벡터 미적분 전체 문제

David and Sandra plan to evaluate line integral ${\int }_{C}^{}\text{F}\cdot d\text{r}$ along a path in the xy-plane from (0, 0) to (1, 1). The force field is $\text{F}(x,y)=(x+2y)\text{i}+(-x+{y}^{2})\text{j}.$ David chooses the path that runs along the x-axis from (0, 0) to (1, 0) and then runs along the vertical line $x=1$ from (1, 0) to the final point (1, 1). Sandra chooses the direct path along the diagonal line $y=x$ from (0, 0) to (1, 1). Whose line integral is larger and by how much?

True or False? If vector field F is conservative on the open and connected region D, then line integrals of F are path independent on D, regardless of the shape of D.

True or False? Function $\text{r}(t)=\text{a}+t(\text{b}-\text{a}),$ where $0\le t\le 1,$ parameterizes the straight-line segment from $\text{a}\,\text{to}\,\text{b}.$

True or False? Vector field $\text{F}(x,y,z)=(y\,\sin z)\text{i}+(x\,\sin z)\text{j}+(xy\,\cos z)\text{k}$ is conservative.

True or False? Vector field $\text{F}(x,y,z)=y\text{i}+(x+z)\text{j}-y\text{k}$ is conservative.

Verify the Fundamental Theorem of Line Integrals for ${\int }_{C}\text{F}\cdot d\text{r}$ by computing the integral using a parameterization and, separately, by finding a potential function for the case when $\text{F}(x,y)=(2x+2y)\text{i}+(2x+2y)\text{j}$ and C is a portion of circle ${x}^{2}+{y}^{2}=25$ oriented counterclockwise from (5, 0) to (3, 4).

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

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

For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.

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

For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.

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

For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.

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

For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.

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

For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.

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

For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function.

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

For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals.

${\int }_{C}(y\text{i}+x\text{j})\cdot d\text{r},$ where C is any path from (0, 0) to (2, 4)

For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals.

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

For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals.

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

Find the conservative vector field for the potential function

$$f(x,y)=5{x}^{2}+3xy+{10y}^{2}.$$

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

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

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

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