### Curl And Divergence

(requires JavaScript)

1. Find the curl and the divergence of the vector field $\mathbf{F}\left(x,y,z\right)=⟨1,x+yz,xy-\sqrt{z}⟩$.
$⟨x-y,-y,1⟩$ and $z-\frac{1}{2\sqrt{z}}$
2. Find the curl and the divergence of the vector field $\mathbf{F}\left(x,y,z\right)={e}^{x}\mathrm{sin}\left(y\right)\mathbf{i}+{e}^{x}\mathrm{cos}\left(y\right)\mathbf{j}+z\mathbf{k}$.
$⟨0,0,0⟩$ and $1$
3. Let $f$ be a scalar field and $\mathbf{F}$ be a vector field. For each of the following expressions, determine whether it is meaningful, and if yes, whether it's a vector or a scalar.

1. $\mathrm{curl}\left(f\right)$
2. $\nabla f$
3. $\mathrm{div}\left(\mathbf{F}\right)$
4. $\mathrm{curl}\left(\nabla f\right)$
5. $\nabla \mathbf{F}$
6. $\nabla \left(\mathrm{div}\left(\mathbf{F}\right)\right)$
7. $\mathrm{div}\left(\nabla f\right)$
8. $\nabla \left(\mathrm{div}\left(f\right)\right)$
9. $\mathrm{curl}\left(\mathrm{curl}\left(\mathbf{F}\right)\right)$
10. $\mathrm{div}\left(\mathrm{div}\left(\mathbf{F}\right)\right)$
11. $\nabla f×\mathrm{div}\left(\mathbf{F}\right)$
12. $\mathrm{div}\left(\mathrm{curl}\left(\nabla f\right)\right)$
4. Determine whether or not the vector field $\mathbf{F}\left(x,y,z\right)=2xy\mathbf{i}+\left({x}^{2}+2yz\right)\mathbf{j}+{y}^{2}\mathbf{k}$ is conservative. If it is, find its potential function.
$f\left(x,y,z\right)={x}^{2}y+{y}^{2}z+K$
5. Determine whether or not the vector field $\mathbf{F}\left(x,y,z\right)=y{e}^{-x}\mathbf{i}+{e}^{-x}\mathbf{j}+2z\mathbf{k}$ is conservative. If it is, find its potential function.
Not conservative.