Div and Curl

By appending a 0 z-component to $\vec{v(x,y)}=<P(x,y),Q(x,y)>$, we can consider the vector field $\vec{w(x,y,z)}=<P(x,y),Q(x,y),0>$ whose divergence is ${\displaystyle \frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}}$ and whose curl is $\hat{k}({\displaystyle \frac{\partial Q}{\partial x}- \frac{\partial P}{\partial y}})$.

The folder :Maple V Release 4:Math 222:Lecture 5 contains small movies of the flows associated to the four vector fields given below. (The flow of a vector field on a set at time t gives a new set representing where at time t the points of the old set would be if each followed a flowline for time t.)

The four examples there are

Nonzero Divergence and Zero Curl:

v(x,y)=<5x+10y,10x+5y>.

Zero Divergence and Zero Curl:

v(x,y)=<5x+10y,10x-5y>.

Zero Divergence and Nonzero Curl:

v(x,y)=<10y,-10x>.

Nonzero Divergence and Zero Curl:

v(x,y)=<5x+15y,-10x+5y>.

For each of these, look at the vector fields and a few integral curves in one of the MacMath programs mentioned above. Can you see a relationship to the flows?