벡터 미적분 전체 문제

For the following exercises, let $\text{F}=x\text{i}+y\text{j},$ $\text{G}=-y\text{i}+x\text{j},$ and $\text{H}=x\text{i}-y\text{j}.$ Match F, G, and H with their graphs.

For the following exercises, let $\text{F}=x\text{i}+y\text{j},$ $\text{G}=-y\text{i}+x\text{j},$ and $\text{H}=x\text{i}-y\text{j}.$ Match F, G, and H with their graphs.

For the following exercises, let $\text{F}=x\text{i}+y\text{j},$ $\text{G}=-y\text{i}+x\text{j},$ and $\text{H}=x\text{i}-y\text{j}.$ Match F, G, and H with their graphs.

For the following exercises, let $\text{F}=x\text{i}+y\text{j},$ $\text{G}=-y\text{i}+x\text{j},$ and $\text{H}=x\text{i}-y\text{j}.$ Match the vector fields in a through d with their graphs.

a. $\text{F}+\text{G}$ b. $\text{F}+\text{H}$ c. $\text{G}+\text{H}$ d. $-\text{F}+\text{G}$

For the following exercises, let $\text{F}=x\text{i}+y\text{j},$ $\text{G}=-y\text{i}+x\text{j},$ and $\text{H}=x\text{i}-y\text{j}.$ Match the vector fields in a through d with their graphs.

a. $\text{F}+\text{G}$ b. $\text{F}+\text{H}$ c. $\text{G}+\text{H}$ d. $-\text{F}+\text{G}$

For the following exercises, let $\text{F}=x\text{i}+y\text{j},$ $\text{G}=-y\text{i}+x\text{j},$ and $\text{H}=x\text{i}-y\text{j}.$ Match the vector fields in a through d with their graphs.

a. $\text{F}+\text{G}$ b. $\text{F}+\text{H}$ c. $\text{G}+\text{H}$ d. $-\text{F}+\text{G}$

For the following exercises, let $\text{F}=x\text{i}+y\text{j},$ $\text{G}=-y\text{i}+x\text{j},$ and $\text{H}=x\text{i}-y\text{j}.$ Match the vector fields in a through d with their graphs.

a. $\text{F}+\text{G}$ b. $\text{F}+\text{H}$ c. $\text{G}+\text{H}$ d. $-\text{F}+\text{G}$

Evaluate ${\int }_{C}\text{F}\cdot d\text{r},$ where $\text{F}(x,y)=-1\text{j},$ and C is the part of the graph of $y=\frac{1}{2}{x}^{3}-x$ from $(2,2)$ to $(-2,-2).$

Force $\text{F}(x,y,z)=zy\text{i}+x\text{j}+{z}^{2}x\text{k}$ acts on a particle that travels from the origin to point (1, 2, 3). Calculate the work done if the particle travels:

a. along the path $(0,0,0)\to (1,0,0)\to (1,2,0)\to (1,2,3)$ along straight-line segments joining each pair of endpoints; b. along the straight line joining the initial and final points. c. Is the work the same along the two paths?

For the following exercises, let $\text{F}(x,y)=2x{y}^{2}\text{i}+(2y{x}^{2}+2y)\text{j}$ and $G(x,y)=(y+x)\text{i}+(y-x)\text{j},$ and let C₁ be the curve consisting of the circle of radius 2, centered at the origin and oriented counterclockwise, and C₂ be the curve consisting of a line segment from (0, 0) to (1, 1) followed by a line segment from (1, 1) to (3, 1).

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

Find the circulation and flux of field $\text{F}=-y\text{i}+x\text{j}$ around and across the closed semicircular path that consists of semicircular arch ${\text{r}}_{1}(t)=(a\cos t)\text{i}+(a\sin t)\text{j},0\le t\le \pi ,$ followed by line segment ${\text{r}}_{2}(t)=t\text{i},-a\le t\le a.$

${\int }_{C}^{}2xydx+(x+y)dy,$ where C is the path from (0, 0) to (1, 1) along the graph of $y={x}^{3}$ and from (1, 1) to (0, 0) along the graph of $y=x$ oriented in the counterclockwise direction

Find the counterclockwise circulation of field $\text{F}(x,y)=xy\text{i}+{y}^{2}\text{j}$ around and over the boundary of the region enclosed by curves $y={x}^{2}$ and $y=x$ in the first quadrant and oriented in the counterclockwise direction.

Evaluate ${\int }_{C}{y}^{3}dx-{x}^{3}dy,$ where C includes the two circles of radius 2 and radius 1 centered at the origin, both with positive orientation.

Evaluate integral ${\int }_{C}({x}^{2}+{y}^{2})dx+2xydy,$ where C is the curve that follows parabola $y={x}^{2}\,\text{from}\,(0,0)\text{ to }(2,4),$ then the line from (2, 4) to (2, 0), and finally the line from (2, 0) to (0, 0).

Evaluate line integral ${\int }_{C}(y-\sin (y)\cos (y))dx+2x\,{\sin }^{2}(y)dy,$ where C is oriented in a counterclockwise path around the region bounded by $x=-1,x=2,y=4-{x}^{2},$ and $y=x-2.$

Find the area of the region enclosed by parametric equation

$$p(\theta )=(\cos (\theta )-{\cos }^{2}(\theta ))\text{i}+(\sin (\theta )-\cos (\theta )\sin (\theta ))\text{j}\,\text{for}\,0\le \theta \le 2\pi .$$

Use Green’s theorem to find the area of the region enclosed by curve

$$\text{r}(t)={t}^{2}\text{i}+\left(\frac{{t}^{3}}{3}-t\right)\text{j},\,-\sqrt{3}\le t\le \sqrt{3}.$$

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.

For the following exercises, consider a rigid body that is rotating about the x-axis counterclockwise with constant angular velocity $\omega =\langle a,b,c\rangle .$ If P is a point in the body located at $\text{r}=x\text{i}+y\text{j}+z\text{k},$ the velocity at P is given by vector field $\text{F}=\omega \,\times \,\text{r}.$

Express F in terms of i, j, and k vectors.