벡터 미적분 전체 문제

Let C be the boundary of square $0\le x\le \pi ,\,0\le y\le \pi ,$ traversed counterclockwise. Use Green’s theorem to find ${\int }_{C}^{}\sin (x+y)dx+\cos (x+y)dy.$

Use Green’s theorem to evaluate line integral ${\int }_{C}^{}\text{F}\cdot d\text{r},$ where $\text{F}(x,y)=({y}^{2}-{x}^{2})\text{i}+({x}^{2}+{y}^{2})\text{j},$ and C is a triangle bounded by $y=0,\,x=3,\,\text{and}\,y=x,$ oriented counterclockwise.

Use Green’s Theorem to evaluate integral ${\int }_{C}^{}\text{F}\cdot d\text{r},$ where $\text{F}(x,y)=(x{y}^{2})\text{i}+x\text{j},$ and C is a unit circle oriented in the counterclockwise direction.

Use Green’s theorem in a plane to evaluate line integral ${\int }_{C}(xy+{y}^{2})dx+{x}^{2}dy,$ where C is a closed curve of a region bounded by $y=x\,\text{and}\,y={x}^{2}$ oriented in the counterclockwise direction.

Calculate the outward flux of $\text{F}=-x\text{i}+2y\text{j}$ over a square with corners $(\pm 1,\pm 1),$ where the unit normal is outward pointing and oriented in the counterclockwise direction.

[T] Let C be circle ${x}^{2}+{y}^{2}=4$ oriented in the counterclockwise direction. Evaluate ${\int }_{C}\left[(3y-{e}^{\tan -{1}_{x}})dx+\left(7x+\sqrt{{y}^{4}+1}\right)dy\right]$ using a computer algebra system.

Find the flux of field $\text{F}=-x\text{i}+y\text{j}$ across ${x}^{2}+{y}^{2}=16$ oriented in the counterclockwise direction.

Let $\text{F}=({y}^{2}-{x}^{2})\text{i}+({x}^{2}+{y}^{2})\text{j},$ and let C be a triangle bounded by $y=0,x=3,$ and $y=x$ oriented in the counterclockwise direction. Find the outward flux of F through C.

[T] Let C be unit circle ${x}^{2}+{y}^{2}=1$ traversed once counterclockwise. Evaluate ${\int }_{C}[-{y}^{3}+\sin (xy)+xy\,\cos (xy)]dx+[{x}^{3}+{x}^{2}\cos (xy)]dy$ by using a computer algebra system.

[T] Find the outward flux of vector field $\text{F}=x{y}^{2}\text{i}+{x}^{2}y\text{j}$ across the boundary of annulus $R=\{(x,y):1\le {x}^{2}+{y}^{2}\le 4\}=\{(r,\theta ):1\le r\le 2,0\le \theta \le 2\pi \}$ using a computer algebra system.

Consider region R bounded by parabolas $y={x}^{2}\,\text{and}\,x={y}^{2}.$ Let C be the boundary of R oriented counterclockwise. Use Green’s theorem to evaluate ${\int }_{C}\left(y+{e}^{\sqrt{x}}\right)dx+(2x+\cos ({y}^{2}))dy.$

All vector fields of the form $\text{F}(x,y,z)=f(x)\text{i}+g(y)\text{j}+h(z)\text{k}$ are conservative.

For the following exercises, find the curl of F.

$\text{F}(x,y,z)=x{y}^{2}{z}^{4}\text{i}+(2{x}^{2}y+z)\text{j}+{y}^{3}{z}^{2}\text{k}$

For the following exercises, find the curl of F.

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

For the following exercises, find the curl of F.

$\text{F}(x,y,z)=3xy{z}^{2}\text{i}+{y}^{2}\sin z\text{j}+x{e}^{2z}\text{k}$

For the following exercises, find the curl of F.

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

For the following exercises, find the curl of F.

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

For the following exercises, find the curl of F.

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

For the following exercises, find the curl of F.

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

For the following exercises, find the curl of F.

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