Vector Calculus

# 40 Conservative Vector Fields

### Learning Objectives

• Describe simple and closed curves; define connected and simply connected regions.
• Explain how to find a potential function for a conservative vector field.
• Use the Fundamental Theorem for Line Integrals to evaluate a line integral in a vector field.
• Explain how to test a vector field to determine whether it is conservative.

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.

### Curves and Regions

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 ((Figure)).

Definition

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.

Types of curves that are simple or not simple and closed or not closed.

Determining Whether a Curve Is Simple and 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 ((Figure)).

A curve that is closed but not simple.

Is the curve given by parameterization a simple closed curve?

Yes

Hint

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.

Definition

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 ((Figure)).

Not all connected regions are simply connected. (a) Simply connected regions have no holes. (b) Connected regions that are not simply connected may have holes but you can still find a path in the region between any two points. (c) A region that is not connected has some points that cannot be connected by a path in the region.

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.

Hint

Consider the definitions.

### Fundamental Theorem for Line Integrals

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.

Evaluating a Line Integral and the Antiderivatives of the Endpoints

Let Calculate where C is the line segment from (0,0) to (2,2)((Figure)).

We use (Figure) to calculate Curve C can be parameterized by Then, and which implies that

The value of line integral depends only on the value of the potential function of F at the endpoints of the curve.

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.

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,

#### Proof

By (Figure),

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.

Applying the Fundamental Theorem

Calculate integral where and C is a curve with parameterization

1. without using the Fundamental Theorem of Line Integrals and
2. using the Fundamental Theorem of Line Integrals.
1. First, let’s calculate the integral without the Fundamental Theorem for Line Integrals and instead use (Figure):

Integral requires integration by parts. Let and Then
and

Therefore,

Thus,

2. Given that 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.

(Figure) 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

Hint

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.

Definition

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.

Path Independence of Conservative Fields

If F is a conservative vector field, then F is independent of path.

#### Proof

Let D denote the domain of F and let and be two paths in D with the same initial and terminal points ((Figure)). 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.

The vector field is conservative, and therefore 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.

The Path Independence Test for Conservative Fields

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.

#### Proof

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 ((Figure)). Then

The first integral does not depend on x, so

If we parameterize by then

By the Fundamental Theorem of Calculus (part 1),

Here, is any path from C to that stays inside D, and is the horizontal line segment from to

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 (Figure) and (Figure), 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.

Showing That a Vector Field Is Not Conservative

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 ((Figure)). 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.

Curves and are both oriented from left to right.

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

Hint

Calculate the corresponding line integrals.

### Conservative Vector Fields and Potential Functions

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.

Finding a Potential Function

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

Find a potential function for

Hint

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:

Problem-Solving Stragegy: Finding a Potential Function for a Conservative Vector Field
1. Integrate P with respect to x. This results in a function of the form where is unknown.
2. Take the partial derivative of with respect to y, which results in the function
3. Use the equation to find
4. Integrate to find
5. Any function of the form 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.

Finding a Potential Function in

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

Find a potential function for

Hint

Following (Figure), begin by integrating with respect to x.

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.

Finding a Potential Function

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

Hint

Use the Problem-Solving Strategy.

### Testing a Vector Field

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 (Figure)). 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.

The Cross-Partial Test for Conservative Fields

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:

Cross-Partial Property of Conservative Fields

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

Determining Whether a Vector Field Is Conservative

Determine whether vector field is conservative.

Note that the domain of F is all of and is simply connected. Therefore, we can use (Figure) to determine whether F is conservative. Let

Since and the vector field is not conservative.

Determining Whether a Vector Field Is 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 (Figure) to determine whether F is conservative. Let

Then and thus F is conservative.

Determine whether is conservative.

It is conservative.

Hint

Use (Figure).

When using (Figure), 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 (Figure), 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 (Figure):

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 (Figure)? 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, (Figure) 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 (Figure).

Using the Fundamental Theorem for Line Integrals

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,

Analysis

Notice that if we hadn’t recognized that F is conservative, we would have had to parameterize C and use (Figure). 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

Hint

Use the Fundamental Theorem for Line Integrals.

Work Done on a Particle

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.

Analysis

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 (Figure). 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

Hint

Use the Fundamental Theorem for Line Integrals.

### Key Concepts

• The theorems in this section require curves that are closed, simple, or both, and regions that are connected or simply connected.
• The line integral of a conservative vector field can be calculated using the Fundamental Theorem for Line Integrals. This theorem is a generalization of the Fundamental Theorem of Calculus in higher dimensions. Using this theorem usually makes the calculation of the line integral easier.
• Conservative fields are independent of path. The line integral of a conservative field depends only on the value of the potential function at the endpoints of the domain curve.
• Given vector field F, we can test whether F is conservative by using the cross-partial property. If F has the cross-partial property and the domain is simply connected, then F is conservative (and thus has a potential function). If F is conservative, we can find a potential function by using the Problem-Solving Strategy.
• The circulation of a conservative vector field on a simply connected domain over a closed curve is zero.

### Key Equations

• Fundamental Theorem for Line Integrals
• Circulation of a conservative field over curve C that encloses a simply connected region

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 (Figure) by showing that

### Glossary

closed curve
a curve that begins and ends at the same point
connected region
a region in which any two points can be connected by a path with a trace contained entirely inside the region
Fundamental Theorem for Line Integrals
the value of line integral depends only on the value of at the endpoints of C:
independence of path
a vector field F has path independence if for any curves and in the domain of F with the same initial points and terminal points
simple curve
a curve that does not cross itself
simply connected region
a region that is connected and has the property that any closed curve that lies entirely inside the region encompasses points that are entirely inside the region