Line Integrals of vector functions and conservative fields

An application: computing work done by a variable force

We first briefly review an application of line integrals of vector fields. In particular, we focus on the concept of work

done by a variable force

along a path

. To compute work done by a force field

, we:

Divide the path into subarcs. See figure (1).
Approximate the work done along the th subarc as , where is the unit tangent vector to the th subarc and is the length of the subarc.
Then, we sum up the work done along all the subarcs to find the approximate work done as .
Finally, we take the limit of number of subarcs approaching :

where

, and

Figure (1): Use the two sliders to visualize the process explained above.

Question (1)

In figure (2), determine whether the work done by the force field along the two paths and is positive, negative, or zero.

Figure (2): Notice the orientation of each curve.

Question (2)

In figure (3), determine whether the work done by the force field along the two paths and is positive, negative, or zero.

Figure (3): Notice the orientation of each curve.

Some important points regarding conservative fields:

If the result of the line integral from point A to point B is the same for all possible paths, then the line integral is said to be path-independent and the field is said to be conservative.
If a vector field can be written as , where is a scalar-valued function, then is a potential function.
There exists a potential function such that if and only if the field is conservative and the line integral is path-independent. Also, the line integral can be computed as .
A vector field is conservative if and only if for every closed curve .

Question (3)

Do the vector fields in figures (2) and (3) appear to be conservative or not? Justify your answer in each case.

Question (4)

Use the parametrized paths to compute the work done by the force fields from point A to point B as shown in figures (4) and (5).
In figures (4) and (5), is it possible to use the fundamental theorem of line integrals to compute the work done by the force fields from point A to point B? If so, use the fundamental theorem of line integrals to verify your answers to the previous part. Remember to fully justify your answers.

Figure (4)

Figure (5)

Question (5)

Compute the line integral of the vector field along the line segment shown in figure (6).

Figure (6)

Question (6)

Compute the line integral of the vector field from point A to point B as shown in figure (7), where the curve is parametrized as .

Figure (7)

Question (7)

Did you notice a relation between the answers to questions (5) and (6)? Did you expect that to be the case?