The Idea of a Line Integral

This section relies heavily on understanding vector fields from , understanding curves in space (from ), and the work interpretation of the dot product from .

A suggestion to check that students understand the material in is to ask them to draw \(-C_2\) and explain why \(C_1-C_2\) does not make sense in that example.

Movement by things like air or water will exert a force on objects, so wind velocity vector fields are related to a force field but the relationship between a moving fluid and the force a fluid exerts on an object in the fluid can be complicated. Specifically, issues like drag/friction and cross sectional area are a critical part of this relationship but are not something that we will focus on in our discussion. In the exploration below, we will look at a simplified version of this situation.

As we discussed in , vector fields are often used to represent forces such as gravity or electromagnetism, as well as the velocity of movement for things like wind or flowing water. We learned in that the dot product of a force vector and a displacement vector tells us how much work the force did on the object as it moved from the start of its displacement vector to the end. However, this calculation assumes that the force is constant in the region of movement and that the object moves in a straight line along the displacement vector. The situation is more complicated than a dot product calculation when an object's movement is not in a straight line and when the force is not uniform throughout the area in which the object moves.

The preview activity uses cardinal directions to specify the direction of displacement vectors. These directions can be described by a Copyright 2013-2026 . The Copyright 2013-2026 given in is an example of a sixteen point rose. Note that directions like ESE are read as east-southeast half way between east and southeast.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

Given our motivation for calculating the work that a force field does on an object as it moves through the field, it is natural to concern ourselves with how the object moves. In particular, in many circumstances it will be different if an object moves from the point \((0,1)\) to the point \((4,3)\) by first going up the \(y\)-axis to \((0,3)\) and then moving horizontally to \((4,3)\) (illustrated by \(C_1\) in ) than if the object moves along the line segment from \((0,1)\) directly to \((4,3)\)(illustrated by \(C_2\) in ). Similarly, given a fixed force field, we would expect the work done to be different (in fact, opposite) if the object moves from \((4,3)\) to \((0,1)\) directly along a line segment (\(C_3\) in ). We say that a curve in \(\R^2\) or \(\R^3\) is oriented if we have specified the direction of travel along the curve. When a curve is given parametrically (including as a vector-valued function), our convention will be that the orientation follows from the smallest allowable value of the parameter to the largest.

In general, there are many ways to parametrize an oriented curve. With line segments, it is common to have the parameter range from \(0\) to \(1\), although there are sometimes good reasons to choose another method. For circles and ellipses, you may find it useful to interchange the placement of \(\cos(t)\) and \(\sin(t)\) to change the orientation, but then careful attention will need to be paid to the start and end points. The interactive graph below allows you to plot parametric curves. You should experiment with the graph below and try to make sense of how changing different elements affects the graph shown below. Remember that you can change the highlighted point using the slider. You should take time now to at least try the following:

1. \(2 \pi\)
2. \(r_1\)\(r_2\)
3. \(t\)\(-t\)
4. \(-t\)\(e^t\)
5. Change the upper and lower bounds to get the same curve plotted as in the earlier parts

showed how we can break up the work done along a path into a sum of work done on each piece. This will be a very helpful idea, especially if we consider the work done by a vector field that is not constant.

For example, let's consider how to measure the work done by \(\vF\), a vector field, along \(C\), the curve shown below that goes from \(P\) to \(Q\).

You can see that there will be parts of \(C\) such that the dot product of the direction of travel and the vector field will be positive and some parts where the dot product is negative. We don't need to consider the output of the vector field except at the points on the curve. Thus, we will look at the following plot of the output of \(\vF\) at a collection of points on the curve \(C\). Remember that when the vector field is plotted on the whole space, the lengths are rescaled to not be visually cluttered. In the actual output vectors are plotted for some points on \(C\).

Since the output of \(\vF\) is changing as you move along the path, we will use a classic calculus approach. For step 1 of our CCA, we will break our region up into smaller pieces to approximate the work done on each piece. shows how we can break \(C\) into \(n\) pieces, which we will call \(C_i\). Note that \(C_i\) goes from the point \(\mathbf{r}_{i-1}\) to \(\mathbf{r}_{i}\).

If we look at one of these smaller pieces (as show in ), we can see that vector field still changes at these points, but the output vectors are very similar. We can also see that the vector \(\Delta \mathbf{r}_i= \mathbf{r}_i-\mathbf{r}_{i-1}\) is a good approximation of the curve piece \(C_i\). Therefore we can approximate the work done by the vector field \(\vF\) on the piece \(C_i\) by \(\vF(\mathbf{r}_i)\cdot \Delta \mathbf{r}_i\). Remember that this is the same idea as in ; Namely, the work done is the dot product of the force and the displacement, but this is done on many small pieces instead of the whole path.

Our approximation with \(n\) pieces of \(C\) would be \[\begin{aligned}\end{aligned}\] which completes step 1 of the CCA.

Note that as we increase \(n\), the number of pieces of \(C\) in our approximation, and make sure the length of each piece, \(C_i\), goes to zero, the vector field \(\vF\) will be nearly constant on each piece. Additionally, the displacement vector, \(\Delta \mathbf{r}_i\), will be a better approximation of \(C_i\) as \(n\) increases. This all combines to ensure that as we increase \(n\), the number of smaller segments, our approximation will improve even more quickly.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

In , we implicitly made use of the idea that if \(C\) can be broken up into two curves \(C_1\) and \(C_2\) such that the terminal point of \(C_1\) is the initial point of \(C_2\), then the line integral of \(\vF\) along \(C\) is the sum of the line integrals of \(\vF\) along \(C_1\) and along \(C_2\). This is a generalization of the property for definite integrals that tells us if \(c\in [a,b]\), then \[\begin{aligned}\end{aligned}\]. We next describe some common ways of breaking line integrals into pieces.

For a constant scalar \(k\), vector fields \(\vF\) and \(\vG\), and oriented curves \(C\), \(C_1\), and \(C_2\), the following properties hold:

1. \(\displaystyle\int_C (k\vF)\cdot d\vr = k\int_C\vF\cdot d\vr\)

2. \(\displaystyle\int_C(\vF+\vG)\cdot d\vr = \int_C \vF\cdot d\vr + \int_C\vG\cdot d\vr\)

3. \(\displaystyle\int_{-C}\vF\cdot d\vr = -\int_C \vF\cdot d\vr\)

4. \(\displaystyle\int_{C_1+C_2} \vF\cdot d\vr = \int_{C_1}\vF\cdot d\vr + \int_{C_2}\vF\cdot d\vr\).

If an oriented curve \(C\) ends at the same point where it started, we say that \(C\) is closed. The line integral of a vector field \(\vF\) along a closed curve \(C\) is called the circulation of \(\vF\) around \(C\). To emphasize the fact that \(C\) is closed, we sometimes write \(\oint_C \vF\cdot d\vr\) for \(\int_C \vF\cdot d\vr\). Circulation serves as a measure of a vector field's tendency to rotate in a manner consistent with the orientation of the (closed) curve and is measured by looking at whether the vector field is working with or against the motion along the path.