Chapter 9

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

Find the gradient of the given function at the indicated point. $$ f(x, y)=x^{2}-4 y^{2} ;(2,4) $$

Graph the curve traced by the given vector function. $$ \mathbf{r}(t)=\left\langle e^{t}, e^{2 t}\right\rangle $$

Find the image of the set \(S\) under the given transformation. $$ \text { S: } 1 \leq u \leq 2,1 \leq v \leq 2 ; x=u v, y=v^{2} $$

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). $$ \begin{aligned} &\mathbf{F}=x^{2} \mathbf{i}+2 y z \mathbf{j}+4 z^{3} \mathbf{k} ; D \text { the region bounded by the paral- }\\\ &\text { lelepiped defined by } 0 \leq x \leq 1,0 \leq y \leq 2,0 \leq z \leq 3 \end{aligned} $$

Evaluate the given iterated integral. $$ \int_{0}^{\sqrt{2}} \int_{\sqrt{y}}^{2} \int_{0}^{e^{x}} x d z d x d y $$

Use Green's theorem to evaluate the given line integral. \(\$_{C}\left(x+y^{2}\right) d x+\left(2 x^{2}-y\right) d y\), where \(C\) is the boundary of the region determined by the graphs of \(y \quad x^{2}, y \quad 4\)

Find the surface area of those portions of the sphere \(x^{2}+y^{2}+z^{2}=2\) that are within the cone \(z^{2}=x^{2}+y^{2}\)

Evaluate the given partial integral. $$ \int_{x^{3}}^{x} e^{2 y / x} d y $$

In Problems, 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. $$ \int_{(1,0)}^{(3,4)} \frac{x d x+y d y}{\sqrt{x^{2}+y^{2}}} \text { on any path not through the origin } $$

Evaluate \(\int_{C} G(x, y, z) d x, \int_{C} G(x, y, z) d y\), \(\int_{C} G(x, y, z) d z\), and \(\int_{C} G(x, y, z) d s\) on the indicated curve \(C\). $$ G(x, y, z)=4 x y z ; x=\frac{1}{3} t^{3}, y=t^{2}, z=2 t, 0 \leq t \leq 1 $$

In Problems \(1-6\), graph some representative vectors in the given vector field. $$ \mathbf{F}(x, y)=x \mathbf{j} $$

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ f(x, y)=\frac{y}{x} ;(2,2) $$

Sketch some of the level curves associated with the given function. $$ f(x, y)=\tan ^{-1}(y-x) $$

In Problems, find the gradient of the given function at the indicated point. $$ f(x, y)=\sqrt{x^{3} y-y^{4}} ;(3,2) $$

In Problems, graph the curve traced by the given vector function. \(\mathbf{r}(t)=\cosh t \mathbf{i}+3 \sinh t \mathbf{j}\)

\( \mathbf{r}(t)\) is the position vector of a moving particle. Graph the curve and the velocity and acceleration vectors at the indicated time. Find the speed at that time. $$ \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t^{3} \mathbf{k} ; t=2 $$

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. $$ \int_{(1,0)}^{(3,4)} \frac{x d x+y d y}{\sqrt{x^{2}+y^{2}}} \text { on any path not through the origin } $$

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

Find the gradient of the given function at the indicated point. $$ f(x, y)=\sqrt{x^{3} y-y^{4}} ;(3,2) $$

Graph the curve traced by the given vector function. $$ \mathbf{r}(t)=\cosh t \mathbf{i}+3 \sinh t \mathbf{j} $$

In Problems \(7-10\), find the Jacobian of the transformation \(T\) from the \(u v\) -plane to the \(x y\) -plane. $$ x=v e^{-u}, y=v e^{u} $$

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). $$ \begin{aligned} &\mathbf{F}=y^{3} \mathbf{i}+x^{3} \mathbf{j}+z^{3} \mathbf{k} ; D \text { the region bounded within by }\\\ &z=\sqrt{4-x^{2}-y^{2}}, x^{2}+y^{2}=3, z=0 \end{aligned} $$

In Problems, find the volume of the solid bounded by the graphs of the given equations. $$ \begin{aligned} &\text { Between } x^{2}+y^{2} \quad 1 \text { and } x^{2}+y^{2} \quad 9 \\ &z \quad \sqrt{16-x^{2}-y^{2}}, z \quad 0 \end{aligned} $$

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

Find the surface area of the portion of the sphere \(x^{2}+y^{2}+z^{2}=25\) that is above the region in the first quadrant bounded by the graphs of \(x=0, y=0,4 x^{2}+y^{2}=25\) [Hint: Integrate first with respect to \(x\).]

In Problems, 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. $$ \int_{(1,2)}^{(3,6)}\left(2 y^{2} x-3\right) d x+\left(2 y x^{2}+4\right) d y $$

Evaluate \(\int_{C}(2 x+y) d x+x y d y\) on the given curve \(C\) between \((-1,2)\) and \((2,5)\). $$ y=x+3 $$

In Problems \(7-16\), find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=x z \mathbf{i}+y z \mathbf{j}+x y \mathbf{k} $$

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ f(x, y)=(x-1)^{2}-y^{2} ;(1,1) $$

Describe the level surfaces but do not graph. $$ F(x, y, z)=\frac{x^{2}}{9}+\frac{z^{2}}{4} $$

In Problems, find the gradient of the given function at the indicated point. $$ F(x, y, z)=x^{2} z^{2} \sin 4 y ;(-2, \pi / 3,1) $$

In Problems, graph the curve traced by the given vector function. \(\mathbf{r}(t)=\langle\sqrt{2} \sin t, \sqrt{2} \sin t, 2 \cos t\rangle ; 0 \leq t \leq \pi / 2\)

\( \mathbf{r}(t)\) is the position vector of a moving particle. Graph the curve and the velocity and acceleration vectors at the indicated time. Find the speed at that time. $$ \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k} ; t=1 $$

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. The integral \(\int_{C}\left(x^{2}+y^{2}\right) d x+2 x y d y\), where \(C\) is given by \(y=x^{3}\) from \((0,0)\) to \((1,1)\), has the same value on the curve \(y=x^{6}\) from \((0,0)\) to \((1,1)\).____

Find the Jacobian of the transformation \(T\) from the \(u v\)-plane to the \(x y\)-plane. $$ x=v e^{-u}, y=v e^{u} $$

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. $$ \int_{(1,2)}^{(3,6)}\left(2 y^{2} x-3\right) d x+\left(2 y x^{2}+4\right) d y $$

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

Find the gradient of the given function at the indicated point. $$ F(x, y, z)=x^{2} z^{2} \sin 4 y ;(-2, \pi / 3,1) $$

Graph the curve traced by the given vector function. $$ \mathbf{r}(t)=\langle\sqrt{2} \sin t, \sqrt{2} \sin t, 2 \cos t\rangle ; 0 \leq t \leq \pi / 2 $$

In Problems \(7-10\), find the Jacobian of the transformation \(T\) from the \(u v\) -plane to the \(x y\) -plane. $$ x=e^{3 u} \sin v, y=e^{3 u} \cos v $$

In Problems, find the volume of the solid bounded by the graphs of the given equations. $$ \text { z } \quad \sqrt{x^{2}+y^{2}}, x^{2}+y^{2} \quad 25, z \quad 0 $$

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

Use Green's theorem to evaluate the given line integral. $$ \begin{aligned} &\oint_{C}(x-3 y) d x+(4 x+y) d y \text { , where } C \text { is the rectangle with }\\\ &\text { vertices }(-2,0),(3,0),(3,2),(-2,2) \end{aligned} $$

Find the surface area of that portion of the graph of \(z=x^{2}-y^{2}\) that is in the first octant within the cylinder \(x^{2}+y^{2}=4\)

In Problems \(7-16, \mathbf{r}(t)\) is the position vector of a moving particle. Find the tangential and normal components of the acceleration at any \(t\). $$ \mathbf{r}(t)=3 \cos t \mathbf{i}+2 \sin t \mathbf{j}+t \mathbf{k} $$

In Problems, 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. $$ \int_{(-1,1)}^{(0,0)}(5 x+4 y) d x+\left(4 x-8 y^{3}\right) d y $$

Evaluate \(\int_{C}(2 x+y) d x+x y d y\) on the given curve \(C\) between \((-1,2)\) and \((2,5)\). $$ y=x^{2}+1 $$

In Problems \(7-16\), find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=10 y z \mathbf{i}+2 x^{2} z \mathbf{j}+6 x^{3} \mathbf{k} $$

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ f(x, y)=\frac{y-1}{\sin x} ;\left(\pi / 6, \frac{3}{2}\right) $$