Capítulo 6

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

Rotación

$\sqrt{65}$ m/s

No.

$\mathrm{1,49063}\times {10}^{\mathrm{-18}},\mathrm{4,96876}\times {10}^{\mathrm{-19}},\mathrm{9,93752}\times {10}^{\mathrm{-19}}\text{N}$

No

$\text{∇}f=\text{v}$

$P_y=x\ne Q_x=\mathrm{–2}xy$

No

$\sqrt{2}$

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

Ambas integrales de línea son iguales $-\frac{1.000\sqrt{30}}{3}.$

$4\sqrt{17}$

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

$\mathrm{-26}$

0

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

3/2

$2\pi$

0

Sí

La región de la figura está conectada. La región de la figura no está simplemente conectada.

2

Si $C_1$ y $C_2$ representan las dos curvas, entonces ${\displaystyle {\int }_{C_1}\text{F}.d\text{r}}\ne {\displaystyle {\int }_{C_2}\text{F}.d\text{r}.}$

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

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

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

Es conservativo.

$\mathrm{–10}\pi$

Negativo

$\frac{45}{2}$

$\frac{4}{3}$

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

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

No

$105\pi$

$y-z^2$

Sí

Todos los puntos en línea $y=1.$

$\text{−}\text{i}$

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

No

Sí

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

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

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

Sí

$\approx \mathrm{43,02}$

Con la parametrización estándar de un cilindro, la Ecuación 6.18 muestra que el área superficial es $2\pi rh.$

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

24

0

$\mathrm{38,401}\pi \approx \mathrm{120,640}$

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

0

400 kg/s/m

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

Ambas integrales dan $-\frac{136}{45}.$

$\text{−}\pi$

$\frac{3}{2}$

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

Ambas integrales son iguales $6\pi .$

30

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

$\approx \mathrm{6,777}\times {10}^9$

Vectores

Falso

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

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

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

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

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

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

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

$E=\frac{c}{{\left|r\right|}^2}r=\frac{c}{\left|r\right|}\frac{r}{\left|r\right|}$

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

H

d. $\text{−}\text{F}+\text{G}$

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

Verdadero

Falso

Falso

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

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

$W=8$

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

$W=\pi$

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

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

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

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

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

$W=\mathrm{-66}$

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

$W=2$

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

$W=2\pi$

${\displaystyle {\int }_C^{}\text{F}.d\text{r}=\frac{25\sqrt{5}+1}{120}}$

${\displaystyle {\int }_C^{}{y}^{2}dx+\left(xy-x^2\right)dy}=\mathrm{6,15}$

${\displaystyle {\int }_{\gamma }^{}x{e}^{y}ds\approx \mathrm{7,157}}$

${\displaystyle {\int }_{\gamma }^{}\left(y^2-xy\right)dx\approx \mathrm{-1,379}}$

${\displaystyle {\int }_C^{}\text{F}.d\text{r}}\approx \mathrm{-1,133}$

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

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

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

$\text{flujo}=0$

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

$W=0$

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

Verdadero

Verdadero

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

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

No es conservatorio

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

Conservativo, $f(x,y)=y{e}^x+x\text{sen}(y)$

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

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

F no es conservativo.

F es conservativo y una función potencial es $f(x,y\text{,}z)=xy{e}^z.$

F es conservativo y una función potencial es $f(x,y,z)=z^2–z–\frac{x}{y}.$

F es conservativo y una función potencial es $f(x,y\text{,}z)=x^{2}y+y^{2}z.$

F es conservativo y una función potencial es $f\left(x,y\right)=e^{x^{2}y}$

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

${\displaystyle {\int }_C\text{F}.dr}=\mathrm{–2}$

${\displaystyle {\oint }_{C_1}\text{G}.d\text{r}}=\mathrm{–8}\pi$

${\displaystyle {\oint }_{C_2}\text{F}.d\text{r}}=7$

${\displaystyle {\int }_C\text{F}}.d\text{r}=150$

${\displaystyle {\int }_C\text{F}.d\text{r}}=\mathrm{–1}$

$4\times {10}^{31}\text{erg}$

${\displaystyle {\int }_C\text{F}.d\text{s}=\mathrm{0,4687}}$

$\text{circulación}=\pi a^2\text{y el flujo}=0$

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

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

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

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

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

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

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

$A=19\pi$

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

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

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

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

${\displaystyle {\int }_C\left(x^{2}y-2xy+y^2\right)ds=–\frac{5}{6}}$

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

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

$W=12\pi$

$W=2\pi$

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

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

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

${\displaystyle {\oint }_C\text{F}.d\text{r}}=2$

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

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

${\displaystyle {\oint }_C\text{F}.d\text{r}}=\frac{15\pi }{4}$

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

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

${\displaystyle {\oint }_C\text{F}.\widehat{n}ds=4}$

${\displaystyle {\oint }_C\text{F}.\text{n}ds}=0$

${\displaystyle {\int }_C\left[\text{−}{y}^3+\text{sen}\left(xy\right)+xy\text{cos}\left(xy\right)\right]dx+\left[x^3+x^2\text{cos}\left(xy\right)\right]dy}=\mathrm{4,7124}$

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

Falso

Verdadero

Verdadero

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

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

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

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

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

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

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

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

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

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

Armónico

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

$\text{div}\text{F}=2r^2$

$\text{rizo}\text{r}=0$

$\text{rizo}\frac{\text{r}}{r^3}=0$

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

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

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

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

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

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

$a=3$

F es conservativo.

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

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

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

$\text{F}\times \text{G}$ no tiene divergencia cero.

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

Verdadero

Verdadero

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

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

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

$A=\mathrm{87,9646}$

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

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

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

$m\approx \mathrm{13,0639}$

$m\approx \mathrm{228,5313}$

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

${\displaystyle {\iint }_S\left(x^2+y-z\right)}d\text{S}\approx \mathrm{0,9617}$

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

${\displaystyle {\iint }_S{x}^{2}zd\text{S}}=\frac{1.023\sqrt{2\pi }}{5}$

${\displaystyle {\iint }_{\text{S}}\left(z+y\right)d\text{S}}\approx \mathrm{10,1}$

$m=\pi a^3$

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

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