Curl and Divergence

Definition of Divergence §

$$\begin{array}{rl}\text{div}\vec{F}& =\nabla \vec{F}\\ & =<\frac{\partial }{\partial x},\frac{\partial }{\partial y}>\cdot <P,Q>\\ & =\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}\end{array}$$

If $\text{div}\vec{F}=0$, $\vec{F}$ is divergence free.

In fluid motion, divergence free vector fields are “incompressible.”

In E&M, divergence free vector field is called “solenoidal.”

Definition of Curl §

$$\begin{array}{rl}\vec{F}& =<P,Q,R>\\ \text{curl}\vec{F}& =\nabla \vec{F}\\ & =\left|\begin{array}{ccc}\frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ P& Q& R\end{array}\right|\end{array}$$

Example 1 §

$\vec{F}=e^x\hat{i}+yz\hat{j}-y{z}^2\hat{k}$

a. Find divergence of $\vec{F}$ at (0, 2, -1)

$$\begin{array}{rl}\text{div}F& =\frac{\partial }{\partial x}[e^x]+\frac{\partial }{\partial y}yz+\frac{\partial }{\partial z}[-y{z}^2]\\ & \\ \text{div}\vec{F}(0,2,-1)& =e^0-1-2(2)(-1)=4\\ & =4\end{array}$$

b. Find the curl of $\vec{F}$

$$\begin{array}{rl}\text{curl}\vec{F}& =\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ e^x& yz& -y{z}^2\end{array}\right|\\ & =(-z^2-y)\hat{i}\\ & \ne 0\end{array}$$

Example 2 §

Find potential function of $\vec{F}=<e^{z}y,e^{z}x,e^z>$

$$\begin{array}{rl}\text{curl}\vec{F}& =\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ \frac{\partial }{\partial x}& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\\ e^{z}y& e^{z}x& e^z\end{array}\right|\\ & =<-e^{z}x,e^{z}y,0>\\ & \ne \vec{0}\end{array}$$

F is not conservative.

Example 3 §

Find the potential function of $\vec{F}(x,y,z)=<\frac{1}{y},-\frac{x}{y^2},i$

$$\begin{array}{rl}f(x,y,z)& =\int \frac{1}{y}dx=\frac{x}{y}+g(y,z)\\ & =\int \frac{-x}{y^2}dy=\frac{x}{y}+h(x,z)\\ & =\int 2zd\end{array}$$