91Ó°ÊÓ

Graph some representative vectors in the given vector field. $$ \mathbf{F}(x, y)=x \mathbf{i}+2 y \mathbf{j} $$

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). $$ \begin{aligned} &\mathbf{F}=4 x \mathbf{i}+y \mathbf{j}+4 z \mathbf{k} ; D \text { the region bounded by the sphere }\\\ &x^{2}+y^{2}+z^{2}=4 \end{aligned} $$

Find the surface area of that portion of the paraboloid \(z=x^{2}+y^{2}\) that is below the plane \(z=2\).

Show that the given line integral is independent of the path. Evaluate in two ways: (a) Find a potential function \(\phi\) and then use Theorem 9.9.1, and (b) Use any convenient path between the endpoints of the path. $$ \int_{(0,0)}^{(\pi / 2,0)} \cos x \cos y d x+(1-\sin x \sin y) d y $$

Evaluate \(\int_{C} G(x, y) d x, \int_{C} G(x, y) d y\), and \(\int_{C} G(x, y) d s\) on the indicated curve \(C\). $$ G(x, y)=x^{2} / y^{3} ; 2 y=3 x^{2 / 3}, 1 \leq x \leq 8 $$

Use a double integral in polar coordinates to find the area of the region bounded by the graphs of the given polar equations. $$ r \quad 8 \sin 4 \theta, \text { one petal } $$

In Problems \(3-6\), find the image of the set \(S\) under the given transformation. $$ S:-1 \leq u \leq 4,1 \leq v \leq 5 ; u=x-y, v=x+2 y $$

In Problems \(1-4\), verify Stokes' thearem. Assume that the surface \(S\) is oriented upward. \(\mathbf{F}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k} ; \boldsymbol{S}\) that portion of the sphere \(x^{2}+y^{2}+z^{2}=1\) for \(z \geq 0\)

Graph the curve traced by the given vector function. $$ \mathbf{r}(t)=4 \mathbf{i}+2 \cos t \mathbf{j}+3 \sin t \mathbf{k} $$

Evaluate the given iterated integral. $$ \int_{0}^{1} \int_{0}^{1-x} \int_{0}^{\sqrt{y}} 4 x^{2} z^{3} d z d y d x $$