91Ó°ÊÓ

In Exercises \(1-8,\) integrate the given function over the given surface. \( G(x, y, z)=z,\) over the cylindrical surface \(y^{2}+z^{2}=4, z \geq 0,1 \leq x \leq 4\)

Find the gradient fields of the functions. $$f(x, y, z)=\ln \sqrt{x^{2}+y^{2}+z^{2}}$$

Which fields are conservative, and which are not? $$\mathbf{F}=y \mathbf{i}+(x+z) \mathbf{j}-y \mathbf{k}$$

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 text.) The first-octant portion of the cone \(z=\) \(\sqrt{x^{2}+y^{2}} / 2\) between the planes \(z=0\) and \(z=3\)

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

Find the curl of each vector field \(\mathbf{F}\). $$\mathbf{F}=(x y+z) \mathbf{i}+(y z+x) \mathbf{j}+(x z+y) \mathbf{k}$$

Find the divergence of the field. $$\mathbf{F}=y e^{x y z} \mathbf{i}+z e^{x y z} \mathbf{j}+x e^{x z} \mathbf{k}$$

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

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

Find the curl of each vector field \(\mathbf{F}\). $$\mathbf{F}=y e^{z} \mathbf{i}+z e^{x} \mathbf{j}-x e^{y} \mathbf{k}$$