Chapter 9

$$ \text { Verify the divergence theorem. } $$ $$ \begin{aligned} &\mathbf{F}=x y \mathbf{i}+y z \mathbf{j}+x z \mathbf{k} ; D \text { the region bounded by the unit cube }\\\ &\text { defined by } 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1 \end{aligned} $$

Consider a transformation \(T\) definedby \(x=4 u-v, y=5 u+4 v\) Find the images of the points \((0,0),(0,2),(4,0)\), and \((4,2)\) in the \(u v\) -plane under \(T\).

In Problems \(1-8\), evaluate the given iterated integral. $$ \int_{2}^{4} \int_{-2}^{2} \int_{-1}^{1}(x+y+z) d x d y d z $$

Verify Green's theorem by evaluating both integrals. $$ \begin{aligned} &\oint_{C}(x-y) d x+x y d y \quad \iint_{R}(y+1) d A, \text { where } C \text { is the triangle }\\\ &\text { with vertices }(0,0),(1,0),(1,3) \end{aligned} $$

In Problems \(1-4\), verify Stokes' thearem. Assume that the surface \(S\) is oriented upward. \(\mathbf{F}=5 y \mathbf{i}-5 x \mathbf{j}+3 \mathbf{k} ; S\) that pottion of theplane \(z=1\) within the cylinder \(x^{2}+y^{2}=4\)

Find the surface area of that portion of the plane \(2 x+3 y+4 z=12\) that is bounded by the coordinate planes in the first octant.

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

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 \(C\). $$ G(x, y)=2 x y ; x=5 \cos t, y=5 \sin t, 0 \leq t \leq \pi / 4 $$

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

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ f(x, y)=x-2 y ;(6,1) $$

Sketch some of the level curves associated with the given function. $$ f(x, y)=x+2 y $$

In Problems, compute the gradient for the given function. $$ f(x, y)=x^{2}-x^{3} y^{2}+y^{4} $$

\( \mathbf{r}(t)\) 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. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+\frac{1}{4} t^{4} \mathbf{j} ; t=1 $$

In Problems, graph the curve traced by the given vector function. \(\mathbf{r}(t)=2 \sin t \mathbf{i}+4 \cos t \mathbf{j}+t \mathbf{k} ; t \geq 0\)

Verify the divergence theorem. \(\mathbf{F}=x y \mathbf{i}+y z \mathbf{j}+x z \mathbf{k} ; D\) the region bounded by the unit cube defined by \(0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\)

Evaluate the given iterated integral. $$ \int_{2}^{4} \int_{-2}^{2} \int_{-1}^{1}(x+y+z) d x d y d z $$

Verify Stokes' thearem. Assume that the surface \(S\) is orienled upwand. \(F=5 y \mathbf{i}-5 x \mathbf{j}+3 \mathbf{k} ; S\) that portion of theplane \(z=1\) within the cylinder \(x^{2}+y^{2}=4\)

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

Compute the gradient for the given function. $$ f(x, y)=x^{2}-x^{3} y^{2}+y^{4} $$

For the given position function, find the unit tangent. $$ \mathbf{r}(t)=(t \cos t-\sin t) \mathbf{i}+(t \sin t+\cos t) \mathbf{j}+t^{2} \mathbf{k}, t>0 $$

Graph the curve traced by the given vector function. $$ \mathbf{r}(t)=2 \sin t \mathbf{i}+4 \cos t \mathbf{j}+t \mathbf{k} ; t \geq 0 $$

$$ \text { Verify the divergence theorem. } $$ $$ \begin{aligned} &\mathbf{F}=6 x y \mathbf{i}+4 y z \mathbf{j}+x e^{-y} \mathbf{k} ; D \text { the region bounded by the three }\\\ &\text { coordinate planes and the plane } x+y+z=1 \end{aligned} $$

Consider a transformation \(T\) defined by \(x=\sqrt{v-u}\) \(y=v+u\). Find the images of the points \((1,1),(1,3)\), and \((\sqrt{2}, 2)\) in the \(x y\) -plane under \(T^{-1}\).

Evaluate the given iterated integral. $$ \int_{1}^{3} \int_{1}^{x} \int_{2}^{x y} 24 x y d z d y d x $$

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. $$ r \quad 2+\cos \theta $$

Verify Green's theorem by evaluating both integrals. $$ \begin{aligned} &\oint_{C} 3 x^{2} y d x+\left(x^{2}-5 y\right) d y \quad \iint_{R}\left(2 x-3 x^{2}\right) d A, \text { where } C \text { is }\\\ &\text { the rectangle with vertices }(-1,0),(1,0),(1,1),(-1,1) \end{aligned} $$

In Problems \(1-4\), verify Stokes' thearem. Assume that the surface \(S\) is oriented upward. \(\mathbf{F}=2 z \mathbf{i}-3 x \mathbf{j}+4 y \mathbf{k} ; \boldsymbol{S}\) that portion of the paraboloid \(z=16-x^{2}-y^{2}\) for \(z \geq 0\)

Find the surface area of that portion of the plane \(2 x+3 y+4 z=12\) that is above the region in the first quadrant bounded by the graph \(r=\sin 2 \theta\).

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. $$ \int_{(1,1)}^{(2,4)} 2 x y d x+x^{2} d y $$

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 \(C\). $$ G(x, y)=x^{3}+2 x y^{2}+2 x ; x=2 t, y=t^{2}, 0 \leq t \leq 1 $$

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

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

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

In Problems, compute the gradient for the given function. $$ f(x, y)=y-e^{-2 x^{2} y} $$

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

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

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. The path of a moving particle whose position vector is \(\mathbf{r}(t)=\left(t^{2}+1\right) \mathbf{i}+4 \mathbf{j}+t^{4} \mathbf{k}\) lies in a plane.____

Verify the divergence theorem. \(\mathbf{F}=6 x y \mathbf{i}+4 y z \mathbf{j}+x e^{-y} \mathbf{k} ; D\) the region bounded by the three coordinate planes and the plane \(x+y+z=1\)

Use a double integral in polar coordinates to find the area of the region bounded by the graphs of the given polar equations. $$ r \quad 2+\cos \theta $$

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

Compute the gradient for the given function. $$ f(x, y)=y-e^{-2 x^{2} y} $$

For the given position function, find the unit tangent. $$ \mathbf{r}(t)=e^{t} \cos t \mathbf{i}+e^{t} \sin t \mathbf{j}+\sqrt{2} e^{t} \mathbf{k} $$

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

Find the image of the set \(S\) under the given transformation. $$ S: 0 \leq u \leq 2,0 \leq v \leq u ; x=2 u+v, y=u-3 v $$

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. $$ \begin{array}{lll} \boldsymbol{r} & 2 \sin \theta, \boldsymbol{r} & 1, \text { common area } \end{array} $$

Evaluate the given iterated integral. $$ \int_{0}^{6} \int_{0}^{6-x} \int_{0}^{6-x-z} d y d z d x $$

Find the surface area of that portion of the cylinder \(x^{2}+z^{2}=16\) that is above the region in the first quadrant bounded on the graphs of \(x=0, x=2, y=0, y=5\)

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

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

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ f(x, y)=y-x^{2} ;(2,5) $$