Curl And Divergence

(requires JavaScript)

1. Find the curl and the divergence of the vector field $\mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=⟨1,x+yz,xy-\sqrt{z}⟩$.
   $⟨x-y,-y,1⟩$ and ${\displaystyle z-\frac{1}{2\sqrt{z}}}$
2. Find the curl and the divergence of the vector field $\mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=e^x\sin <!--FUNCTION\; APPLICATION-->\left(y\right)\mathbf{i}+e^x\cos <!--FUNCTION\; APPLICATION-->\left(y\right)\mathbf{j}+z\mathbf{k}$.
   $⟨0,0,0⟩$ 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. $\mathrm{curl}<!--FUNCTION\; APPLICATION-->\left(f\right)$
   2. $\nabla f$
   3. $\mathrm{div}<!--FUNCTION\; APPLICATION-->\left(\mathbf{F}\right)$
   4. $\mathrm{curl}<!--FUNCTION\; APPLICATION-->\left(\nabla f\right)$
   5. $\nabla \mathbf{F}$
   6. $\nabla \left(\mathrm{div}<!--FUNCTION\; APPLICATION-->\left(\mathbf{F}\right)\right)$
   7. $\mathrm{div}<!--FUNCTION\; APPLICATION-->\left(\nabla f\right)$
   8. $\nabla \left(\mathrm{div}<!--FUNCTION\; APPLICATION-->\left(f\right)\right)$
   9. $\mathrm{curl}<!--FUNCTION\; APPLICATION-->\left(\mathrm{curl}<!--FUNCTION\; APPLICATION-->\left(\mathbf{F}\right)\right)$
   10. $\mathrm{div}<!--FUNCTION\; APPLICATION-->\left(\mathrm{div}<!--FUNCTION\; APPLICATION-->\left(\mathbf{F}\right)\right)$
   11. $\nabla f\times \mathrm{div}<!--FUNCTION\; APPLICATION-->\left(\mathbf{F}\right)$
   12. $\mathrm{div}<!--FUNCTION\; APPLICATION-->\left(\mathrm{curl}<!--FUNCTION\; APPLICATION-->\left(\nabla f\right)\right)$
4. Determine whether or not the vector field $\mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=2xy\mathbf{i}+\left(x^2+2yz\right)\mathbf{j}+y^2\mathbf{k}$ is conservative. If it is, find its potential function.
   $f<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=x^{2}y+y^{2}z+K$
5. Determine whether or not the vector field $\mathbf{F}<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=y{e}^{-x}\mathbf{i}+e^{-x}\mathbf{j}+2z\mathbf{k}$ is conservative. If it is, find its potential function.
   Not conservative.