6.3 Conservative vector fields  (Page 10/16)

Justify the Fundamental Theorem of Line Integrals for ${\displaystyle {\int }_C\text{F}·d\text{r}}$ in the case when $\text{F}\left(x,y\right)=\left(2x+2y\right)\text{i}+\left(2x+2y\right)\text{j}$ and C is a portion of the positively oriented circle $x^2+y^2=25$ from (5, 0) to (3, 4).

[T] Find ${\displaystyle {\int }_C^{}\text{F}·d\text{r}},\text{,]}$ where $\text{F}(x,y)=\left(y{e}^{xy}+\text{cos}x\right)\text{i}+\left(x{e}^{xy}+\frac{1}{y^2+1}\right)\text{j}$ and C is a portion of curve $y=\text{sin}x$ from $x=0$ to $x=\frac{\pi }{2}.$

[T] Evaluate line integral ${\displaystyle {\int }_C^{}\text{F}·d\text{r}},$ where $\text{F}(x,y)=\left(e^x\text{sin}y-y\right)\text{i}+\left(e^x\text{cos}y-x-2\right)\text{j},$ and C is the path given by $r(t)=\left[t^3\text{sin}\frac{\pi t}{2}\right]\text{i}-\left[\frac{\pi }{2}\text{cos}\left(\frac{\pi t}{2}+\frac{\pi }{2}\right)\right]\text{j}$ for $0\le t\le 1.$

$\text{F}(x,y)=2x{y}^3\text{i}+3y^2{x}^2\text{j}$

$\text{F}(x,y)=\left(\text{−}y+e^x\text{sin}y\right)\text{i}+\left[\left(x+2\right)e^x\text{cos}y\right]\text{j}$

$\text{F}(x,y)=\left(e^{2x}\text{sin}y\right)\text{i}+\left[e^{2x}\text{cos}y\right]\text{j}$

$\text{F}(x,y)=(6x+5y)\text{i}+(5x+4y)\text{j}$

$\text{F}(x,y)=\left[2x\text{cos}(y)-y\text{cos}(x)\right]\text{i}+\left[\text{−}{x}^2\text{sin}(y)-\text{sin}(x)\right]\text{j}$

$\text{F}(x,y)=\left[y{e}^x+\text{sin}(y)\right]\text{i}+\left[e^x+x\text{cos}(y)\right]\text{j}$

${\displaystyle {\oint }_C(y\text{i}+x\text{j})·d\text{r}},$ where C is any path from (0, 0) to (2, 4)

${\displaystyle {\oint }_C(2ydx+2xdy)},$ where C is the line segment from (0, 0) to (4, 4)

[T] ${\displaystyle {\oint }_C\left[\text{arctan}\frac{y}{x}-\frac{xy}{x^2+y^2}\right]dx+\left[\frac{x^2}{x^2+y^2}+e^{\text{−}y}(1-y)\right]dy},$ where C is any smooth curve from (1, 1) to $\left(\mathrm{-1},2\right)$

Find the conservative vector field for the potential function

$\text{F}(x,y)=\left(12xy\right)\text{i}+6\left(x^2+y^2\right)\text{j}$

$\text{F}(x,y)=\left(e^x\text{cos}y\right)\text{i}+6\left(e^x\text{sin}y\right)\text{j}$

$\text{F}(x,y)=\left(2xy{e}^{x^{2}y}\right)\text{i}+6\left(x^2{e}^{x^{2}y}\right)\text{j}$

$\text{F}(x,y\text{,}z)=\left(y{e}^z\right)\text{i}+\left(x{e}^z\right)\text{j}+\left(xy{e}^z\right)\text{k}$

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

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

$\text{F}(x,y,z)=3z^2\text{i}-\text{cos}y\text{j}+2xz\text{k}$

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

$\text{F}\left(x,y\right)=\left(e^x\text{cos}y\right)\text{i}+6\left(e^x\text{sin}y\right)\text{j}$

$\text{F}\left(x,y\right)=\left(2xy{e}^{x^{2}y}\right)\text{i}+6\left(x^2{e}^{x^{2}y}\right)\text{j}$

Evaluate ${\displaystyle {\int }_C\text{F}·dr},$ where $f(x,y\text{,}z)=\text{cos}(\pi x)+\text{sin}(\pi y)-xyz$ and C is any path that starts at $\left(1,\frac{1}{2},2\right)$ and ends at $\left(2,1,\mathrm{-1}\right).$

[T] Evaluate ${\displaystyle {\int }_C\text{F}·dr},$ where $f(x,y)=xy+e^x$ and C is a straight line from $\left(0,0\right)$ to $\left(2,1\right).$

[T] Evaluate ${\displaystyle {\int }_C\text{F}·ds},$ where $f(x,y)=x^{2}y-x$ and C is any path in a plane from (1, 2) to (3, 2).

Evaluate ${\displaystyle {\int }_C\text{F}·dr},$ where $f(x,y\text{,}z)=xy{z}^2-yz$ and C has initial point (1, 2) and terminal point (3, 5).

Calculate the line integral of F over C ₁ .

Calculate the line integral of G over C ₁ .

Calculate the line integral of F over C ₂ .

Calculate the line integral of G over C ₂ .

[T] Let $\text{F}(x,y,z)=x^2\text{i}+z\text{sin}(yz)\text{j}+y\text{sin}(yz)\text{k}.$ Calculate ${\displaystyle {\oint }_C\text{F}·dr},$ where C is a path from $A=(0,0,1)$ to $B=(3,1,2).$

[T] Find line integral ${\displaystyle {\oint }_C\text{F}·dr}$ of vector field $\text{F}(x,y,z)=3x^{2}z\text{i}+z^2\text{j}+\left(x^3+2yz\right)\text{k}$ along curve C parameterized by $r(t)=\left(\frac{\text{ln}t}{\text{ln}2}\right)\text{i}+t^{3\text{/}2}\text{j}+t\text{cos}\left(\pi t\right)\text{,}1\le t\le 4.$

$\text{F}=\left(x{y}^2+3x^{2}y\right)\text{i}+\left(x+y\right)x^2\text{j};$ C is the curve consisting of line segments from $(1,1)$ to $(0,2)$ to $(3,0).$

$\text{F}=\frac{2x}{y^2+1}\text{i}-\frac{2y\left(x^2+1\right)}{{\left(y^2+1\right)}^2}\text{j};$ C is parameterized by $x=t^3-1,y=t^6-t,0\le t\le 1.$

[T] $\text{F}=\left[\text{cos}\left(x{y}^2\right)-x{y}^2\text{sin}\left(x{y}^2\right)\right]\text{i}-2x^{2}y\text{sin}\left(x{y}^2\right)\text{j};$ C is curve $\left(e^t,e^{t+1}\right),\mathrm{-1}\le t\le 0.$

The mass of Earth is approximately $6\times {10}^{27}\text{g}$ and that of the Sun is 330,000 times as much. The gravitational constant is $6.7\times {10}^{\mathrm{-8}}{\text{cm}}^3\text{/}{\text{s}}^2·\text{g}.$ The distance of Earth from the Sun is about $1.5\times {10}^{12}\text{cm}.$ Compute, approximately, the work necessary to increase the distance of Earth from the Sun by $1\text{cm}.$

[T] Let $\text{F}=(x,y,z)=\left(e^x\text{sin}y\right)\text{i}+\left(e^x\text{cos}y\right)\text{j}+z^2\text{k}.$ Evaluate the integral ${\displaystyle {\int }_C\text{F}·ds},$ where $\text{c}(t)=\left(\sqrt{t},t^3,e^{\sqrt{t}}\right),0\le t\le 1.$

[T] Let $\text{c}:\left[1,2\right]\to {ℝ}^2$ be given by $x=e^{t-1},y=\text{sin}\left(\frac{\pi }{t}\right).$ Use a computer to compute the integral ${\displaystyle {\int }_C\text{F}·d\text{s}={\displaystyle {\int }_{C}2x\text{cos}ydx-x^2\text{sin}ydy,}}$ where $\text{F}=\left(2x\text{cos}y\right)\text{i}-\left(x^2\text{sin}y\right)\text{j}.$

[T] Use a computer algebra system to find the mass of a wire that lies along curve $\text{r}(t)=\left(t^2-1\right)\text{j}+2t\text{k},0\le t\le 1,$ if the density is $\frac{3}{2}t.$

Find the circulation and flux of field $\text{F}=\text{−}y\text{i}+x\text{j}$ around and across the closed semicircular path that consists of semicircular arch ${\text{r}}_1(t)=\left(a\text{cos}t\right)\text{i}+\left(a\text{sin}t\right)\text{j},0\le t\le \pi ,$ followed by line segment ${\text{r}}_2(t)=t\text{i},\text{−}a\le t\le a.$

Compute ${\displaystyle {\int }_C\text{cos}x\text{cos}ydx-\text{sin}x\text{sin}ydy},$ where $\text{c}(t)=\left(t,t^2\right),0\le t\le 1.$

Complete the proof of [link] by showing that $f_y=Q\left(x,y\right).$