91Ó°ÊÓ

Find the \(\mathbf{k}\) -component of \(\operatorname{curl}(\mathbf{F})\) for the following vector fields on the plane. \(\mathbf{F}=\left(x^{2} y\right) \mathbf{i}+\left(x y^{2}\right) \mathbf{j}\)

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

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

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

Give a formula \(\mathbf{F}=M(x, y) \mathbf{i}+N(x, y) \mathbf{j}\) for the vector field in the plane that has the property that \(\mathbf{F}\) points toward the origin with magnitude inversely proportional to the square of the distance from \((x, y)\) to the origin. (The field is not defined at \((0,0) . )\)

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

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.) $$ \begin{array}{l}{\text { Spherical cap The cap cut from the sphere } x^{2}+y^{2}+z^{2}=9} \\ {\text { by the cone } z=\sqrt{x^{2}+y^{2}}}\end{array} $$

In Exercises \(1-8,\) integrate the given function over the given surface. Portion of plane \(F(x, y, z)=z,\) over the portion of the plane \(x+y+z=4\) that lies above the square \(0 \leq x \leq 1\) \(0 \leq y \leq 1,\) in the \(x y\) -plane

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

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.) $$ \begin{array}{l}{\text { Spherical cap The portion of the sphere } x^{2}+y^{2}+z^{2}=4 \text { in }} \\ {\text { the first octant between the } x y \text { -plane and the cone } z=\sqrt{x^{2}+y^{2}}}\end{array} $$