17. Divergence, Curl and Potentials

d. Differential Identities

2. Second Order Differential Operators and Identities

\(\underline{\vec\nabla\cdot\vec\nabla f}\) \(\underline{\vec\nabla\times\vec\nabla f}\) \(\underline{\vec\nabla\cdot\vec\nabla\times\vec F}\) \(\underline{\vec\nabla\times\vec\nabla\times\vec F}\)

c. The Divergence of a Curl (\(\vec\nabla\cdot\vec\nabla\times\vec F\))

Let's again start with an example and an exercise.

Compute the divergence of the curl of the vector field \(\vec F=\left\langle x^2y+yz,y^2+xz^2,z^3-x^2y \right\rangle\).

The curl is \[\begin{aligned} \vec\nabla\times\vec F &=\begin{vmatrix} \hat\imath & \hat\jmath & \hat k \\[2pt] \partial_x & \partial_y & \partial_z \\[2pt] x^2y+yz & y^2+xz^2 & z^3-x^2y \end{vmatrix} \\[2pt] &=\hat\imath (-x^2-2xz) -\hat\jmath (-2xy-y) +\hat k (z^2-x^2-z) \\[2pt] &=\left\langle-x^2-2xz, 2xy+y, z^2-x^2-z \right\rangle \end{aligned}\] So the divergence of the curl is \[\begin{aligned} \vec\nabla\cdot\vec\nabla\times\vec F &=\partial_x(-x^2-2xz)+\partial_y(2xy+y)+\partial_z(z^2-x^2-z) \\ &=(-2x-2z)+(2x+1)+(2z-1)=0 \end{aligned}\]

Compute the divergence of the curl of the vector field \(\vec F=\left\langle x^2y,y^2+z^2,z^3-x^2 \right\rangle\).

\(\vec\nabla\cdot\vec\nabla\times\vec F=0\)

The curl is \[\begin{aligned} \vec\nabla\times\vec F &=\begin{vmatrix} \hat\imath & \hat\jmath & \hat k \\[2pt] \partial_x & \partial_y & \partial_z \\[2pt] x^2y & y^2+z^2 & z^3-x^2 \end{vmatrix} \\[2pt] &=\hat\imath(0-2z)-\hat\jmath(-2x-0)+\hat k(0-x^2) \\ &=\left\langle -2z,2x,-x^2\right\rangle \end{aligned}\] So the divergence of the curl is \[ \vec\nabla\cdot\vec\nabla\times\vec F =\partial_x(-2z)+\partial_y(2x)+\partial_z(-x^2)=0 \]

It is no coincidence that in both problems the answer was \(0\). As long as the assumptions of Clairaut's Theorem are satisfied (so that mixed partial derivatives are equal) the divergence of a curl is always \(0\).

If all second partial derivatives of the components of \(\vec F\) are continuous functions, then \[ \vec\nabla\cdot\vec\nabla\times\vec F=0 \]

The general curl is \[\begin{aligned} \vec\nabla\times\vec F &=\begin{vmatrix} \hat\imath & \hat\jmath & \hat k \\[2pt] \partial_x & \partial_y & \partial_z \\[2pt] F_1 & F_2 & F_3 \end{vmatrix} \\[1pt] &=\hat\imath (\partial_yF_3-\partial_zF_2) -\hat\jmath (\partial_xF_3-\partial_zF_1) +\hat k (\partial_xF_2-\partial_yF_1) \\[1pt] &=\left\langle \partial_yF_3-\partial_zF_2, \partial_zF_1-\partial_xF_3, \partial_xF_2-\partial_yF_1\right\rangle \end{aligned}\] So the divergence of the curl is \[\begin{aligned} \vec\nabla\cdot\vec\nabla\times\vec F &=\partial_x(\partial_yF_3-\partial_zF_2) +\partial_y(\partial_zF_1-\partial_xF_3) +\partial_z(\partial_xF_2-\partial_yF_1) \\ &=\partial_x\partial_yF_3-\partial_x\partial_zF_2 +\partial_y\partial_zF_1-\partial_y\partial_xF_3 +\partial_z\partial_xF_2-\partial_z\partial_yF_1 \\ &=0 \end{aligned}\] provided the mixed partial derivatives are equal.

In this class, we will always be dealing with continuous functions. So you can always assume the divergence of a curl is zero.

This identity is used to help determine when a vector field has a vector potential as discussed at the end of this chapter. This identity is also used to solve the equations of electromagnetism.