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 Figure 12.2.2) than if the object moves along the line segment from $(0,1)$ directly to $(4,3)$(illustrated by $C_2$ in Figure 12.2.2). 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 Figure 12.2.2). We say that a curve in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ 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.

Preview Activity 12.2.1 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 $\mathbf{F}\text{,}$ a vector field, along $C\text{,}$ the curve shown below that goes from $P$ to $Q\text{.}$

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 $\mathbf{F}$ at a collection of points on the curve $C\text{.}$ Remember that when the vector field is plotted on the whole space, the lengths are rescaled to not be visually cluttered. In Figure 12.2.5 the actual output vectors are plotted for some points on $C\text{.}$

Since the output of $\mathbf{F}$ is changing as you move along the path, we will use a type of argument that has come up repeatedly throughout our study of integral calculus. Here we will break our region up into smaller pieces to approximate the work done on each piece. Figure 12.2.6 shows how we can break $C$ into $n$ pieces, which we will call $C_i\text{.}$ Note that $C_i$ goes from the point ${\mathbf{r}}_{i-1}$ to ${\mathbf{r}}_i\text{.}$

If we look at one of these smaller pieces (as show in Figure 12.2.7), 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 $\mathrm{\Delta }{\mathbf{r}}_i={\mathbf{r}}_i-{\mathbf{r}}_{i-1}$ is a good approximation of the curve piece $C_i\text{.}$ Therefore we can approximate the work done by the vector field $\mathbf{F}$ on the piece $C_i$ by $\mathbf{F}({\mathbf{r}}_i)\cdot \mathrm{\Delta }{\mathbf{r}}_i\text{.}$ Remember that this is the same idea as in Section 9.3; 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.

We are able to say a bit here about conditions that guarantee the limit in Definition 12.2.9 exists. First, we require that $C$ is a relatively “nice” curve. We say that a curve is smooth provided that it can be parameterized with functions that are infinitely differentiable. We do not require that $C$ be smooth, but only that it be piecewise smooth, which means that $C$ can be separated into parts which are individually smooth. The other requirement is that $\mathbf{F}$ is a continuous vector field, by which we mean that each component function of $\mathbf{F}$ is continuous as a function of $2$ or $3$ variables (for a large enough region around our curve $C$).

Because the dot products in the definition of the line integral ${\int }_C\mathbf{F}\cdot d\mathbf{r}$ can each be viewed as the work done by $\mathbf{F}$ as an object moves along the (very small) vector $\mathrm{\Delta }{\mathbf{r}}_i\text{,}$ the line integral gives the total work done by the vector field on an object that moves along $C$ (in the direction of its orientation).

If we are trying to determine how much a wind current helps or hinders an aircraft flying along a path determined by the curve, then calculating the dot product $\mathbf{F}({\mathbf{r}}_i)\cdot \mathrm{\Delta }{\mathbf{r}}_i$ makes sense for the local amount of help or hindrance. Note here that our vector $\mathrm{\Delta }{\mathbf{r}}_i$ is the displacement from ${\mathbf{r}}_{i-1}$ to ${\mathbf{r}}_i\text{.}$ If the displacement vector, $\mathrm{\Delta }{\mathbf{r}}_i\text{,}$ and the force field vector, $\mathbf{F}({\mathbf{r}}_i)\text{,}$ point in directions with an acute angle between them, the dot product will be positive.

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We are abusing notation here a tiny bit, since technically the domain of $\mathbf{F}$ is points in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3\text{,}$ and ${\mathbf{r}}_i$ is a vector. By $\mathbf{F}(\mathbf{r})\text{,}$ we mean $\mathbf{F}(r_1,r_2)\text{,}$ where $\mathbf{r}=⟨r_1,r_2⟩\text{.}$

On the other hand, if the angle between them is obtuse, the dot product will be negative. In this case, we also note that the force field is hindering the aircraft’s progress. This interpretation carries through with our limit argument above. Specifically, the line integral over a curve will be positive if more of the vector field is in the direction of travel than against the direction of travel. You will need to consider the value of the dot product and not just the sign when assessing whether a line integral will be positive/neagtive/zero.

The next several sections will be devoted to determining ways to efficiently calculate line integrals. As with the limits in the definition of every other type of integral we’ve studied so far, the limit in the definition of the line integral is is cumbersome to work with in most cases. However, in the case where the oriented curve $C$ is composed of horizontal and vertical line segments, we can make a rather quick reduction to a single-variable integral, as the following example shows.

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 $\mathbf{F}$ along a closed curve $C$ is called the circulation of $\mathbf{F}$ around $C\text{.}$ To emphasize the fact that $C$ is closed, we sometimes write ${\oint }_C\mathbf{F}\cdot d\mathbf{r}$ for ${\int }_C\mathbf{F}\cdot d\mathbf{r}\text{.}$ 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.

This section relies heavily on understanding vector fields from Section 12.1, understanding curves in space (from Section 9.6), and the work interpretation of the dot product from Section 9.3.