Chapter 9

In Problems, compute the gradient for the given function. $$ F(x, y, z)=\frac{x y^{2}}{z^{3}} $$

In Problems, graph the curve traced by the given vector function. \(\mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}+\cos t \mathbf{k} ; t \geq 0\)

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. The binormal vector is perpendicular to the osculating plane.____

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). \(\mathbf{F}=x^{3} \mathbf{i}+y^{3} \mathbf{j}+z^{3} \mathbf{k} ; D\) the region bounded by the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\)

Verify Stokes' thearem. Assume that the surface \(S\) is orienled upwand. \(\mathbf{F}=z \mathbf{i}+x \mathbf{j}+y \mathbf{k} ; S\) that portion of theplane \(2 x+y+2 z=6\) in the first octant

Use a double integral in polar coordinates to find the area of the region bounded by the graphs of the given polar equations. $$ 2 \sin \theta, r \quad 1, \text { common area } $$

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. $$ \int_{(1,0)}^{(3,2)}(x+2 y) d x+(2 x-y) d y $$

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

Compute the gradient for the given function. $$ F(x, y, z)=\frac{x y^{2}}{z^{3}} $$

\(\mathbf{r}(t)\) 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. $$ \mathbf{r}(t)=-\cosh 2 t \mathbf{i}+\sinh 2 t \mathbf{j} ; t=0 $$

Graph the curve traced by the given vector function. $$ \mathbf{r}(t)=t \mathbf{i}+2 t \mathbf{j}+\cos t \mathbf{k} ; t \geq 0 $$

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

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

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

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 \(z \geq 0\)

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

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 \(C\). $$ G(x, y)=x^{2} / y^{3} ; 2 y=3 x^{2 / 3}, 1 \leq x \leq 8 $$

In Problems \(1-6\), graph some representative vectors in the given vector field. $$ \mathbf{F}(x, y)=x \mathbf{i}+2 y \mathbf{j} $$

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ f(x, y)=x^{2}+y^{2} ;(-1,3) $$

Sketch some of the level curves associated with the given function. $$ f(x, y)=\sqrt{36-4 x^{2}-9 y^{2}} $$

In Problems, compute the gradient for the given function. $$ F(x, y, z)=x y \cos y z $$

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

\( \mathbf{r}(t)\) 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. $$ \mathbf{r}(t)=2 \cos t \mathbf{i}+(1+\sin t) \mathbf{j} ; t=\pi / 3 $$

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

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

Evaluate the given partial integral. $$ \int_{\sqrt{y}}^{y^{3}}\left(8 x^{3} y-4 x y^{2}\right) d x $$

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. $$ \int_{(0,0)}^{(\pi / 2,0)} \cos x \cos y d x+(1-\sin x \sin y) d y $$

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

Compute the gradient for the given function. $$ F(x, y, z)=x y \cos y z $$

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

Find the image of the set \(S\) under the given transformation. $$ S: 0 \leq u \leq 1,0 \leq v \leq 2 ; x=u^{2}-v^{2}, y=u v $$

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). $$ \begin{aligned} &\mathbf{F}=y^{2} \mathbf{i}+x z^{3} \mathbf{j}+(z-1)^{2} \mathbf{k} ; D \text { the region bounded by the }\\\ &\text { cylinder } x^{2}+y^{2}=16 \text { and the planes } z=1, z=5 \end{aligned} $$

In Problems, find the volume of the solid bounded by the graphs of the given equations. $$ \text { One petal of } r \quad 5 \cos 3 \theta, z \quad 0, z \quad 4 $$

Evaluate the given iterated integral. $$ \int_{0}^{\pi / 2} \int_{0}^{r^{2}} \int_{0}^{y} \cos \left(\frac{x}{y}\right) d z d x d y $$

Use Green's theorem to evaluate the given line integral. $$ \oint_{C} 2 y d x+5 x d y, \text { where } C \text { is the circle }(x-1)^{2}+(y+3)^{2} \quad 25 $$

In Problems \(5-12\), use Stokes' theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\). Assume \(C\) is oriented counterclockwise as viewed from above. \(\mathbf{F}=(2 z+x) \mathbf{i}+(y-z) \mathbf{j}+(x+y) \mathbf{k} ; C\) the triangle with vertices \((1,0,0),(0,1,0),(0,0,1)$$\mathbf{F}=(2 z+x) \mathbf{i}+(y-z) \mathbf{j}+(x+y) \mathbf{k} ; C\) the triangle with vertices \((1,0,0),(0,1,0),(0,0,1)\)

Find the surface area of that portion of the paraboloid \(z=4-x^{2}-y^{2}\) that is above the \(x y\) -plane.

Evaluate the given partial integral. $$ \int_{0}^{2 x} \frac{x y}{x^{2}+y^{2}} d y $$

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. $$ \int_{(4,1)}^{(4,4)} \frac{-y d x+x d y}{y^{2}} \text { on any path not crossing the } x \text { -axis } $$

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 \(C\). $$ G(x, y, z)=z ; x=\cos t, y=\sin t, z=t, 0 \leq t \leq \pi / 2 $$

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

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ f(x, y)=\frac{x^{2}}{4}+\frac{y^{2}}{9} ;(-2,-3) $$

Sketch some of the level curves associated with the given function. $$ f(x, y)=e^{y-x^{2}} $$

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

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

\( \mathbf{r}(t)\) 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. $$ \mathbf{r}(t)=2 \mathbf{i}+(t-1)^{2} \mathbf{j}+t \mathbf{k} ; t=2 $$

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. $$ \nabla z \text { is perpendicular to the graph of } z=f(x, y) \text {. } $$___

Use Stokes' theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\). Assume \(C\) is oriented counterclockwise as viewed from above. \(\mathbf{F}=(2 z+x) \mathbf{i}+(y-z) \mathbf{j}+(x+y) \mathbf{k} ; C\) the triangle with vertices \((1,0,0),(0,1,0),(0,0,1)\)

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. $$ \int_{(4,1)}^{(4,4)} \frac{-y d x+x d y}{y^{2}} \text { on any path not crossing the } x \text {-axis } $$