91Ó°ÊÓ

In Exercises \(1-8,\) find the divergence of the field. $$ \mathbf{F}=(x \ln y) \mathbf{i}+(y \ln z) \mathbf{j}+(z \ln x) \mathbf{k} $$

In Exercises \(1-16,\) find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) $$ \text {The paraboloid }z=9-x^{2}-y^{2}, z \geq 0 $$

In Exercises \(1-8,\) find the divergence of the field. $$ \mathbf{F}=y e^{x y z} \mathbf{i}+z e^{x y z} \mathbf{j}+x e^{x y z} \mathbf{k} $$

Find the \(\mathbf{k}\) -component of \(\operatorname{curl}(\mathbf{F})\) for the following vector fields on the plane. \(\mathbf{F}=\left(x e^{y}\right) \mathbf{i}+\left(y e^{x}\right) \mathbf{j}\)

In Exercises \(1-8,\) integrate the given function over the given surface. Sphere \(\quad G(x, y, z)=x^{2},\) over the unit sphere \(x^{2}+y^{2}+z^{2}=1\)

In Exercises \(1-16,\) find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) $$ \begin{array}{l}{\text { Cone frustum The first-octant portion of the cone } z=} \\ {\sqrt{x^{2}+y^{2}} / 2 \text { between the planes } z=0 \text { and } z=3}\end{array} $$

Find the gradient fields of the functions in Exercises \(1-4\) $$g(x, y, z)=e^{z}-\ln \left(x^{2}+y^{2}\right)$$

In Exercises \(1-6,\) find the curl of each vector field \(\mathbf{F}\) $$\mathbf{F}=(x y+z) \mathbf{i}+(y z+x) \mathbf{j}+(x z+y) \mathbf{k}$$

In Exercises \(1-6,\) find the curl of each vector field \(\mathbf{F}\) $$\mathbf{F}=y e^{2} \mathbf{i}+z e^{x} \mathbf{j}-x e^{y} \mathbf{k}$$

Find the gradient fields of the functions in Exercises \(1-4\) $$g(x, y, z)=x y+y z+x z$$