The Curl of a Vector Field

This section relies heavily on understanding vector fields from . If the calculation of the curl is your primary purpose for using this section, then you can skip many of the details of and . The details have been made as bite-sized as possible, but the particulars of curl typically aren't easily understood without some of the calculations of circulation density. Additionally, the visualization of vector fields in 3D makes geometric interpretations harder to demonstrate.

In , we examined how the strength of a vector field in two (or more) dimensions changed in different regions. In particular, we developed the divergence of a vector field as a local (or density) measurement for how the strength of the vector field changes. The key ideas when interpreting divergence are:

• A positive divergence means that the vector field is growing in strength.
• A negative divergence means that the vector field is decreasing in strength.
• A zero divergence means that the vector field is not changing in strength.

In many physical settings, it is also useful to measure the rotational strength of a vector field at a local scale. For instance, the vector field on the left of Figure shows a vector field which flows in a counterclockwise fashion around the origin. The vector field on the right of Figure shows a vector field which does not have a rotational aspect to its flow.

These global ideas of rotation are nice, but are not always visually apparent or may only appear in some regions. For instance, the vector field on the left of seems to have rotation around at least a couple of different points. In each quadrant, the vector field on the right of has different rotational patterns.

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

In this subsection, we will develop the measurement of the circulation density for a two-dimensional vector field. We will use this measurement to generalize to a notion of rotational strength in higher dimensional cases in the next subsection.

We will start by measuring the circulation of a vector field on a path around the point \((a,b)\) and use this measurement to define circulation density. Specifically, we will measure the circulation of a vector field as we move around a square centered at \((a,b)\). Using this measurement, we will calculate the circulation density by dividing our measurement by the area enclosed. This will allow us to compare our measurement across regions of different sizes. By taking the limit of this circulation density as the square's side length goes to zero, we will have the circulation density at the point \((a,b)\). Just as in our discussion of , we will look at a two dimensional setting first, then examine how our argument can be generalized to higher dimensions.

Let's start by measuring the circulation around a square with side length \(2h\) centered at a point \((a,b)\). Namely, we will look at the line integral of our vector field as we move along the square curve shown in .

We can parametrize the top edge of the square using the parametrization \[\begin{aligned}\end{aligned}\] for \(-h\leq t\leq h\). Similarly, the bottom, right, and left can be parametrized (over the same range of values for \(t\)) as follows: \[\begin{aligned}\vr_{\text{bottom}}(t) \amp= \langle a+t,b-h\rangle \\ \vr_{\text{right}}(t) \amp= \langle a+h,b+t\rangle \\ \vr_{\text{left}}(t) \amp= \langle a-h,b-t\rangle\end{aligned}\]

A larger square is likely to have a larger total circulation since there is more distance to accumulate how much the vector field is moving in the same direction as our path. This is why the side length \(2h\) of the square appears as a factor in the formula for our total circulation. In order to compare our rotational or circulation ideas over different sizes of squares, we will now look at the circulation density (strength of rotation per unit area), which can be computed directly from our total circulation measurement. We can take the limit of the circulation density as we shrink the square to the central point \((a,b)\) and get a measurement rotational strength of our vector field at a point.

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

The previous subsection showed how we can measure the circulation density, or strength of rotation, at a point for a two-dimensional vector field. In this subsection, we will look at how this two-dimensional measurement can be used to define circulation density for three dimensional vector fields.

We first need to consider how we want to represent and measure rotation in three dimensions. When we developed the circulation density on the \(xy\)-plane, we measured the rotational strength at a point \((a,b)\) with an axis of rotation coming out of the \(xy\)-plane. This corresponds to the axis of rotation being given by the blue vector in . Remember that positive rotation corresponds to counterclockwise rotation in the \(xy\)-plane. We can generalize this idea to think of rotation in three dimensions as being represented by a vector where

• the direction of the vector represents the axis of rotation and
• the magnitude of the vector represents the strength of the rotation.

If you look carefully at , you can see that the vector shown in blue will be parallel to \(\vk\) and will represent rotations on planes of the form \(z=c\). Similarly, the vector shown in yellow will be parallel to \(\vi\) and will represent rotations on planes of the form \(x=a\). When looking down the blue and yellow vectors (from the terminal to the initial point), you can see the positive coordinate axes as pointing to the right and upWhen placing the \(x\)-axis horizontally for the blue vector and placing the \(y\)-axis horizontally for the yellow vector (as we would expect on a two-dimensional plot). In contrast, when you look down the magenta vector (from the terminal to the initial point), the positive coordinate axes (the \(x\) and \(z\) axes) point to the left and up.When placing the \(x\)-axis horizontally. In fact, when we look down the magenta vector the \(xz\)-plane is flipped. A positive rotation on a plane of the form \(y=b\) will correspond to a rotation vector that is in the direction of \(-\vj\). This is a consequence of the right-handed coordinate system and our right-handed idea of rotation, as reflected in the relationships amongst the vectors \(\vi,\vj\,\vk\) recalled below: \[\begin{aligned}\end{aligned}\]

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

The previous subsection discussed how we can view the amount of rotation of a three-dimensional vector field in a plane parallel to the \(xy\)-, \(xz\)-, or \(yz\)-plane and recalled that vectors in \(\R^3\) can be written as a linear combination of the vectors \(\vi\), \(\vj\), and \(\vk\). We use our definition of curl for a two-dimensional vector field to measure the amount of rotation in the appropriate planes, which leads to the following definition.

As with divergence, there is an alternate notation for curl that uses the del operator \(\nabla = \langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \rangle\). Specifically, for a three-dimensional vector field \(\vF\), \(\curl(\vF) = \nabla\times \vF\).

The curl is exactly the rotational description described at the end of . When evaluated at a point \((u,v,w)\), the first component of the curl will measure the circulation density of the vector field restricted to the plane \(x=u\). Similarly, the second and third components of the curl will measure the circulation density of the vector field restricted to the planes \(y=v\) and \(z=w\), respectively.

We next consider an example where the trace of the vector field in planes parallel to coordinate planes is not as easy to visualize as in the previous example.

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

It is worth making explicit a fact that we have used implicitly throughout this section: the curl of a vector field is itself a vector field! That is, evaluating \(\curl(\vF)\) at a point gives a vector. As we saw earlier in this section, the vector output of \(\curl(\vF)\) represents the rotational strength of the vector field \(\vF\) as a linear combination of rotational strengths (or circulation densities) from two-dimensional planar descriptions. From our description of vectors as a representation of rotations, We can think of rotation in three dimensions as being represented by a vector where

• the direction of the vector represents the axis of rotation and
• the magnitude of the vector represents the strength of the rotation.

With these conventions, the output vector of the curl evaluated at a point \(P\), written as \(\curl(\vF)(P)\), will have the following properties:

1. \(\curl(\vF)(P)\)
2. \(\curl(\vF)(P)\)\(P\)\(\curl(\vF)(P)\)

The following theorem allows us to use \(\curl(\vF)\) to measure the rotational strength of \(\vF\) around an arbitrary axis. This will be particularly useful to us in later sections.

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