Multivariable calculus and vector analysis

Divergence and curl notation

For $\mathbf{F}:{\mathbf{R}}^3\to {\mathbf{R}}^3$, the formulas for the divergence and curl are

$$\begin{array}{llll}\hfill \mathrm{div}\mathbf{F}& =\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}& \hfill & \\ \hfill \mathrm{curl}\mathbf{F}& =\left(\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z},\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x},\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right).& \hfill & \end{array}$$

These formulas are easy to memorize using a tool called the “del” operator, denoted by $\nabla$. Think of $\nabla$ as a “fake” vector composed of all the partial derivatives that we use just to help us remember the formulas:

$$\begin{array}{lll}\hfill \nabla =\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right).& & \hfill \end{array}$$

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 $\mathbf{F}=\left(F_1,F_2,F_3\right)$:

$$\begin{array}{llll}\hfill \nabla \cdot \mathbf{F}& =\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)\cdot \left(F_1,F_2,F_3\right)& \hfill & \\ \hfill & =\frac{\partial }{\partial x}{F}_1+\frac{\partial }{\partial y}{F}_2+\frac{\partial }{\partial z}{F}_3& \hfill & \end{array}$$

If we think of each “multiplication” in the dot product as instead being the derivative of the corresponding $F$, 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

$$\begin{array}{lll}\hfill \mathrm{div}\mathbf{F}=\nabla \cdot \mathbf{F}=\frac{\partial F_1}{\partial x}+\frac{\partial F_2}{\partial y}+\frac{\partial F_3}{\partial z}.& & \hfill \end{array}$$

This notation is also helpful because you will always know that $\nabla \cdot \mathbf{F}$ 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 $\mathrm{curl}\mathbf{F}$ is equal to $\nabla \times \mathbf{F}$. To see this, let’s take the cross product of the $\nabla$ vector with $\mathbf{F}$.

$$\begin{array}{llll}\hfill \nabla \times \mathbf{F}& =\left(\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}\right)\times \left(F_1,F_2,F_3\right)& \hfill & \\ \hfill & =\left|\begin{array}{ccc}\hfill \mathbf{i}\hfill & \hfill \mathbf{j}\hfill & \hfill \mathbf{k}\hfill \\ \hfill \frac{\partial }{\partial x}\hfill & \hfill \frac{\partial }{\partial y}\hfill & \hfill \frac{\partial }{\partial z}\hfill \\ \hfill F_1\hfill & \hfill F_2\hfill & \hfill F_3\hfill \end{array}\right|& \hfill & \\ \hfill & =\mathbf{i}\left(\frac{\partial }{\partial y}{F}_3-\frac{\partial }{\partial z}{F}_2\right)-\mathbf{j}\left(\frac{\partial }{\partial x}{F}_3-\frac{\partial }{\partial z}{F}_1\right)+\mathbf{k}\left(\frac{\partial }{\partial x}{F}_2-\frac{\partial }{\partial y}{F}_1\right)& \hfill & \\ \hfill & =\left(\frac{\partial F_3}{\partial y}-\frac{\partial F_2}{\partial z}\right)\mathbf{i}+\left(\frac{\partial F_1}{\partial z}-\frac{\partial F_3}{\partial x}\right)\mathbf{j}+\left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)\mathbf{k}& \hfill & \end{array}$$

This is exactly the formula we gave above. So if you can use the rule that “multiplication” by $\frac{\partial }{\partial x}$ 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

$$\begin{array}{lll}\hfill \mathrm{curl}\mathbf{F}=\nabla \times \mathbf{F}.& & \hfill \end{array}$$

Contents by topic

1. Geometry in higher dimensions

2. Differential Calculus

3. Integral calculus

4. Vector calculus

4a. Vector field basics

• Vector fields
• The idea of divergence and curl
• The components of the curl
• More details on the components of the curl
• Divergence and curl notation
• Divergence and curl example
• Subtleties about divergence
• Subtleties about curl

4b. Parametrized curves

4c. Parametrized surfaces

4d. Path integrals and line integrals

4e. Surface integrals

5. The fundamental theorems of vector calculus

6. Review material