Chapter 6

$12\text{i}-\text{j}$

Rotational

$\sqrt{65}$ m/sec

No.

$\mathrm{–1.49063}\times {10}^{\mathrm{-18}},4.96876\times {10}^{\mathrm{-19}},\mathrm{–9.93752}\times {10}^{\mathrm{-19}}\text{N}$

No

$\text{∇}f=\text{v}$

$P_y=x\ne Q_x=\mathrm{-2}xy$

No

$\sqrt{2}$

$2\sqrt{10}\mathrm{\pi }+2\sqrt{10}{\mathrm{\pi }}^2$

Both line integrals equal $-\frac{1000\sqrt{30}}{3}.$

$4\sqrt{17}$

${\displaystyle {\int }_C\text{F}·\text{T}}ds$

$\mathrm{-26}$

0

$18\sqrt{2}{\pi }^2$ kg

3/2

$2\pi$

0

Yes

The region in the figure is connected. The region in the figure is not simply connected.

2

If $C_1$ and $C_2$ represent the two curves, then ${\displaystyle {\int }_{C_1}\text{F}·d\text{r}}\ne {\displaystyle {\int }_{C_2}\text{F}·d\text{r}.}$

$f(x,y)=e^x{y}^3+xy$

$f\left(x,y,z\right)=4x^3+\text{sin}y\text{cos}z+z$

$f(x,y,z)=\frac{G}{\sqrt{x^2+y^2+z^2}}$

It is conservative.

$\mathrm{-10}\pi$

Negative

$\frac{45}{2}$

$\frac{2}{3}$

$\frac{3\pi }{2}$

$g\left(x,y\right)=\text{−}x\text{cos}y$

No

$105\pi$

$y-z^2$

Yes

All points on line $y=1.$

$\text{−}\text{i}$

$\text{curl}\text{v}=0$

No

Yes

Cylinder $x^2+y^2=4$

Cone $x^2+y^2=z^2$

$\text{r}(u,v)=\langle u\text{cos}v,u\text{sin}v,u\rangle ,$ $0<u<\infty ,0\le v<\frac{\pi }{2}$

Yes

$\approx 43.02$

With the standard parameterization of a cylinder, Equation 6.18 shows that the surface area is $2\pi rh.$

$2\pi \left(\sqrt{2}+{\text{sinh}}^{\mathrm{-1}}(1)\right)$

24

0

$38.401\pi \approx 120.640$

$\text{N}(x,y)=\langle \frac{\text{−}y}{\sqrt{1+x^2+y^2}},\frac{\text{−}x}{\sqrt{1+x^2+y^2}},\frac{1}{\sqrt{1+x^2+y^2}}\rangle$

0

400 kg/sec/m

$-\frac{440\pi }{3}$

Both integrals give $0$

$\text{−}\pi$

$\frac{3}{2}$

$\text{curl}\text{E}=\langle x,y,\mathrm{-2}z\rangle$

Both integrals equal $6\pi .$

30

$9\text{ln}\left(16\right)$

$\approx 6.777\times {10}^9$

Vectors

False

$\text{F}(x,y)=\text{sin}(y)\text{i}+(x\text{cos}y-\text{sin}y)\text{j}$

$\text{F}(x,y,z)=(2xy+y)\text{i}+(x^2+x+2yz)\text{j}+y^2\text{k}$

$\text{F}(x,y)=\left(\frac{2x}{1+x^2+2y^2}\right)\text{i}+\left(\frac{4y}{1+x^2+2y^2}\right)\text{j}$

$\text{F}(x,y)=\frac{(1-x)\text{i}-y\text{j}}{\sqrt{{(1-x)}^2+y^2}}$

$\text{F}(x,y)=\frac{-x\text{i}-y\text{j}}{\sqrt{x^2+y^2}}$

$\text{F}(x,y)=y\text{i}-x\text{j}$

$\text{F}(x,y)=\frac{\mathrm{-10}}{{(x^2+y^2)}^{3\text{/}2}}\left(x\text{i}+y\text{j}\right)$

$||\mathbf{E}||=\frac{c}{x^2+y^2}\sqrt{x^2+y^2}=\frac{c}{\sqrt{x^2+y^2}}=\frac{c}{r}$

$\text{c}\text{′}\left(t\right)=\left(\text{cos}t,\text{−}\text{sin}t,e^{\text{−}t}\right)=\text{F}\left(\text{c}\left(t\right)\right)$

H

d. $\text{−}\text{F}+\text{G}$

a. $\text{F}+\text{G}$

True

False

False

${\displaystyle {\int }_C^{}(x-y)ds}=10$

${\displaystyle {\int }_C^{}x{y}^{4}ds}=\frac{8192}{5}$

$W=8$

$W=\frac{3\pi }{4}$

$W=\pi$

${\displaystyle {\int }_C\text{F}·d\text{r}}=4$

${\displaystyle {\int }_C^{}yzdx+xzdy+xydz}=\mathrm{-1}$

${\displaystyle {\int }_C^{}\left(y^2\right)dx+(x)dy}=\frac{245}{6}$

${\displaystyle {\int }_C^{}xydx+ydy=\frac{190}{3}}$

${\displaystyle {\int }_C\frac{y}{2x^2-y^2}ds=\sqrt{2}\text{ln}5}$

$W=\mathrm{-66}$

$W=\mathrm{-10}{\pi }^2$

$W=2$

a. $W=11;$ b. $W=\frac{39}{4};$ c. No

$W=2\pi$

${\displaystyle {\int }_C^{}xy ds=\frac{25\sqrt{5}+1}{120}}$

${\displaystyle {\int }_C^{}{y}^{2}dx+\left(xy-x^2\right)dy}=6.15$

${\displaystyle {\int }_{\gamma }^{}x{e}^{y}ds\approx 7.157}$

${\displaystyle {\int }_{\gamma }^{}\left(y^2-xy\right)dx\approx \mathrm{-1.379}}$

${\displaystyle {\int }_C^{}\text{F}·d\text{r}}\approx \mathrm{-1.133}$

${\displaystyle {\int }_C^{}\text{F}·d\text{r}\approx 22.857}$

$\text{flux}=-\frac{1}{3}$

$\text{flux}=\mathrm{-20}$

$\text{flux}=0$

$m=4\pi \rho \sqrt{5}$

$W=0$

$W=\frac{k}{2}$

True

True

${\displaystyle {\int }_C^{}\text{F}·d\text{r}}=24$

${\displaystyle {\int }_C^{}\text{F}·d\text{r}}=e-\frac{3\pi }{2}$

Not conservative

Conservative, $f(x,y)=3x^2+5xy+2y^2$

Conservative, $f(x,y)=y{e}^x+x\text{sin}(y)$

${\displaystyle {\int }_C(2ydx+2xdy)}=32$

$\text{F}(x,y)=(10x+3y)\text{i}+(3x+20y)\text{j}$

F is not conservative.

F is conservative and a potential function is $f(x,y\text{,}z)=xy{e}^z.$

F is conservative and a potential function is $f(x,y,z)=z^2–z–\frac{x}{y}.$

F is conservative and a potential function is $f(x,y\text{,}z)=x^{2}y+y^{2}z.$

F is conservative and a potential function is $f\left(x,y\right)=e^{x^{2}y}$

${\displaystyle {\int }_C\text{F}·dr}=e^2+1$

${\displaystyle {\int }_C\text{F}·dr}=-2$

${\displaystyle {\int }_{C_1}\text{G}·d\text{r}}=\mathrm{-8}\pi$

${\displaystyle {\int }_{C_2}\text{G}·d\text{r}}=7$

${\displaystyle {\int }_C\text{F}}·d\text{r}=159$

${\displaystyle {\int }_C\text{F}·d\text{r}}=\mathrm{-1}$

$4\times {10}^{29}\text{erg}$

${\displaystyle {\int }_C\text{F}·d\text{r}\approx 2.9923}$

$\text{circulation}=\pi a^2\text{and flux}=0$

${\displaystyle {\int }_C^{}2xydx+(x+y)dy}=\frac{32}{3}$

${\displaystyle {\int }_C^{}\text{sin}x\text{cos}ydx+(xy+\text{cos}x\text{sin}y)dy}=\frac{1}{12}$

${\displaystyle {\int }_C(\text{−}ydx+xdy)}=\pi$

${\displaystyle {\int }_{C}x{e}^{\mathrm{-2}x}dx+\left(x^4+2x^2{y}^2\right)dy=0}$

${\displaystyle {\int }_C{y}^{3}dx-x^{3}ydy=\mathrm{-20}\pi }$

${\displaystyle {\int }_C\text{−}{x}^{2}ydx+x{y}^{2}dy}=8\pi$

${\displaystyle {\int }_C\left(x^2+y^2\right)dx+2xydy}=0$

$A=19\pi$

$A=\frac{3\pi }{8}$

${\displaystyle {\int }_{C+}\left(y^2+x^3\right)dx+x^{4}dy=0}$

$A=\frac{9\pi }{8}$

$A=\frac{8\sqrt{3}}{5}$

${\displaystyle {\int }_C\left(x^{2}y-2xy+y^2\right)ds=\frac{1}{2}}$

${\displaystyle {\int }_C^{}\frac{xdx+ydy}{x^2+y^2}}=0$

$W=\frac{225}{2}$

$W=12\pi$

$W=2\pi$

${\displaystyle {\int }_C{y}^{2}dx+x^{2}dy=\frac{1}{3}}$

${\displaystyle {\int }_C^{}\sqrt{1+x^3}dx+2xydy}=-3$

${\displaystyle {\int }_C^{}\left(3y-e^{\text{sin}x}\right)dx}+\left(7x+\sqrt{y^4+1}\right)dy=36\pi$

${\displaystyle {\int }_C\text{F}·d\text{r}}=2$

${\displaystyle {\int }_C(y+x)dx+(x+\text{sin}y)dy}=0$

${\displaystyle {\int }_{C}xydx+x^3{y}^{3}dy=\frac{22}{21}}$

${\displaystyle {\int }_C\text{F}·d\text{r}}=\frac{15\pi }{4}$

${\displaystyle {\int }_C^{}\text{sin}(x+y)dx+\text{cos}(x+y)dy}=4$

${\displaystyle {\int }_C^{}\text{F}·d\text{r}}=\pi$

${\displaystyle {\int }_C\text{F}·\widehat{N}ds=4}$

${\displaystyle {\int }_C\text{F}·\text{N}ds}=0$

${\displaystyle {\int }_C\left[\text{−}{y}^3+\text{sin}\left(xy\right)+xy\text{cos}\left(xy\right)\right]dx+\left[x^3+x^2\text{cos}\left(xy\right)\right]dy}=4.7124$

${\displaystyle {\int }_C\left(y+e^{\sqrt{x}}\right)dx+\left(2x+\text{cos}\left(y^2\right)\right)dy=\frac{1}{3}}$

False

True

True

$\text{curl}\text{F}=\text{i}+x^2\text{j}+y^2\text{k}$

$\text{curl}\text{F}=\left(x{z}^2-x{y}^2\right)\text{i}+\left(x^{2}y-y{z}^2\right)\text{j}+\left(y^{2}z-x^{2}z\right)\text{k}$

$\text{curl}\text{F}=\text{i}+\text{j}+\text{k}$

$\text{curl}\text{F}=\text{−}y\text{i}-z\text{j}-x\text{k}$

$\text{curl}\text{F}=\mathbf{0}$

$\text{div}\text{F}=3y{z}^2+2y\text{sin}z+2x{e}^{2z}$

$\text{div}\text{F}=2(x+y+z)$

$\text{div}\text{F}=\frac{1}{\sqrt{x^2+y^2}}$

$\text{div}\text{F}=a+b$

$\text{div}\text{F}=x+y+z$

Harmonic

$\text{div}(\text{F}\times \text{G})=2z+3x$

$\text{div}\text{F}=2(x^2 + y^2 + z^2)$

$\text{curl}\text{r}=\mathbf{0}$

$\text{curl}\frac{\text{r}}{r^3}=\mathbf{0}$

$\text{curl}\text{F}=\frac{2x}{x^2+y^2}\text{k}$

$\text{div}\text{F}=0$

$\text{div}\text{F}=2-2e^{\mathrm{-6}}$

$\text{div}\text{F}=0$

$\text{curl}\text{F}=\text{j}-3\text{k}$

$\text{curl}\text{F}=2\text{j}-\text{k}$

$a=3$

F is conservative.

$\text{div}\text{F}=\text{cosh}x+\text{sinh}y-xy$

$(bz-cy)\text{i}+(cx-az)\text{j}+(ay-bx)\text{k}$

$\text{curl}\text{F}=2\omega$

$\text{F}\times \text{G}$ does not have zero divergence.

$\nabla ·\text{F}=\mathrm{-200}k\left[1+2\left(x^2+y^2+z^2\right)\right]e^{\text{−}{x}^2+y^2+z^2}$

True

True

$\text{r}\left(u,v\right)=\langle u,v,2-3u+2v\rangle$ for $\text{−}\infty \le u<\infty$ and $\text{−}\infty \le v<\infty .$

$\text{r}(u,v)=\langle u,v,\frac{1}{3}\left(16-2u+4v\right)\rangle$ for $\left|u\right|<\infty$ and $\left|v\right|<\infty .$

$\text{r}(u,v)=\langle 3\text{cos}u,3\text{sin}u,v\rangle$ for $0\le u\le \frac{\pi }{2},0\le v\le 3$

$A=28\mathrm{\pi }=87.9646$

${\displaystyle {\iint }_{S}zdS}=8\pi$

${\displaystyle {\iint }_S\left(x^2+y^2\right)zdS}=16\pi$

${\displaystyle {\iint }_S\text{F}·\text{N}dS=\frac{4\pi }{3}}$

$m\approx 13.0639$

$m\approx 228.5313$

${\displaystyle {\iint }_{S}gdS}=3\sqrt{14}$

${\displaystyle {\iint }_S\left(x-y^2+z\right)}dS\approx 0.9617$

${\displaystyle {\iint }_S\left(x^2+y^2\right)dS}=\frac{4\pi }{3}$

${\displaystyle {\iint }_S{x}^{2}zdS}=\frac{1023\pi \sqrt{2}}{5}$

${\displaystyle {\iint }_S\left(z+y\right)dS}\approx 10.1$

$m=\pi a^3$

${\displaystyle {\iint }_S\text{F}·\text{N}dS=\frac{13}{24}}$

${\displaystyle {\iint }_S\text{F}·\text{N}dS=\frac{3}{4}}$