Multivariable calculus and vector analysis

Divergence and curl example

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

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

(The formula for curl was somewhat motivated in an earlier reading.)

Given these formulas, there isn’t a whole lot to computing the divergence and curl. Just “plug and chug,” as they say.

Example

Calculate the divergence and curl of $\mathbf{F}=\left(-y,xy,z\right)$.

Solution: Since

$$\begin{array}{lll}\hfill \frac{\partial F_1}{\partial x}=0,\frac{\partial F_2}{\partial y}=x,\frac{\partial F_3}{\partial z}=1& & \hfill \end{array}$$

we calculate that

$$\begin{array}{lll}\hfill \text{ div}\left(\mathbf{F}\right)=0+x+1=x+1.& & \hfill \end{array}$$

Since

$$\begin{array}{lll}\hfill \frac{\partial F_1}{\partial y}=-1,\frac{\partial F_2}{\partial x}=y,& & \hfill \\ \hfill \frac{\partial F_1}{\partial z}=\frac{\partial F_2}{\partial z}=\frac{\partial F_3}{\partial x}=\frac{\partial F_3}{\partial y}=0,& & \hfill \end{array}$$

we calculate that

$$\begin{array}{lll}\hfill \text{ curl}\left(\mathbf{F}\right)=\left(0-0,0-0,y+1\right)=\left(0,0,y+1\right).& & \hfill \end{array}$$

Good things we can do this with math. If you can figure out the divergence or curl from the picture of the vector field (below), you doing better than I can.

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