The Divergence of a Vector Field

1. Let $\vF=\langle{F_1,F_2,F_3}\rangle$ and let Show that $\divg(\vG)=\vec{0}$.

2. Vector fields with a zero divergence everywhere in their domain are called divergence-free or incompressible vector fields. Which of the following vector fields are divergence-free?

   1. $\vF=\langle{-y,z,x}\rangle$
   2. $\vF=\langle{\cos(yz),3xe^{z-x},6(x+y+z)^3}\rangle$
   3. $\vF=\langle{4xyz,y^2z,yz^2}\rangle$
   4. $\vF=\nabla f$ where $f$ is a scalar function of $x$, $y$, and $z$
3. Let $\vF_1=\langle{3(x-z)^2,2\cos(x)+3yz+y,-(z-1)^2+e^{xy}}\rangle$. Calculate the divergence of $\vF_1$ and give a point where $\divg(\vF_1)=0$.
4. Is $\vF_1$ a divergence free vector field?

In this exercise, we will look at how different forms of component function's dependence on our variables will change the divergence of the vector field. For each of the vector fields given below, decide if the divergence of the vector field must be zero and write a sentence or two to justify your decision.