but its been a while and its important in this section so well give itagain. The remaining definitions are new.

Definitions

First suppose that is a continuous vector field in some domain D.

1. is a conservative vector field if there is a function fsuch that

. The function fis called a potential function for

the vector field. We first saw this definition in the first section of this chapter.

2. is independent of path if for any two paths and

in D with the same initial and final points the integral has the same

value.

3. A path Cis called closed if its initial and final points are the same point. For


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example a circle is a closed path.

4. A path Cis simple if it doesnt cross itself. A circle is a simple curve while afigure 8 type curve is not simple.

5. A region D is open if it doesnt contain any of its boundary points.

6. A region D is connected if we can connect any two points in the region with apath that lies completely inD.

7. A region D is simply-connected if it is connected and it contains no holes. We

wont need this one until the next section, but it fits in with all the other

definitions given here so this was a natural place to put the definition.

With these definitions we can now give some nice facts.

Facts

1. is independent of path.

This is easy enough to prove since all we need to do is look at the theorem above.

The theorem tells us that in order to evaluate this integral all we need are the initialand final points of the curve. This in turn tells us that the line integral must be

independent of path.

2.

If is a conservative vector field then is independent ofpath.

This fact is also easy enough to prove. If is conservative then it has a

potential function, f, and so the line integral becomes

. Then using the first fact we know that this line integral

must be independent of path.

3. If is a continuous vector field on an open connected region D and if

is independent of path (for any path in D) then is a

conservative vector field on D.


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4. If is independent of path then for every

closed path C.

5. If for every closed path Cthen is

independent of path.

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.

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Fundamental Theorem for Line Integrals