Line Integrals

In the module on Using Geometry we studied work and we saw that

In one dimension, a force can be described by a real number F that is positive if the force is pushing from left to right and is negative if the force is pushing from right to left. When an object is moved from x = a to x = b by a force F the amount of work done by the force is
F (b - a).
In two or three dimensions, a force can be described by a vector whose magnitude is the strength of the force and whose direction is the direction in which the force is pushing -- for example, the vector 2 i represents a force whose strength is two units and that is pushing from west to east. If an object moves under the influence of a force, F, and its motion is represented by the vector S as shown in the figure below
then the work done by the force is

It is often useful to think of ordinary integration as "advanced multiplication." For example, the area of a simple rectangle whose height is H and whose base goes from x = a to x = b as shown in the figure below

is H (b - a) and the area of the figure below whose height is given by the function y = f(x)

When height is a simple constant, area is height multiplied by width but when the height varies simple multiplication is replaced by integration or "advanced multiplication."

In the same way when an object moves from the point x = a to the point x = b under the influence of a force that is not constant and is given by a function f(x), then the work done by the force is given by

and the simple multiplication of our first formula for work is replaced by integration or "advanced multiplication."

In this module we look at a two-dimensional or three dimensional situation in which the force is not constant -- at different points the force may point in different directions and it may have different strangths. The figure below shows an example. Notice the force is always pointing toward the origin in this particular example.

Not only does the force vary from point-to-point but there is another complication -- we are interested in an object traveling along a curved path as shown in the movie below.

Because the force varies from one place to another we call it a force field. A force field is represented mathematically by a vector field. We use the notation

F(x, y) = P(x, y) i + Q(x, y) j

to describe a force field in R² and the notation

F(x, y, z) = P(x, y, z) i + Q(x, y, z) j + R(x, y, z) k

to describe a force field in R³.

Now suppose that C is a curve in R² or R³ like the curve shown in the figure below. This example is the path followed by the object in the movie above. We are interested in the work done by a force F acting on an object as it travels along a curve C in a particular direction. In the example below the object travels along the curve C from left to right.

We estimate this work in the obvious way. First, we approximate the curve C by a polygonal path -- a path made up of straight line segements -- as shown in the two figures below.

We choose points s₀, s₁, s₂, ... s_n along the path C and then connect these points as shown in the figure above. Then we estimate the work done on the i-th segment of the path by

and the amount of work done on the whole path by

By using a large number of small segments we can obtain a very good estimate for the amount of work done. The exact amount of work done is obtained by taking the limit of these estimates. This limit is called the line integral of the vector field F over the path C and is denoted

In practice we usually describe the curve C by a function

and we divide the time interval [a, b] up into n subintervals. by letting

and

and then

We use four examples of force fields to illustrate our work.

The first force field is a two-dimensional force field and is pointing directly toward the origin. At the origin it is zero but everywhere else its strength is one unit.
The second force field is a three-dimensional force field and is pointing directly toward the origin. At each point its strength is equal to the distance from the origin.
The third force field is a three-dimensional force field. At every point the force is the same -- one unit pointing straight down.
The last force field is a two-dimensional force field. Above the x-axis it is pointing from right to left with a strength of one unit. Below the x-axis, it is zero.

Before going on, you should think about the amount of work that would be done by these four force fields acting on objects traveling along various different paths.

We will look at objects traveling along various different paths. Here are some examples.

A straight line from one point A to another point B. Such a path can be decsribed by a function
S(t) = (1 - t) A + t B
with 0 <= t <= 1.
A circular path, centered at the origin, with radius R. Such a path can be described by a function
S(t) = (R cos t) i + (R sin t) j
with 0 <= t <= 2 pi.
Where does this particular path start? Is the path being covered clockwise or counterclockwise? How would you describe a circular path that started at the point (0, R) and made two complete circles around the origin in the clockwise direction?
A helical path -- for example,
S(t) = (R cos t) i + (R sin t) j + m k
where R and m are constants.

For each problem below, estimate the answer first based on your physical intuition and then estimate the answer numerically using your CAS window. The force fields F, H, L, and M are the force fields described above.

Suppose that an object travels from (0, 0) to (1, 2) along a straight line. How much work would be done by the force field F?
Suppose that an object travels from (0, 0) to (1, 2) along a straight line. How much work would be done by the force field M?
Suppose that an object travels from (-1, -1, -1) to (2, 3, 4). How much work would be done by the force field H?
Suppose that an object travels from (-1, -1, -1) to (2, 3, 4). How much work would be done by the force field L?
Suppose that an object travels around the circle of radius 3 centered at the origin, counterclockwise, sttarting at the point (-3, 0). How much work would be done by the force F?
Suppose that an object travels around the circle of radius 3 centered at the origin, counterclockwise, sttarting at the point (-3, 0). How much work would be done by the force M?
Suppose that an object travels along the helical path
S(t) = (2 cos t) i + (2 sin t) j + 0.5 t k
for 0 <= t <= 6 pi. How much work would be done by the force H?
Suppose that an object travels along the helical path
S(t) = (2 cos t) i + (2 sin t) j + 0.5 t k
for 0 <= t <= 6 pi. How much work would be done by the force L?

As usual, whenever we have a situation in which we can get arbitrarily good estimates for some quantity, we can find the exact value of the quantity by taking the limit of the estimates. The details are worked out in Multivariable Calculus, Linear Algebra, and Differential Equations in a Real and Complex World. The final results are.

where we use the following notation in R²

and the following notation in R³

Repeat the same problems in the exercise set above using your CAS window to find the answers by integration as described above.

Diffusion Into a Region

We end this section with one long homework problem. We are interested in the way that a substance like ink or a pollutant diffuses into a two-dimensional region. Because this problem is mathematically similar to computing work, it makes a good homework problem at this time.

The way that a substance diffuses in a medium is determined by many factors -- for example, temperature, magnetic and electric fields, and the concentration of the substance itself. At each point (x, y) in a two-dimensional medium the diffusion of the substance can be described by a vector

F(x, y)

that is measured in units of mass per unit of length per unit of time. For example, if

F(x, y) = (2 mg per cm per minute) i

then the the substance is flowing in the direction of the vector i at a rate that would result in 2 milligrams of the substance crossing a vertical line whose length was 1 centimeter in one minute. Notice that if the line was horizontal -- that is, parallel to the direction of flow -- then none of the substance would cross the line.

Suppose that a region is enclosed by a curve C. How would you compute the rate at which the substance is flowing into the region?
Suppose that the flow of a pollutant through an aquifer is described by the vector field
F(x, y) = 5 j
measured in units of milligrams per foot per day. At what rate is it flowing into the triangle whose vertices are at the points (0, 0), (0, 4), and (3, 0)?
Suppose that the flow of a pollutant through an aquifer is described by the vector field
F(x, y) = 5 j
measured in units of milligrams per foot per day. At what rate is it flowing into the disk of radius 4 whose center is at the origin?
Suppose that the flow of a pollutant through an aquifer is described by the vector field
F(x, y) = 5 x j
measured in units of milligrams per foot per day. At what rate is it flowing into the triangle whose vertices are at the points (0, 0), (0, 4), and (3, 0)?
Suppose that the flow of a pollutant through an aquifer is described by the vector field
F(x, y) = 5 x j
measured in units of milligrams per foot per day. At what rate is it flowing into the disk of radius 4 whose center is at the origin?
Suppose that the flow of a pollutant through an aquifer is described by the vector field
F(x, y) =2 y i + 5 x j
measured in units of milligrams per foot per day. At what rate is it flowing into the triangle whose vertices are at the points (0, 0), (0, 4), and (3, 0)?
Suppose that the flow of a pollutant through an aquifer is described by the vector field
F(x, y) = 2 y i + 5 x j
measured in units of milligrams per foot per day. At what rate is it flowing into the disk of radius 4 whose center is at the origin?