3 The curl of a vector field

The curl of the vector field given by $\underset{̲}{F}=F_1\underset{̲}{i}+F_2\underset{̲}{j}+F_3\underset{̲}{k}$ is defined as the vector field

$\text{curl}\underset{̲}{F}=\underset{̲}{\nabla }\times \underset{̲}{F}=\left|\begin{array}{ccc}\hfill \underset{̲}{i}\hfill & \hfill \underset{̲}{j}\hfill & \hfill \underset{̲}{k}\hfill \\ \hfill \hfill \\ \hfill \frac{\partial }{\partial x}\hfill & \hfill \frac{\partial }{\partial y}\hfill & \hfill \frac{\partial }{\partial z}\hfill \\ \hfill \hfill \\ \hfill F_1\hfill & \hfill F_2\hfill & \hfill F_3\hfill \end{array}\right|$

$=\left(\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z}\right)\underset{̲}{i}+\left(\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x}\right)\underset{̲}{j}+\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)\underset{̲}{k}$

Physical significance of curl

The divergence of a vector field represents the outflow rate from a point; however the curl of a vector field represents the rotation at a point.

Consider the flow of water down a river (Figure 18). The surface velocity $\underset{̲}{v}$ of the water is revealed by watching a light floating object such as a leaf. You will notice two types of motion. First the leaf floats down the river following the streamlines of $\underset{̲}{v}$ , but it may also rotate. This rotation may be quite fast near the bank, but slow or zero in midstream. Rotation occurs when the velocity, and hence the drag, is greater on one side of the leaf than the other.

Figure 18:

Note that for a two-dimensional vector field, such as $\underset{̲}{v}$ described here, curl $\underset{̲}{v}$ is perpendicular to the motion, and this is the direction of the axis about which the leaf rotates. The magnitude of curl $\underset{̲}{v}$ is related to the speed of rotation.

For motion in three dimensions a particle will tend to rotate about the axis that points in the direction of curl $\underset{̲}{v}$ , with its magnitude measuring the speed of rotation.

If, at any point P, curl $\underset{̲}{v}=\underset{̲}{0}$ then there is no rotation at P and $\underset{̲}{v}$ is said to be irrotational at P . If curl $\underset{̲}{v}=\underset{̲}{0}$ at all points of the domain of $\underset{̲}{v}$ then the vector field is an irrotational vector field .

Example 13

Find curl $\underset{̲}{v}$ for the following two-dimensional vector fields

1. $\underset{̲}{v}=x\underset{̲}{i}+2\underset{̲}{j}$
2. $\underset{̲}{v}=-y\underset{̲}{i}+x\underset{̲}{j}$

   If $\underset{̲}{v}$ represents the surface velocity of the flow of water, describe the motion of a floating leaf.

Solution

1. $$\begin{array}{rcll}\underset{̲}{\nabla }\times \underset{̲}{v}& =& \left|\begin{array}{ccc}\hfill \underset{̲}{i}\hfill & \hfill \underset{̲}{j}\hfill & \hfill \underset{̲}{k}\hfill \\ \hfill \hfill \\ \hfill \frac{\partial }{\partial x}\hfill & \hfill \frac{\partial }{\partial y}\hfill & \hfill \frac{\partial }{\partial z}\hfill \\ \hfill \hfill \\ \hfill x\hfill & \hfill 2\hfill & \hfill 0\hfill \end{array}\right|& \\ & =& \left(\frac{\partial }{\partial y}\left(0\right)-\frac{\partial }{\partial z}\left(2\right)\right)\underset{̲}{i}+\left(\frac{\partial }{\partial z}\left(x\right)-\frac{\partial }{\partial x}\left(0\right)\right)\underset{̲}{j}+\left(\frac{\partial }{\partial x}\left(2\right)-\frac{\partial }{\partial y}\left(x\right)\right)\underset{̲}{k}=\underset{̲}{0}& \end{array}$$

   A floating leaf will travel along the streamlines without rotating.

2. $$\begin{array}{rcll}\underset{̲}{\nabla }\times \underset{̲}{v}& =& \left|\begin{array}{ccc}\hfill \underset{̲}{i}\hfill & \hfill \underset{̲}{j}\hfill & \hfill \underset{̲}{k}\hfill \\ \hfill \hfill \\ \hfill \frac{\partial }{\partial x}\hfill & \hfill \frac{\partial }{\partial y}\hfill & \hfill \frac{\partial }{\partial z}\hfill \\ \hfill \hfill \\ \hfill -y\hfill & \hfill x\hfill & \hfill 0\hfill \end{array}\right|& \\ & =& \left(\frac{\partial }{\partial y}\left(0\right)-\frac{\partial }{\partial z}\left(x\right)\right)\underset{̲}{i}+\left(\frac{\partial }{\partial z}\left(-y\right)-\frac{\partial }{\partial x}\left(0\right)\right)\underset{̲}{j}+\left(\frac{\partial }{\partial x}\left(x\right)-\frac{\partial }{\partial y}\left(-y\right)\right)\underset{̲}{k}& \\ & =& 0\underset{̲}{i}+0\underset{̲}{j}+2\underset{̲}{k}=2\underset{̲}{k}& \end{array}$$

   A floating leaf will travel along the streamlines (anti-clockwise around the origin ) and will rotate anticlockwise (as seen from above).
   
   An analogy of the right-hand screw rule is that a positive (anti-clockwise) rotation in the $xy$ plane represents a positive $z$ -component of the curl. Similar results apply for the other components.

Example 14

1. Find the curl of $\underset{̲}{u}=x^2\underset{̲}{i}+y^2\underset{̲}{j}$ . When is $\underset{̲}{u}$ irrotational?
2. Given $\underset{̲}{F}=\left(xy-xz\right)\underset{̲}{i}+3x^2\underset{̲}{j}+yz\underset{̲}{k}$ , find curl $\underset{̲}{F}$ at the origin $\left(0,0,0\right)$ and at the point $P=\left(1,2,3\right)$ .

Solution

1. $$\begin{array}{rcll}\text{curl}\underset{̲}{u}& =& \underset{̲}{\nabla }\times \underset{̲}{F}=\left|\begin{array}{ccc}\hfill \underset{̲}{i}\hfill & \hfill \underset{̲}{j}\hfill & \hfill \underset{̲}{k}\hfill \\ \hfill \hfill \\ \hfill \frac{\partial }{\partial x}\hfill & \hfill \frac{\partial }{\partial y}\hfill & \hfill \frac{\partial }{\partial z}\hfill \\ \hfill \hfill \\ \hfill x^2\hfill & \hfill y^2\hfill & \hfill 0\hfill \end{array}\right|& \\ & =& \left(\frac{\partial }{\partial y}\left(0\right)-\frac{\partial }{\partial z}\left(y^2\right)\right)\underset{̲}{i}+\left(\frac{\partial }{\partial z}\left(x^2\right)-\frac{\partial }{\partial x}\left(0\right)\right)\underset{̲}{j}+\left(\frac{\partial }{\partial x}\left(y^2\right)-\frac{\partial }{\partial y}\left(x^2\right)\right)\underset{̲}{k}& \\ & =& 0\underset{̲}{i}+0\underset{̲}{j}+0\underset{̲}{k}=\underset{̲}{0}& \end{array}$$

   
   
   curl $\underset{̲}{u}=\underset{̲}{0}$ so $\underset{̲}{u}$ is irrotational everywhere.

2. $$\begin{array}{rcll}\text{curl}\underset{̲}{F}& =& \underset{̲}{\nabla }\times \underset{̲}{F}=\left|\begin{array}{ccc}\hfill \underset{̲}{i}\hfill & \hfill \underset{̲}{j}\hfill & \hfill \underset{̲}{k}\hfill \\ \hfill \hfill \\ \hfill \frac{\partial }{\partial x}\hfill & \hfill \frac{\partial }{\partial y}\hfill & \hfill \frac{\partial }{\partial z}\hfill \\ \hfill \hfill \\ \hfill xy-xz\hfill & \hfill 3x^2\hfill & \hfill yz\hfill \end{array}\right|& \\ & =& \left(\frac{\partial }{\partial y}\left(yz\right)-\frac{\partial }{\partial z}\left(3x^2\right)\right)\underset{̲}{i}+\left(\frac{\partial }{\partial z}\left(xy-xz\right)-\frac{\partial }{\partial x}\left(yz\right)\right)\underset{̲}{j}& \\ & +& \left(\frac{\partial }{\partial x}\left(3x^2\right)-\frac{\partial }{\partial y}\left(xy-xz\right)\right)\underset{̲}{k}& \\ & =& z\underset{̲}{i}-x\underset{̲}{j}+5x\underset{̲}{k}& \end{array}$$

   At the point $\left(0,0,0\right)$ , curl $\underset{̲}{F}=\underset{̲}{0}$ . At the point $\left(1,2,3\right)$ , curl $\underset{̲}{F}=3\underset{̲}{i}-\underset{̲}{j}+5\underset{̲}{k}$ .