Chapter 9

(a) Find the image of the region \(S: 0 \leq u \leq 1,0 \leq v \leq 1\) under the transformation \(x=u-u v, y=u v\). (b) Explain why the transformation is not one-to-one on the boundary of \(S\).

Use Stokes' theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\). Assume \(C\) is oriented counterclockwise as viewed from above. $$ \begin{aligned} &\mathbf{F}=x \mathbf{i}+x^{3} y^{2} \mathbf{j}+z \mathbf{k} ; C \text { the boundary of the semi-ellipsoid }\\\ &z=\sqrt{4-4 x^{2}-y^{2}} \text { in the plane } z=0 \end{aligned} $$

Use Green's theorem to evaluate the given line integral. $$ \begin{aligned} &\oint_{C} x y d x+x^{2} d y \text { , where } C \text { is the boundary of the region }\\\ &\begin{array}{lll} \text { determined by the graphs of } x & 0, x^{2}+y^{2} & 1, x \geq 0 \end{array} \end{aligned} $$

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

Sketch the region of integration for the given iterated integral. $$ \int_{-1}^{3} \int_{0}^{\sqrt{16-y^{2}}} f(x, y) d x d y $$

In Problems, determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=\left(4 x^{3} y^{3}+3\right) \mathbf{i}+\left(3 x^{4} y^{2}+1\right) \mathbf{j} $$

Evaluate \(\int_{C} y d x+x d y\) on the given curve \(C\) between \((0,0)\) and \((1,1)\). $$ y=x^{2} $$

In Problems \(7-16\), find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=3 x^{2} y \mathbf{i}+2 x z^{3} \mathbf{j}+y^{4} \mathbf{k} $$

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ F(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2} ;}(3,4,0) $$

$$ \begin{aligned} &\text { Graph some of the level surfaces associated with } F(x, y, z)=\\\ &x^{2}+y^{2}-z^{2} \text { for } c=0, c>0, \text { and } c<0 \end{aligned} $$

A shell is fired from ground level with an initial speed of \(480 \mathrm{ft} / \mathrm{s}\) at an angle of elevation of \(30^{\circ}\). Find: (a) a vector function and parametric equations of the shell's trajectory, (b) the maximum altitude attained, (c) the range of the shell, and (d) the speed at impact.

In Problems, find the directional derivative of the given function at the given point in the indicated direction. $$ f(x, y)=5 x^{3} y^{6} ;(-1,1), \theta=\pi / 6 $$

In Problems, find the vector function that describes the curve \(C\) of intersection between the given surfaces. Sketch the curve \(C\). Use the indicated parameter. $$ z=x^{2}+y^{2}, y=x ; x=t $$

Change the indicated order of integration to each of the other five orders. $$ \int_{0}^{2} \int_{0}^{4-2 y} \int_{x+2 y}^{4} F(x, y, z) d z d x d y $$

Determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=\left(4 x^{3} y^{3}+3\right) \mathbf{i}+\left(3 x^{4} y^{2}+1\right) \mathbf{j} $$

Find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=3 x^{2} y \mathbf{i}+2 x z^{3} \mathbf{j}+y^{4} \mathbf{k} $$

Graph some of the level surfaces associated with \(F(x, y, z)=\) \(x^{2}+y^{2}-z^{2}\) for \(c=0, c>0\), and \(c<0\).

\(\mathbf{r}(t)\) is the position vector of a moving particle. Find the tangential and normal components of the acceleration at any \(t\). $$ \mathbf{r}(t)=2 t \mathbf{i}+t^{2} \mathbf{j} $$

Find the vector function that describes the curve \(C\) of intersection between the given surfaces. Sketch the curve \(C\). Use the indicated parameter. $$ z=x^{2}+y^{2}, y=x ; x=t $$

Use the divergence theorem to find the outward flux \(\iint_{S}(\mathbf{F} \cdot \mathbf{n}) d S\) of the given vector field \(\mathbf{F}\). $$ \begin{aligned} &\mathbf{F}=15 x^{2} y \mathbf{i}+x^{2} z \mathbf{j}+y^{4} \mathbf{k} ; D \text { the region bounded by } x+y=2\\\ &z=x+y, z=3, x=0, y=0 \end{aligned} $$

Use Stokes' theorem to evaluate \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\). Assume \(C\) is oriented counterclockwise as viewed from above. \(\mathbf{F}=z \mathbf{i}+x \mathbf{j}+y \mathbf{k} ; \boldsymbol{C}\) the curve of intersection of the plane \(x+y+z=0\) and the sphere \(x^{2}+y^{2}+z^{2}=1[\) Hint \(:\) Recall that the area of an ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) is \(\pi a b\). \(]\)

Use Green's theorem to evaluate the given line integral. $$ \begin{aligned} &\oint_{C} e^{x^{2}} d x+2 \tan ^{-1} x d y \text { , where } C \text { is the triangle with vertices }\\\ &(0,0),(0,1),(-1,1) \end{aligned} $$

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 \(t\). $$ \mathbf{r}(t)=\tan ^{-1} t \mathbf{i}+\frac{1}{2} \ln \left(1+t^{2}\right) \mathbf{j} $$

Sketch the region of integration for the given iterated integral. $$ \int^{2} \int^{x^{2}+1} f(x, y) d y d x $$

In Problems, determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=2 x y^{3} \mathbf{i}+3 y^{2}\left(x^{2}+1\right) \mathbf{j} $$

Evaluate \(\int_{C} y d x+x d y\) on the given curve \(C\) between \((0,0)\) and \((1,1)\). $$ y=x $$

In Problems \(7-16\), find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=5 y^{3} \mathbf{i}+\left(\frac{1}{2} x^{3} y^{2}-x y\right) \mathbf{j}-\left(x^{3} y z-x z\right) \mathbf{k} $$

Sketch the level curve or surface passing through the indicated point. Sketch the gradient at the point. $$ F(x, y, z)=x^{2}-y^{2}+z ;(0,-1,1) $$

Given that $$ F(x, y, z)=\frac{x^{2}}{16}+\frac{y^{2}}{4}+\frac{z^{2}}{9} $$ find the \(x-, y\) -, and \(z\) -intercepts of the level surface that passes through \((-4,2,-3)\)

In Problems, find the directional derivative of the given function at the given point in the indicated direction. $$ f(x, y)=4 x+x y^{2}-5 y ;(3,-1), \theta=\pi / 4 $$

In Problems, find the vector function that describes the curve \(C\) of intersection between the given surfaces. Sketch the curve \(C\). Use the indicated parameter. $$ x^{2}+y^{2}-z^{2}=1, y=2 x ; x=t $$

Fill in the blank or answer true/false. Where appropriate, assume continuity of \(P, O\), and their first partial derivatives. In a conservative force field \(\mathrm{F}\), the work done by \(\mathrm{F}\) around a simple closed curve is zero.___

Determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=2 x y^{3} \mathbf{i}+3 y^{2}\left(x^{2}+1\right) \mathbf{j} $$

Find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=5 y^{3} \mathbf{i}+\left(\frac{1}{2} x^{3} y^{2}-x y\right) \mathbf{j}-\left(x^{3} y z-x z\right) \mathbf{k} $$

Find the directional derivative of the given function at the given point in the indicated direction. $$ f(x, y)=4 x+x y^{2}-5 y ;(3,-1), \quad \theta=\pi / 4 $$

\(\mathbf{r}(t)\) is the position vector of a moving particle. Find the tangential and normal components of the acceleration at any \(t\). $$ \mathbf{r}(t)=\tan ^{-1} t \mathbf{i}+\frac{1}{2} \ln \left(1+t^{2}\right) \mathbf{j} $$

Find the vector function that describes the curve \(C\) of intersection between the given surfaces. Sketch the curve \(C\). Use the indicated parameter. $$ x^{2}+y^{2}-z^{2}=1, y=2 x ; x=t $$

Evaluate the given integral by means of the indicated change of variables. $$ \begin{aligned} &\iint_{R}(x+y) d A \text { , where } R \text { is the region bounded by the graphs of }\\\ &x-2 y=-6, x-2 y=6, x+y=-1, x+y=3 ; u=x-2 y\\\ &v=x+y \end{aligned} $$

In Problems 13-16, use Stokes' theorem to evaluate \(\iint_{S}\) (curl \(\left.\mathbf{F}\right) \cdot \mathbf{n} d S\). Assume that the surface \(S\) is oriented upward. $$ \begin{aligned} &\mathbf{F}=6 y z \mathbf{i}+5 x \mathbf{j}+y z e^{x^{2}} \mathbf{k} ; S \text { that portion of the paraboloid }\\\ &z=\frac{1}{4} x^{2}+y^{2} \text { for } 0 \leq z \leq 4 \end{aligned} $$

Use Green's theorem to evaluate the given line integral. \(\oint_{C} \frac{1}{3} y^{3} d x+\left(x y+x y^{2}\right) d y\), where \(C\) is the boundary of the region in the first quadrant determined by the graphs of \(y \quad 0\), \(x \quad y^{2}, x \quad 1-y^{2}\)

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 \(t\). $$ \mathbf{r}(t)=2 \cos t 1+2 \sin t \mathrm{~J} $$

Find the surface area of that portion of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) that is bounded between \(y=c_{1}\) and \(y=c_{2}\), \(0

Evaluate the double integral over the region \(R\) that is bounded by the graphs of the given equations. Choose the most convenient order of integration. $$ \iint_{R} x^{3} y^{2} d A ; y=x, y=0, x=1 $$

In Problems, determine whether the given vector field is a conservative field. If so, find a potential function \(\phi\) for \(\mathbf{F}\). $$ \mathbf{F}(x, y)=y^{2} \cos x y^{2} \mathbf{i}-2 x y \sin x y^{2} \mathbf{j} $$

Evaluate \(\int_{C} y d x+x d y\) on the given curve \(C\) between \((0,0)\) and \((1,1)\). \(C\) consists of the line segments from \((0,0)\) to \((0,1)\) and from \((0,1)\) to \((1,1)\).

In Problems \(7-16\), find the curl and the divergence of the given vector field. $$ \mathbf{F}(x, y, z)=x e^{-z} \mathbf{i}+4 y z^{2} \mathbf{j}+3 y e^{-z} \mathbf{k} $$

Find the points on the given surface at which the gradient is parallel to the indicated vector. $$ z=x^{2}+y^{2} ; 4 \mathbf{i}+\mathbf{j}+\frac{1}{2} \mathbf{k} $$

Find the first partial derivatives of the given function. $$ z=x^{2}-x y^{2}+4 y^{5} $$

A used car is pushed off an 81 -ft-high sheer seaside cliff with a speed of \(4 \mathrm{ft} / \mathrm{s}\). Find the speed at which the car hits the water.

In Problems, find the directional derivative of the given function at the given point in the indicated direction. $$ f(x, y)=\tan ^{-1} \frac{y}{x} ;(2,-2), \mathbf{i}-3 \mathbf{j} $$