Section 3.4 Fundamental Theorem of line integrals
Objectives
You should be able to:
State the Fundamental Theorem of line integrals for line integrals of exact one-forms, and use it to evaluate line integrals.
Show that the Fundamental Theorem of line integrals implies that line integrals of exact one-forms only depend on the starting and ending points of the curve.
Show that the Fundamental Theorem of line integrals implies that line integrals of exact one-forms over closed curves vanish.
State these results in terms of conservative vector fields and their associated potential functions.
Use the Fundamental Theorem of line integrals and its consequences to show that a given one-form cannot be exact.
Subsection 3.4.1 The Fundamental Theorem of line integrals
Recall from Section 2.2 that an exact one-form is a one-form that can be written as the differential of a function:Theorem 3.4.1. The Fundamental Theorem of line integrals.
Let
The integral thus only depends on the starting and ending points of the image curve
Proof.
You have probably noticed that this theorem is similar in flavour to the Fundamental Theorem of Calculus for definite integrals; in fact it follows from it, as we will see.
First, by the definition of line integrals, we have:
Next, we can use one of the fundamental properties of the pullback, which is that
If we introduce a parameter
But then, the right-hand-side is just a standard definite integral of the derivative of a function. By the Fundamental Theorem of Calculus (part 2), we know that the right-hand-side is simply equal to
Corollary 3.4.2. The line integrals of an exact form along two curves that start and end at the same points are equal.
If
Corollary 3.4.3. The line integral of an exact one-form along a closed curve vanishes.
Let
We sometimes write
Example 3.4.4. An example of a line integral of an exact one-form.
Suppose that you want to integrate the one-form
which is
What is neat as well is that you know that the line integral of
Neat!
One thing that we did not explain however here: how did we know that
Subsection 3.4.2 The Fundamental Theorem of line integrals for vector fields
To end this section, let us rewrite the Fundamental Theorem for line integrals in terms of the associated vector fields.Theorem 3.4.5. The Fundamental Theorem of line integrals for vector fields.
Let
In the notation introduced in Remark 3.3.8, we can rewrite this integral as
where
the line integral of a conservative vector field does not depend on the path chosen between two points;
the line integral of a conservative vector field along a closed curve is always zero.
Exercises 3.4.3 Exercises
1.
Consider the one-form
To show that it is exact, we simply find a function
which is indeed
Using the Fundamental Theorem of line integrals, we can integrate
2.
Recall from Example 2.2.13 (see also Example 3.6.5) that the one-form
We parametrize the unit circle as usual by
The line integral thus simply becomes:
In particular, it is non-zero. This proves that
3.
Suppose that
It should be
4.
Consider the one-form
The one-form
If we write
Now we want this integral to be zero. Thus we want
But we don't want a closed curve, so we must choose our curve such that
Then
Thus
for any parametric curve that starts at
5.
Let
for any two curves
for any two curves
In other words, if the line integral of a one-form between two given points is path independent, then it is path independent everywhere.
The proof is fairly intuitive. Fix
On the one hand, the curve
On the other hand, the curve
But we know that
Equating the two expressions for these line integrals, and simplifying, we end up with the statement that
Since this must be true for any points