벡터 미적분 전체 문제

Evaluate line integral ${\int }_{C}{y}^{2}dx+{x}^{2}dy,$ where C is the boundary of a triangle with vertices $(0,0),(1,1),\,\text{and}\,(1,0),$ with the counterclockwise orientation.

Use Green’s theorem to evaluate line integral ${\int }_{C}\text{h}\cdot d\text{r}$ if $\text{h}(x,y)={e}^{y}\text{i}-\sin \,\pi x\text{j},$ where C is a triangle with vertices (1, 0), (0, 1), and (–1, 0) traversed counterclockwise.

Use Green’s theorem to evaluate line integral ${\int }_{C}^{}\sqrt{1+{x}^{3}}dx+2xydy$ where C is a triangle with vertices (0, 0), (1, 0), and (1, 3) oriented clockwise.

Use Green’s theorem to evaluate line integral ${\int }_{C}^{}{x}^{2}ydx-x{y}^{2}dy$ where C is a circle ${x}^{2}+{y}^{2}=4$ oriented counterclockwise.

Use Green’s theorem to evaluate line integral ${\int }_{C}^{}(3y-{e}^{\sin \,x})dx+\left(7x+\sqrt{{y}^{4}+1}\right)dy$ where C is circle ${x}^{2}+{y}^{2}=9$ oriented in the counterclockwise direction.

Use Green’s theorem to evaluate line integral ${\int }_{C}^{}(3x-5y)dx+(x-6y)dy,$ where C is ellipse $\frac{{x}^{2}}{4}+{y}^{2}=1$ and is oriented in the counterclockwise direction.

Let C be a triangular closed curve from (0, 0) to (1, 0) to (1, 1) and finally back to (0, 0). Let $\text{F}(x,y)=4y\text{i}+6{x}^{2}\text{j}.$ Use Green’s theorem to evaluate ${\int }_{C}\text{F}\cdot d\text{r}.$

Use Green’s theorem to evaluate line integral ${\int }_{C}ydx-xdy,$ where C is circle ${x}^{2}+{y}^{2}={a}^{2}$ oriented in the clockwise direction.

Use Green’s theorem to evaluate line integral ${\int }_{C}(y+x)dx+(x+\sin \,y)dy,$ where C is any smooth simple closed curve joining the origin to itself oriented in the counterclockwise direction.

Use Green’s theorem to evaluate line integral ${\int }_{C}(y-\ln ({x}^{2}+{y}^{2}))dx+\left(2\,\arctan \,\frac{y}{x}\right)dy,$ where C is the positively oriented circle ${(x-2)}^{2}+{(y-3)}^{2}=1.$

Use Green’s theorem to evaluate ${\int }_{C}xydx+{x}^{3}{y}^{3}dy,$ where C is a triangle with vertices (0, 0), (1, 0), and (1, 2) with positive orientation.

Use Green’s theorem to evaluate line integral ${\int }_{C}^{}\sin \,ydx+x\,\cos \,ydy,$ where C is ellipse ${x}^{2}+xy+{y}^{2}=1$ oriented in the counterclockwise direction.

Let $\text{F}(x,y)=(\cos ({x}^{5}))-\frac{1}{3}{y}^{3}\text{i}+\frac{1}{3}{x}^{3}\text{j}.$ Find the counterclockwise circulation ${\int }_{C}\text{F}\cdot d\text{r},$ where C is a curve consisting of the line segment joining $(-2,0)\,\text{and}\,(-1,0),$ half circle $y=\sqrt{1-{x}^{2}},$ the line segment joining (1, 0) and (2, 0), and half circle $y=\sqrt{4-{x}^{2}}.$

Use Green’s theorem to evaluate line integral ${\int }_{C}^{}\sin ({x}^{3})dx+2y{e}^{{x}^{2}}dy,$ where C is a triangular closed curve that connects the points (0, 0), (2, 2), and (0, 2) counterclockwise.

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.