17. Divergence, Curl and Potentials

Homework

1. Compute the gradient of \(f=xy^2+yz^2+zx^2\), i.e. \(\vec\nabla f\).

2. Compute the divergence of \(\vec F=\left\langle x^2yz, xy^2z, xyz^2\right\rangle\), i.e. \(\vec\nabla \cdot F\).

3. Compute the curl of \(\vec F=\left\langle x^2yz, xy^2z, xyz^2\right\rangle\), i.e. \(\vec\nabla\times F\).

4. Compute the divergence of the gradient of \(f=xy^2+yz^2+zx^2\), i.e. \(\vec\nabla\cdot\vec\nabla f\). This is also called the Laplacian of \(f\), i.e. \(\text{Lap}f=\vec\nabla\cdot\vec\nabla f\).

5. Compute the curl of the gradient of \(f=xy^2+yz^2+zx^2\), i.e. \(\vec\nabla\times\vec\nabla f\).

6. Compute the divergence of the curl of \(\vec F=\left\langle x^2yz, xy^2z, xyz^2\right\rangle\), i.e. \(\vec\nabla\cdot\vec\nabla\times\vec F\).

7. Compute the curl of the curl of \(\vec F=\left\langle x^2yz, xy^2z, xyz^2\right\rangle\), i.e. \(\vec\nabla\times\vec\nabla\times\vec F\).

8. If \(f=xyz\) and \(\vec G=\langle x^2yz,xy^2z,xyz^2\rangle\), compute \(\vec\nabla\cdot (f\vec G)\) in two ways.

   1. Find \((\vec\nabla f)\cdot\vec G\) and \(f\vec\nabla\cdot\vec G\) and add them to get \(\vec\nabla\cdot(f\vec G)\).

   2. Find \(f\vec G\) and then \(\vec\nabla\cdot(f\vec G)\).

9. If \(f=xyz\) and \(\vec G=\langle x^2yz,xy^2z,xyz^2\rangle\), compute \(\vec\nabla\times (f\vec G)\) in two ways.

   1. Find \((\vec\nabla f)\times\vec G\) and \(f\vec\nabla\times\vec G\) and add them to get \(\vec\nabla\times(f\vec G)\).

   2. Find \(f\vec G\) and then \(\vec\nabla\times(f\vec G)\).

10. At a point \(P\), we know: \[\begin{aligned} f&=6\qquad& \vec\nabla f&=\langle 4,-2,3\rangle \\ \vec G&=\langle -2,-3,2\rangle\qquad& \vec\nabla\cdot\vec G&=4 \end{aligned}\] Find \(\vec\nabla\cdot(f\vec G)\) at \(P\).

11. At a point \(P\), we know: \[\begin{aligned} f&=6\qquad& \vec\nabla f&=\langle 4,-2,3\rangle \\ \vec G&=\langle -2,-3,2\rangle\qquad& \vec\nabla\times\vec G&=\left\langle 2,4,-3\right\rangle \end{aligned}\] Find \(\vec\nabla\times(f\vec G)\) at \(P\).

12. Find a scalar potential for the vector field: \[ \vec F=\langle 2xy^2+2xz^2,2x^2y+2yz^2,2x^2z+2y^2z+2z\rangle \] or show one does not exist.

13. Find a scalar potential for the vector field: \[ \vec F=\langle 2xy^2-2xz^2,2x^2y-2yz^2,2x^2z-2y^2z+2z\rangle \] or show one does not exist.

14. Show the vector field \(\vec A=\langle \cos z, -\sin z, x\sin z+y\cos z\rangle\) is a vector potential for which of the following vector fields.
    a. \(\vec G=\langle2\sin z, 2\cos z, 0\rangle\)
    b. \(\vec H=\langle2\sin z, -2\cos z, 0\rangle\)
    c. \(\vec J=\langle2\cos z, 2\sin z, 0\rangle\)
    d. \(\vec K=\langle2\cos z, -2\sin z, 0\rangle\)
    e. another vector field but not one of these
    f. \(\vec A\) cannot be a vector potential for any vector field.