In this section, we continue the study of conservative vector fields. We examine the Fundamental Theorem for Line Integrals, which is a useful generalization of the Fundamental Theorem of Calculus to line integrals of conservative vector fields. We also discover show how to test whether a given vector field is conservative, and determine how to build a potential function for a vector field known to be conservative.
Before continuing our study of conservative vector fields, we need some geometric definitions. The theorems in the subsequent sections all rely on integrating over certain kinds of curves and regions, so we develop the definitions of those curves and regions here.
We first define two special kinds of curves: closed curves and simple curves. As we have learned, a closed curve is one that begins and ends at the same point. A simple curve is one that does not cross itself. A curve that is both closed and simple is a simple closed curve ([link]).
Curve C is a closed curve if there is a parameterization
of C such that the parameterization traverses the curve exactly once and
Curve C is a simple curve if C does not cross itself. That is, C is simple if there exists a parameterization
of C such that r is one-to-one over
It is possible for
meaning that the simple curve is also closed.
Is the curve with parameterization
a simple closed curve?
Note that
therefore, the curve is closed. The curve is not simple, however. To see this, note that
and therefore the curve crosses itself at the origin ([link]).
Is the curve given by parameterization
a simple closed curve?
Yes
Sketch the curve.
Many of the theorems in this chapter relate an integral over a region to an integral over the boundary of the region, where the region’s boundary is a simple closed curve or a union of simple closed curves. To develop these theorems, we need two geometric definitions for regions: that of a connected region and that of a simply connected region. A connected region is one in which there is a path in the region that connects any two points that lie within that region. A simply connected region is a connected region that does not have any holes in it. These two notions, along with the notion of a simple closed curve, allow us to state several generalizations of the Fundamental Theorem of Calculus later in the chapter. These two definitions are valid for regions in any number of dimensions, but we are only concerned with regions in two or three dimensions.
A region D is a connected region if, for any two points
and
there is a path from
to
with a trace contained entirely inside D. A region D is a simply connected region if D is connected for any simple closed curve C that lies inside D, and curve C can be shrunk continuously to a point while staying entirely inside D. In two dimensions, a region is simply connected if it is connected and has no holes.
All simply connected regions are connected, but not all connected regions are simply connected ([link]).
Is the region in the below image connected? Is the region simply connected?
The region in the figure is connected. The region in the figure is not simply connected.
Consider the definitions.
Now that we understand some basic curves and regions, let’s generalize the Fundamental Theorem of Calculus to line integrals. Recall that the Fundamental Theorem of Calculus says that if a function
has an antiderivative F, then the integral of
from a to b depends only on the values of F at a and at b—that is,
If we think of the gradient as a derivative, then the same theorem holds for vector line integrals. We show how this works using a motivational example.
Let
Calculate
where C is the line segment from (0,0) to (2,2)([link]).
We use [link] to calculate
Curve C can be parameterized by
Then,
and
which implies that
Notice that
where
If we think of the gradient as a derivative, then
is an “antiderivative” of F. In the case of single-variable integrals, the integral of derivative
is
where a is the start point of the interval of integration and b is the endpoint. If vector line integrals work like single-variable integrals, then we would expect integral F to be
where
is the endpoint of the curve of integration and
is the start point. Notice that this is the case for this example:
and
In other words, the integral of a “derivative” can be calculated by evaluating an “antiderivative” at the endpoints of the curve and subtracting, just as for single-variable integrals.
The following theorem says that, under certain conditions, what happened in the previous example holds for any gradient field. The same theorem holds for vector line integrals, which we call the Fundamental Theorem for Line Integrals.
Let C be a piecewise smooth curve with parameterization
Let
be a function of two or three variables with first-order partial derivatives that exist and are continuous on C. Then,
By [link],
By the chain rule,
Therefore, by the Fundamental Theorem of Calculus,
□
We know that if F is a conservative vector field, there are potential functions
such that
Therefore
In other words, just as with the Fundamental Theorem of Calculus, computing the line integral
where F is conservative, is a two-step process: (1) find a potential function (“antiderivative”)
for F and (2) compute the value of
at the endpoints of C and calculate their difference
Keep in mind, however, there is one major difference between the Fundamental Theorem of Calculus and the Fundamental Theorem for Line Integrals. A function of one variable that is continuous must have an antiderivative. However, a vector field, even if it is continuous, does not need to have a potential function.
Calculate integral
where
and C is a curve with parameterization
Integral
requires integration by parts. Let
and
Then
and
Therefore,
Thus,
is a potential function for F, let’s use the Fundamental Theorem for Line Integrals to calculate the integral. Note that
This calculation is much more straightforward than the calculation we did in (a). As long as we have a potential function, calculating a line integral using the Fundamental Theorem for Line Integrals is much easier than calculating without the theorem.
[link] illustrates a nice feature of the Fundamental Theorem of Line Integrals: it allows us to calculate more easily many vector line integrals. As long as we have a potential function, calculating the line integral is only a matter of evaluating the potential function at the endpoints and subtracting.
Given that
is a potential function for
calculate integral
where C is the lower half of the unit circle oriented counterclockwise.
2
The Fundamental Theorem for Line Intervals says this integral depends only on the value of
at the endpoints of C.
The Fundamental Theorem for Line Integrals has two important consequences. The first consequence is that if F is conservative and C is a closed curve, then the circulation of F along C is zero—that is,
To see why this is true, let
be a potential function for F. Since C is a closed curve, the terminal point r(b) of C is the same as the initial point r(a) of C—that is,
Therefore, by the Fundamental Theorem for Line Integrals,
Recall that the reason a conservative vector field F is called “conservative” is because such vector fields model forces in which energy is conserved. We have shown gravity to be an example of such a force. If we think of vector field F in integral
as a gravitational field, then the equation
follows. If a particle travels along a path that starts and ends at the same place, then the work done by gravity on the particle is zero.
The second important consequence of the Fundamental Theorem for Line Integrals is that line integrals of conservative vector fields are independent of path—meaning, they depend only on the endpoints of the given curve, and do not depend on the path between the endpoints.
Let F be a vector field with domain D. The vector field F is independent of path (or path independent) if
for any paths
and
in D with the same initial and terminal points.
The second consequence is stated formally in the following theorem.
If F is a conservative vector field, then F is independent of path.
Let D denote the domain of F and let
and
be two paths in D with the same initial and terminal points ([link]). Call the initial point
and the terminal point
Since F is conservative, there is a potential function
for F. By the Fundamental Theorem for Line Integrals,
Therefore,
and F is independent of path.
□
To visualize what independence of path means, imagine three hikers climbing from base camp to the top of a mountain. Hiker 1 takes a steep route directly from camp to the top. Hiker 2 takes a winding route that is not steep from camp to the top. Hiker 3 starts by taking the steep route but halfway to the top decides it is too difficult for him. Therefore he returns to camp and takes the non-steep path to the top. All three hikers are traveling along paths in a gravitational field. Since gravity is a force in which energy is conserved, the gravitational field is conservative. By independence of path, the total amount of work done by gravity on each of the hikers is the same because they all started in the same place and ended in the same place. The work done by the hikers includes other factors such as friction and muscle movement, so the total amount of energy each one expended is not the same, but the net energy expended against gravity is the same for all three hikers.
We have shown that if F is conservative, then F is independent of path. It turns out that if the domain of F is open and connected, then the converse is also true. That is, if F is independent of path and the domain of F is open and connected, then F is conservative. Therefore, the set of conservative vector fields on open and connected domains is precisely the set of vector fields independent of path.
If F is a continuous vector field that is independent of path and the domain D of F is open and connected, then F is conservative.
We prove the theorem for vector fields in
The proof for vector fields in
is similar. To show that
is conservative, we must find a potential function
for F. To that end, let X be a fixed point in D. For any point
in D, let C be a path from X to
Define
by
(Note that this definition of
makes sense only because F is independent of path. If F was not independent of path, then it might be possible to find another path
from X to
such that
and in such a case
would not be a function.) We want to show that
has the property
Since domain D is open, it is possible to find a disk centered at
such that the disk is contained entirely inside D. Let
with
be a point in that disk. Let C be a path from X to
that consists of two pieces:
and
The first piece,
is any path from C to
that stays inside D;
is the horizontal line segment from
to
([link]). Then
The first integral does not depend on x, so
If we parameterize
by
then
By the Fundamental Theorem of Calculus (part 1),
A similar argument using a vertical line segment rather than a horizontal line segment shows that
Therefore
and F is conservative.
□
We have spent a lot of time discussing and proving [link] and [link], but we can summarize them simply: a vector field F on an open and connected domain is conservative if and only if it is independent of path. This is important to know because conservative vector fields are extremely important in applications, and these theorems give us a different way of viewing what it means to be conservative using path independence.
Use path independence to show that vector field
is not conservative.
We can indicate that F is not conservative by showing that F is not path independent. We do so by giving two different paths,
and
that both start at
and end at
and yet
Let
be the curve with parameterization
and let
be the curve with parameterization
([link]). Then
and
Since
the value of a line integral of F depends on the path between two given points. Therefore, F is not independent of path, and F is not conservative.
Show that
is not path independent by considering the line segment from
to
and the piece of the graph of
that goes from
to
If
and
represent the two curves, then
Calculate the corresponding line integrals.
As we have learned, the Fundamental Theorem for Line Integrals says that if F is conservative, then calculating
has two steps: first, find a potential function
for F and, second, calculate
where
is the endpoint of C and
is the starting point. To use this theorem for a conservative field F, we must be able to find a potential function
for F. Therefore, we must answer the following question: Given a conservative vector field F, how do we find a function
such that
Before giving a general method for finding a potential function, let’s motivate the method with an example.
Find a potential function for
thereby showing that F is conservative.
Suppose that
is a potential function for F. Then,
and therefore
Integrating the equation
with respect to x yields the equation
Notice that since we are integrating a two-variable function with respect to x, we must add a constant of integration that is a constant with respect to x, but may still be a function of y. The equation
can be confirmed by taking the partial derivative with respect to x:
Since
is a potential function for F,
and therefore
This implies that
so
Therefore, any function of the form
is a potential function. Taking, in particular,
gives the potential function
To verify that
is a potential function, note that
The logic of the previous example extends to finding the potential function for any conservative vector field in
Thus, we have the following problem-solving strategy for finding potential functions:
where
is unknown.
with respect to y, which results in the function
to find
to find
where C is a constant, is a potential function for F.
We can adapt this strategy to find potential functions for vector fields in
as shown in the next example.
Find a potential function for
thereby showing that F is conservative.
Suppose that
is a potential function. Then,
and therefore
Integrating this equation with respect to x yields the equation
for some function g. Notice that, in this case, the constant of integration with respect to x is a function of y and z.
Since
is a potential function,
Therefore,
Integrating this function with respect to y yields
for some function
of z alone. (Notice that, because we know that g is a function of only y and z, we do not need to write
Therefore,
To find
, we now must only find h. Since
is a potential function,
This implies that
so
Letting
gives the potential function
To verify that
is a potential function, note that
We can apply the process of finding a potential function to a gravitational force. Recall that, if an object has unit mass and is located at the origin, then the gravitational force in
that the object exerts on another object of unit mass at the point
is given by vector field
where G is the universal gravitational constant. In the next example, we build a potential function for F, thus confirming what we already know: that gravity is conservative.
Find a potential function
for
Suppose that
is a potential function. Then,
and therefore
To integrate this function with respect to x, we can use u-substitution. If
then
so
for some function
Therefore,
Since
is a potential function for F,
Since
also equals
Therefore,
which implies that
Thus, we can take
to be any constant; in particular, we can let
The function
is a potential function for the gravitational field F. To confirm that
is a potential function, note that
Find a potential function
for the three-dimensional gravitational force
Use the Problem-Solving Strategy.
Until now, we have worked with vector fields that we know are conservative, but if we are not told that a vector field is conservative, we need to be able to test whether it is conservative. Recall that, if F is conservative, then F has the cross-partial property (see [link]). That is, if
is conservative, then
and
So, if F has the cross-partial property, then is F conservative? If the domain of F is open and simply connected, then the answer is yes.
If
is a vector field on an open, simply connected region D and
and
throughout D, then F is conservative.
Although a proof of this theorem is beyond the scope of the text, we can discover its power with some examples. Later, we see why it is necessary for the region to be simply connected.
Combining this theorem with the cross-partial property, we can determine whether a given vector field is conservative:
Let
be a vector field on an open, simply connected region D. Then
and
throughout D if and only if F is conservative.
The version of this theorem in
is also true. If
is a vector field on an open, simply connected domain in
then F is conservative if and only if
Determine whether vector field
is conservative.
Note that the domain of F is all of
and
is simply connected. Therefore, we can use [link] to determine whether F is conservative. Let
Since
and
the vector field is not conservative.
Determine vector field
is conservative.
Note that the domain of F is the part of
in which
Thus, the domain of F is part of a plane above the x-axis, and this domain is simply connected (there are no holes in this region and this region is connected). Therefore, we can use [link] to determine whether F is conservative. Let
Then
and thus F is conservative.
When using [link], it is important to remember that a theorem is a tool, and like any tool, it can be applied only under the right conditions. In the case of [link], the theorem can be applied only if the domain of the vector field is simply connected.
To see what can go wrong when misapplying the theorem, consider the vector field from [link]:
This vector field satisfies the cross-partial property, since
and
Since F satisfies the cross-partial property, we might be tempted to conclude that F is conservative. However, F is not conservative. To see this, let
be a parameterization of the upper half of a unit circle oriented counterclockwise (denote this
and let
be a parameterization of the lower half of a unit circle oriented clockwise (denote this
Notice that
and
have the same starting point and endpoint. Since
and
Therefore,
Thus,
and
have the same starting point and endpoint, but
Therefore, F is not independent of path and F is not conservative.
To summarize: F satisfies the cross-partial property and yet F is not conservative. What went wrong? Does this contradict [link]? The issue is that the domain of F is all of
except for the origin. In other words, the domain of F has a hole at the origin, and therefore the domain is not simply connected. Since the domain is not simply connected, [link] does not apply to F.
We close this section by looking at an example of the usefulness of the Fundamental Theorem for Line Integrals. Now that we can test whether a vector field is conservative, we can always decide whether the Fundamental Theorem for Line Integrals can be used to calculate a vector line integral. If we are asked to calculate an integral of the form
then our first question should be: Is F conservative? If the answer is yes, then we should find a potential function and use the Fundamental Theorem for Line Integrals to calculate the integral. If the answer is no, then the Fundamental Theorem for Line Integrals can’t help us and we have to use other methods, such as using [link].
Calculate line integral
where
and C is any smooth curve that goes from the origin to
Before trying to compute the integral, we need to determine whether F is conservative and whether the domain of F is simply connected. The domain of F is all of
which is connected and has no holes. Therefore, the domain of F is simply connected. Let
so that
Since the domain of F is simply connected, we can check the cross partials to determine whether F is conservative. Note that
Therefore, F is conservative.
To evaluate
using the Fundamental Theorem for Line Integrals, we need to find a potential function
for F. Let
be a potential function for F. Then,
and therefore
Integrating this equation with respect to x gives
for some function h. Differentiating this equation with respect to y gives
which implies that
Therefore, h is a function of z only, and
To find h, note that
Therefore,
and we can take
A potential function for F is
Now that we have a potential function, we can use the Fundamental Theorem for Line Integrals to evaluate the integral. By the theorem,
Notice that if we hadn’t recognized that F is conservative, we would have had to parameterize C and use [link]. Since curve C is unknown, using the Fundamental Theorem for Line Integrals is much simpler.
Calculate integral
where
and C is a semicircle with starting point
and endpoint
Use the Fundamental Theorem for Line Integrals.
Let
be a force field. Suppose that a particle begins its motion at the origin and ends its movement at any point in a plane that is not on the x-axis or the y-axis. Furthermore, the particle’s motion can be modeled with a smooth parameterization. Show that F does positive work on the particle.
We show that F does positive work on the particle by showing that F is conservative and then by using the Fundamental Theorem for Line Integrals.
To show that F is conservative, suppose
were a potential function for F. Then,
and therefore
and
Equation
implies that
Deriving both sides with respect to y yields
Therefore,
and we can take
If
then note that
and therefore
is a potential function for F.
Let
be the point at which the particle stops is motion, and let C denote the curve that models the particle’s motion. The work done by F on the particle is
By the Fundamental Theorem for Line Integrals,
Since
and
by assumption,
Therefore,
and F does positive work on the particle.
Notice that this problem would be much more difficult without using the Fundamental Theorem for Line Integrals. To apply the tools we have learned, we would need to give a curve parameterization and use [link]. Since the path of motion C can be as exotic as we wish (as long as it is smooth), it can be very difficult to parameterize the motion of the particle.
Let
and suppose that a particle moves from point
to
along any smooth curve. Is the work done by F on the particle positive, negative, or zero?
Negative
Use the Fundamental Theorem for Line Integrals.
True or False? If vector field F is conservative on the open and connected region D, then line integrals of F are path independent on D, regardless of the shape of D.
True
True or False? Function
where
parameterizes the straight-line segment from
True or False? Vector field
is conservative.
True
True or False? Vector field
is conservative.
Justify the Fundamental Theorem of Line Integrals for
in the case when
and C is a portion of the positively oriented circle
from (5, 0) to (3, 4).
[T] Find
where
and C is a portion of curve
from
to
[T] Evaluate line integral
where
and C is the path given by
for
For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.
Not conservative
Conservative,
Conservative,
For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals.
where C is any path from (0, 0) to (2, 4)
where C is the line segment from (0, 0) to (4, 4)
[T]
where C is any smooth curve from (1, 1) to
Find the conservative vector field for the potential function
For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.
F is not conservative.
F is conservative and a potential function is
F is conservative and a potential function is
F is conservative and a potential function is
For the following exercises, determine whether the given vector field is conservative and find a potential function.
F is conservative and a potential function is
For the following exercises, evaluate the integral using the Fundamental Theorem of Line Integrals.
Evaluate
where
and C is any path that starts at
and ends at
[T] Evaluate
where
and C is a straight line from
to
[T] Evaluate
where
and C is any path in a plane from (1, 2) to (3, 2).
Evaluate
where
and C has initial point (1, 2) and terminal point (3, 5).
For the following exercises, let
and
and let C1 be the curve consisting of the circle of radius 2, centered at the origin and oriented counterclockwise, and C2 be the curve consisting of a line segment from (0, 0) to (1, 1) followed by a line segment from (1, 1) to (3, 1).
Calculate the line integral of F over C1.
Calculate the line integral of G over C1.
Calculate the line integral of F over C2.
Calculate the line integral of G over C2.
[T] Let
Calculate
where C is a path from
to
[T] Find line integral
of vector field
along curve C parameterized by
For the following exercises, show that the following vector fields are conservative by using a computer. Calculate
for the given curve.
C is the curve consisting of line segments from
to
to
C is parameterized by
[T]
C is curve
The mass of Earth is approximately
and that of the Sun is 330,000 times as much. The gravitational constant is
The distance of Earth from the Sun is about
Compute, approximately, the work necessary to increase the distance of Earth from the Sun by
[T] Let
Evaluate the integral
where
[T] Let
be given by
Use a computer to compute the integral
where
[T] Use a computer algebra system to find the mass of a wire that lies along curve
if the density is
Find the circulation and flux of field
around and across the closed semicircular path that consists of semicircular arch
followed by line segment
Compute
where
Complete the proof of [link] by showing that
depends only on the value of
at the endpoints of C:
for any curves
and
in the domain of F with the same initial points and terminal points
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