Suppose a particle moves along a curve under the influence of an electromagnetic force described by a vector field . Since a force is inducing motion, work is performed. How can we calculate how much work is performed?
Recall that when moving in a straight line, if represents a constant force and represents the direction and length of travel, then work is simply . However, we generally want to be able to calculate work even if is not constant and is not a straight line.
As we have practiced many times before, we can calculate work by first approximating, then refining our approximation through a limit that leads to integration.
Assume as we did in Section 15.1 that can be parametrized by the arc length parameter . Over a short piece of the curve with length , the curve is approximately straight and our force is approximately constant. The straight-line direction of this short length of curve is given by , the unit tangent vector; let , which gives the direction and magnitude of a small section of . Thus work over this small section of is .
Summing up all the work over these small segments gives an approximation of the work performed. By taking the limit as goes to zero, and hence the number of segments approaches infinity, we can obtain the exact amount of work. Following the logic presented at the beginning of this chapter in the Integration Review, we see that
This line integral is beautiful in its simplicity, yet is not so useful in making actual computations (largely because the arc length parameter is so difficult to work with). To compute actual work, we need to parametrize with another parameter via a vector-valued function . As stated in Section 15.1, , and recall that . Thus
Subsection15.3.1Evaluating Line Integrals over Vector Fields
These integrals are known as line integrals over vector fields. By contrast, the line integrals we dealt with in Section 15.1 are sometimes referred to as line integrals over scalar fields. Just as a vector field is defined by a function that returns a vector, a scalar field is a function that returns a scalar, such as . We waited until now to introduce this terminology so we could contrast the concept with vector fields.
Definition15.3.2.Line Integral Over A Vector Field.
Let be a vector field with continuous components defined on a smooth curve , parametrized by , and let be the unit tangent vector of . The line integral over along is
In Definition 15.3.2, note how the dot product is just a scalar. Therefore, this new line integral is really just a special kind of line integral found in Section 15.1; letting , the right-hand side simply becomes , and we can use the techniques of that section to evaluate the integral. We combine those techniques, along with parts of Equation (15.3.1), to clearly state how to evaluate a line integral over a vector field in the following Key Idea.
An important concept implicit in this Key Idea: we can use any continuously differentiable parametrization of that preserves the orientation of : there isn’t a “right” one. In practice, choose one that seems easy to work with.
Notation note: the above Definition and Key Idea implicitly evaluate along the curve , which is parametrized by . For instance, if and , then evaluating along means substituting the - and -components of in for and , respectively, in . Therefore, along ,. Since we are substituting the output of for the input of , we write this as . This is a slight abuse of notation as technically the input of is to be a point, not a vector, but this shorthand is useful.
Example15.3.5.Evaluating a line integral over a vector field: computing work.
Two particles move from to under the influence of the force field . One particle follows , the line ; the other follows , the curve , as shown in Figure 15.3.6. Force is measured in newtons and distance is measured in meters. Find the work performed by each particle.
Figure15.3.6.Paths through a vector field in Example 15.3.5
Solution.
To compute work, we need to parametrize each path. We use to parametrize , and let parametrize ; for each, .
Along the straight-line path, . We find . The integral that computes work is:
joules .
Along the curve ,. We find . The work performed along this path is
joules .
Note how differing amounts of work are performed along the different paths. This should not be too surprising: the force is variable, one path is longer than the other, etc.
Example15.3.7.Evaluating a line integral over a vector field: computing work.
Two particles move from to under the influence of a force field . One moves along the curve , the parabola defined by . The other particle moves along the curve , the bottom half of the circle defined by , as shown in Figure 15.3.8. Force is measured in pounds and distances are measured in feet. Find the work performed by moving each particle along its path.
Figure15.3.8.Paths through a vector field in Example 15.3.7
Solution.
We start by parametrizing : the parametrization is straightforward, giving . On ,.
Computing the work along , we have:
ft-lbs .
For , it is probably simplest to parametrize the half circle using sine and cosine. Recall that is a parametrization of the unit circle on ; we add 1 to the second component to shift the circle up one unit, then restrict the domain to to obtain only the lower half, giving ,, and hence and .
Computing the work along , we have:
ft-lbs .
Note how the work along and in this example is the same. We’ll address why later in this section when conservative fields and path independence are discussed.
Subsection15.3.2Properties of Line Integrals Over Vector Fields
Line integrals over vector fields share the same properties as line integrals over scalar fields, with one important distinction. The orientation of the curve matters with line integrals over vector fields, whereas it did not matter with line integrals over scalar fields.
It is relatively easy to see why. Let be the unit circle. The area under a surface over is the same whether we traverse the circle in a clockwise or counterclockwise fashion, hence the line integral over a scalar field on is the same irrespective of orientation. On the other hand, if we are computing work done by a force field, direction of travel definitely matters. Opposite directions create opposite signs when computing dot products, so traversing the circle in opposite directions will create line integrals that differ by a factor of .
As is piecewise smooth, we break it into two components and , where follows the curve and follows the curve .
We parametrize with on , with . We will use .
While we always have unlimited ways in which to parametrize a curve, there are 2 “direct” methods to choose from when parametrizing . The parametrization , traces the correct line segment but with the wrong orientation. Using Property 3 of Theorem 15.3.10, we can use this parametrization and negate the result.
Another choice is to use the techniques of Section 11.5 to create the line with the orientation we desire. We wish to start at and travel in the direction for one length of , giving equation on .
Either choice is fine; we choose to practice using line integral properties. We find and .
Evaluating the line integral (note how we subtract the integral over as the orientation of is opposite):
.
If we interpret this integral as computing work, the negative work implies that the motion is mostly against the direction of the force, which seems plausible when we look at Figure 15.3.13.
Subsection15.3.3The Fundamental Theorem of Line Integrals
We are preparing to make important statements about the value of certain line integrals over special vector fields. Before we can do that, we need to define some terms that describe the domains over which a vector field is defined.
A region in the plane is connected if any two points in the region can be joined by a piecewise smooth curve that lies entirely in the region. In Figure 15.3.19, sets and are connected; set is not connected, though it is composed of two connected subregions.
A region is simply connected if every simple closed curve that lies entirely in the region can be continuously deformed (shrunk) to a single point without leaving the region. (A curve is simple if it does not cross itself.) In Figure 15.3.19, only set is simply connected. Region is not simply connected as any closed curve that goes around the “hole” in cannot be continuously shrunk to a single point. As is not even connected, it cannot be simply connected, though again it consists of two simply connected subregions.
We have applied these terms to regions of the plane, but they can be extended intuitively to domains in space (and hyperspace). In Figure 15.3.20.(a), the domain bounded by the sphere (at left) and the domain with a subsphere removed (at right) are both simply connected. Any simple closed path that lies entirely within these domains can be continuously deformed into a single point. In Figure 15.3.20.(a), neither domain is simply connected. A left, the ball has a hole that extends its length and the pictured closed path cannot be deformed to a point. At right, two paths are illustrated on the torus that cannot be shrunk to a point.
Recall how in Example 15.3.7 particles moved from to along two different paths, wherein the same amount of work was performed along each path. It turns out that regardless of the choice of path from to , the amount of work performed under the field is the same. Since our expectation is that differing amounts of work are performed along different paths, we give such special fields a name.
Let be a vector field defined on an open, connected domain in the plane or in space containing points and . If the line integral has the same value for all choices of paths starting at and ending at , then
When is a conservative field, the line integral from points to is sometimes written as to emphasize the independence of its value from the choice of path; all that matters are the beginning and ending points of the path.
How can we tell if a field is conservative? To show a field is conservative using the definition, we need to show that all line integrals from points to have the same value. It is equivalent to show that all line integrals over closed paths are 0. Each of these tasks are generally nontrivial.
There is a simpler method. Consider the surface defined by . We can compute the gradient of this function: . Note that this is the field from Example 15.3.7, which we have claimed is conservative. We will soon give a theorem that states that a field is conservative if, and only if, it is the gradient of some scalar function . To show is conservative, we need to determine whether or not for some function . (We’ll later see that there is a yet simpler method). To recognize the special relationship between and in this situation, is given a name.
Let be a differentiable function defined on a domain in the plane or in space (i.e., or ) and let , the gradient of . Then is a potential function of .
Theorem15.3.23.Fundamental Theorem of Line Integrals.
Let be a vector field whose components are continuous on a connected domain in the plane or in space, let and be any points in , and let be any path in starting at and ending at .
Once again considering Example 15.3.7, we have , and . In that example, we evaluated two line integrals from to and found the value of each was 2. Note that is a potential function for . Following the Fundamental Theorem of Line Integrals, consider :
Example15.3.24.Using the Fundamental Theorem of Line Integrals.
Let , and . Use the first part of the Fundamental Theorem of Line Integrals to show that is conservative, then choose any path from to and confirm the second part of the theorem.
To show is conservative, we need to find such that . That is, we need to find such that and . As all we know about are its partial derivatives, we recover by integration:
.
Note how the constant of integration is more than “just a constant”: it is anything that acts as a constant when taking a derivative with respect to . Any function that is a function of (containing no ’s) acts as a constant when deriving with respect to .
Integrating in this example gives:
.
Likewise, integrating with respect to gives:
.
These two results should be equal with appropriate choices of and :
and .
We find , a potential function of . (If were not conservative, no choice of and would give equality.)
By the Fundamental Theorem of Line Integrals, regardless of the path from to ,
.
To illustrate the validity of the Fundamental Theorem, we pick a path from to . The line between these two points would be simple to construct; we choose a slightly more complicated path by choosing the parabola . This leads to the parametrization ,, with . Thus
The Fundamental Theorem of Line Integrals states that we can determine whether or not is conservative by determining whether or not has a potential function. This can be difficult. A simpler method exists if the domain of is simply connected (not just connected as needed in the Fundamental Theorem of Line Integrals), which is a reasonable requirement. We state this simpler method as a theorem.
Let be a vector field whose components have continuous partial derivatives on a simply connected domain in the plane or in space. Then is conservative if and only if or , in 2D or 3D, respectively.
In Example 15.3.24, we showed that is conservative by finding a potential function for . Using the above theorem, we can show that is conservative much more easily by computing its curl: