벡터 미적분 전체 문제

For the following exercises, use Green’s theorem to calculate the work done by force F on a particle that is moving counterclockwise around closed path C.

$\text{F}(x,y)=xy\text{i}+(x+y)\text{j},$ $C:{x}^{2}+{y}^{2}=4$

For the following exercises, use Green’s theorem to calculate the work done by force F on a particle that is moving counterclockwise around closed path C.

$\text{F}(x,y)=({x}^{3/2}-3y)\text{i}+\left(6x+5\sqrt{y}\right)\text{j},$ C : boundary of a triangle with vertices (0, 0), (5, 0), and (0, 5)

Evaluate ${\int }_{C}(2{x}^{3}-{y}^{3})dx+({x}^{3}+{y}^{3})dy,$ where C is a unit circle oriented in the counterclockwise direction.

A particle starts at point $(-2,0),$ moves along the x-axis to (2, 0), and then travels along semicircle $y=\sqrt{4-{x}^{2}}$ to the starting point. Use Green’s theorem to find the work done on this particle by force field $\text{F}(x,y)=x\text{i}+({x}^{3}+3x{y}^{2})\text{j}.$

David and Sandra are skating on a frictionless pond in the wind. David skates on the inside, going along a circle of radius 2 in a counterclockwise direction. Sandra skates once around a circle of radius 3, also in the counterclockwise direction. Suppose the force of the wind at point $(x,y)$ $(x,y)$ $(x,y)$ is $\text{F}(x,y)=({x}^{2}y+10y)\text{i}+({x}^{3}+2x{y}^{2})\text{j}.$ Use Green’s theorem to determine who does more work.

Use Green’s theorem to find the work done by force field $\text{F}(x,y)=(3y-4x)\text{i}+(4x-y)\text{j}$ when an object moves once counterclockwise around ellipse $4{x}^{2}+{y}^{2}=4.$

Use Green’s theorem to evaluate line integral ${\int }_{C}{e}^{2x}\sin \,2ydx+{e}^{2x}\cos \,2ydy,$ where C is ellipse $9{(x-1)}^{2}+4{(y-3)}^{2}=36$ oriented counterclockwise.

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.

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.