91Ó°ÊÓ

Evaluate the surface integral. \(\iint_{S}\left(x^{2}+y^{2}+z^{2}\right) d S\) \(S\) is the part of the cylinder \(x^{2}+y^{2}=9\) between the planes \(z=0\) and \(z=2,\) together with its top and bottom disks

Evaluate the line integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(C\) is given by the vector function \(\mathbf{r}(t)\) $$ \begin{array}{l}{\mathbf{F}(x, y, z)=\left(x+y^{2}\right) \mathbf{i}+x z \mathbf{j}+(y+z) \mathbf{k}} \\ {\mathbf{r}(t)=t^{2} \mathbf{i}+t^{3} \mathbf{j}-2 t \mathbf{k}, \quad 0 \leqslant t \leqslant 2}\end{array} $$

If a circle \(C\) with radius 1 rolls along the outside of the circle \(x^{2}+y^{2}=16,\) a fixed point \(P\) on \(C\) traces out a curve called an epicycloid, with parametric equations \(x=5 \cos t-\cos 5 t, y=5 \sin t-\sin 5 t .\) Graph the epicycloid and use (5) to find the area it encloses.

Evaluate the line integral \(\int_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(C\) is given by the vector function \(\mathbf{r}(t)\) $$ \begin{array}{l}{\mathbf{F}(x, y, z)=\sin x \mathbf{i}+\cos y \mathbf{j}+x z \mathbf{k}} \\ {\mathbf{r}(t)=t^{3} \mathbf{i}-t^{2} \mathbf{j}+t \mathbf{k}, \quad 0 \leqslant t \leqslant 1}\end{array} $$

(a) If \(C\) is the line segment connecting the point \(\left(x_{1}, y_{1}\right)\) to the point \(\left(x_{2}, y_{2}\right)\), show that $$\int_{c} x d y-y d x=x_{1} y_{2}-x_{2} y_{1}$$ (b) If the vertices of a polygon, in counterclockwise order. are \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right) \ldots \ldots\left(x_{n}, y_{n}\right)\), show that the area of the polygon is $$\begin{aligned} A=& \frac{1}{2}\left[\left(x_{1} y_{2}-x_{2} y_{1}\right)+\left(x_{2} y_{3}-x_{3} y_{2}\right)+\right.\\\ &\left.+\left(x_{n-1} y_{n}-x_{n} y_{n-1}\right)+\left(x_{n} y_{1}-x_{1} y_{n}\right)\right] \end{aligned}$$ (c) Find the area of the pentagon with vertices \((0,0),(2,1)\), \((1,3),(0,2)\), and \((-1,1).\)

Plot the vector field and guess where \(\operatorname{div} \mathbf{F}>0\) and where div \(\mathbf{F}<0 .\) Then calculate div \(\mathbf{F}\) to check your guess. \(\mathbf{F}(x, y)=\left\langle x y, x+y^{2}\right\rangle\)

Find a parametric representation for the surface. The part of the hyperboloid \(4 x^{2}-4 y^{2}-z^{2}=4\) that lies in front of the \(y z\) -plane

\(21-24\) Find the gradient vector field of \(f\) $$ f(x, y)=y \sin (x y) $$

Show that any vector field of the form $$ \mathbf{F}(x, y, z)=f(x) \mathbf{i}+g(y) \mathbf{j}+h(z) \mathbf{k} $$ where \(f, g, h\) are differentiable functions, is irrotational.

Show that any vector field of the form $$ \mathbf{F}(x, y, z)=f(y, z) \mathbf{i}+g(x, z) \mathbf{j}+h(x, y) \mathbf{k} $$ is incompressible.