# Math Insight

### Divergence and curl notation

For $\dlvf: \R^3 \to \R^3$ (confused?), the formulas for the divergence and curl are \begin{align*} \div \dlvf &= \pdiff{\dlvfc_1}{x} + \pdiff{\dlvfc_2}{y} + \pdiff{\dlvfc_3}{z}\\ \curl \dlvf &= \left(\pdiff{\dlvfc_3}{y}-\pdiff{\dlvfc_2}{z}, \pdiff{\dlvfc_1}{z} - \pdiff{\dlvfc_3}{x}, \pdiff{\dlvfc_2}{x} - \pdiff{\dlvfc_1}{y} \right). \end{align*}

These formulas are easy to memorize using a tool called the “del” operator, denoted by the nabla symbol $\nabla$. Think of $\nabla$ as a “fake” vector composed of all the partial derivatives that we use just to help us remember the formulas: \begin{align*} \nabla = \left(\pdiff{}{x}, \pdiff{}{y}, \pdiff{}{z}\right). \end{align*} Although it may not seem to make sense to just have the partial derivatives without them acting on a function, we won't worry about that. This is just notation.

Now, let's take the dot product of the $\nabla$ vector with $\dlvf=(\dlvfc_1,\dlvfc_2, \dlvfc_3)$: \begin{align*} \nabla \cdot \dlvf &= \left(\pdiff{}{x}, \pdiff{}{y}, \pdiff{}{z}\right) \cdot (\dlvfc_1,\dlvfc_2, \dlvfc_3)\\ &= \pdiff{}{x}\dlvfc_1 + \pdiff{}{y}\dlvfc_2 + \pdiff{}{z}\dlvfc_3 \end{align*} If we think of each “multiplication” in the dot product as instead being the derivative of the corresponding $\dlvfc$, then we have the formula for the divergence. So, if you can remember the del operator $\nabla$ and how to take a dot product, you can easily remember the formula for the divergence \begin{align*} \div \dlvf = \nabla \cdot \dlvf = \pdiff{\dlvfc_1}{x} + \pdiff{\dlvfc_2}{y} + \pdiff{\dlvfc_3}{z}. \end{align*}

This notation is also helpful because you will always know that $\nabla \cdot \dlvf$ is a scalar (since, of course, you know that the dot product is a scalar product).

The curl, on the other hand, is a vector. We know one product that gives a vector: the cross product. And, yes, it turns out that $\curl \dlvf$ is equal to $\nabla \times \dlvf$. To see this, let's take the cross product of the $\nabla$ vector with $\dlvf$. \begin{align*} \nabla \times \dlvf &= \left(\pdiff{}{x}, \pdiff{}{y}, \pdiff{}{z}\right) \times (\dlvfc_1,\dlvfc_2, \dlvfc_3)\\ &= \left| \begin{array}{ccc} \vc{i} & \vc{j} & \vc{k}\\ \pdiff{}{x} & \pdiff{}{y} & \pdiff{}{z}\\ \dlvfc_1 & \dlvfc_2 & \dlvfc_3 \end{array} \right| \\ &= \vc{i} \left(\pdiff{}{y}\dlvfc_3 - \pdiff{}{z}\dlvfc_2\right) - \vc{j} \left(\pdiff{}{x} \dlvfc_3 -\pdiff{}{z}\dlvfc_1\right) + \vc{k} \left(\pdiff{}{x}\dlvfc_2 - \pdiff{}{y}\dlvfc_1\right) \\ &= \left(\pdiff{\dlvfc_3}{y} - \pdiff{\dlvfc_2}{z}\right)\vc{i} + \left(\pdiff{\dlvfc_1}{z} -\pdiff{\dlvfc_3}{x}\right)\vc{j} + \left(\pdiff{\dlvfc_2}{x} - \pdiff{\dlvfc_1}{y}\right)\vc{k} \end{align*}

This is exactly the formula we gave above. So if you can use the rule that “multiplication” by $\pdiff{}{x}$ is the same as taking the partial derivative with respect to $x$ (and similar for the other derivatives), then you can remember the curl formula by \begin{align*} \curl \dlvf = \nabla \times \dlvf. \end{align*}