벡터 미적분 전체 문제

[T] Use a CAS to evaluate line integral ${\int }_{C}^{}xydx+ydy$ over path C given by $x=2t,\,y=10t,$ where $0\le t\le 1.$

Evaluate line integral ${\int }_{C}^{}(2x-y)dx+(x+3y)dy,$ where C lies along the x-axis from $x=0\,\text{to}\,x=5.$

For the following exercises, evaluate the line integrals.

[T] Use a CAS to evaluate ${\int }_{C}^{}\frac{y}{2{x}^{2}-{y}^{2}}ds,$ where C is $x=t,\,y=t,\,1\le t\le 5.$

For the following exercises, evaluate the line integrals.

[T] Use a CAS to evaluate ${\int }_{C}xyds,$ where C is $x={t}^{2},y=4t,0\le t\le 1.$

In the following exercises, find the work done by force field F on an object moving along the indicated path.

$\text{F}(x,y)=-x\text{i}-2y\text{j}$

$C:\,y={x}^{3}\,\text{from (0, 0) to (2, 8)}$

In the following exercises, find the work done by force field F on an object moving along the indicated path.

$\text{F}(x,\,y)=2xi+y\text{j}$

C: counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 1)

In the following exercises, find the work done by force field F on an object moving along the indicated path.

$\text{F}(x,\,y,\,z)=x\text{i}+y\text{j}-5z\text{k}$

$\text{C}:\,\text{r}(t)=2\,\cos \,t\text{i}+2\,\sin \,t\text{j}+t\text{k},\,0\le t\le 2\pi$

Let F be vector field $\text{F}(x,y)=({y}^{2}+2x{e}^{y}+1)\text{i}+(2xy+{x}^{2}{e}^{y}+2y)\text{j}.$ Compute the work of integral ${\int }_{C}^{}\text{F}\cdot d\text{r},$ where C is the path $\text{r}(t)=\sin \,t\text{i}+\cos \,t\text{j},\,0\le t\le \frac{\pi }{2}.$

Compute the work done by force $\text{F}(x,y,z)=2x\text{i}+3y\text{j}-z\text{k}$ along path $\text{r}(t)=t\text{i}+{t}^{2}\text{j}+{t}^{3}\text{k},$ where $0\le t\le 1.$

Evaluate ${\int }_{C}^{}\text{F}\cdot d\text{r},$ where $\text{F}(x,y)=\frac{1}{x+y}\text{i}+\frac{1}{x+y}\text{j}$ and C is the segment of the unit circle going counterclockwise from $(1,0)$ to (0, 1).

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?

Find the work done by vector field $\text{F}(x,y,z)=x\text{i}+3xy\text{j}-(x+z)\text{k}$ on a particle moving along a line segment that goes from (1, 4, 2) to (0, 5, 1).

How much work is required to move an object in vector field $\text{F}(x,y)=y\text{i}+3x\text{j}$ along the upper part of ellipse $\frac{{x}^{2}}{4}+{y}^{2}=1$ from (2, 0) to $(-2,0)?$

A vector field is given by $\text{F}(x,y)=(2x+3y)\text{i}+(3x+2y)\text{j}.$ Evaluate the line integral of the field around a circle of unit radius traversed in a clockwise fashion.

Evaluate the line integral of scalar function $xy$ along parabolic path $y={x}^{2}$ connecting the origin to point (1, 1).

In the following exercises, find the work done by force field F on an object moving along the indicated path.

Find ${\int }_{C}^{}{y}^{2}dx+(xy-{x}^{2})dy$ along C: $y=3x$ from (0, 0) to (1, 3).

In the following exercises, find the work done by force field F on an object moving along the indicated path.

Find ${\int }_{C}^{}{y}^{2}dx+(xy-{x}^{2})dy$ along C: ${y}^{2}=9x$ from (0, 0) to (1, 3).

For the following exercises, use a CAS to evaluate the given line integrals.

[T] Evaluate $\text{F}(x,y,z)={x}^{2}z\text{i}+6y\text{j}+y{z}^{2}\text{k},$ where C is represented by $\text{r}(t)=t\text{i}+{t}^{2}\text{j}+\ln \,t\text{k},\,1\le t\le 3.$

[T] Evaluate line integral ${\int }_{\gamma }^{}x{e}^{y}ds$ where, $\gamma$ is the arc of curve $x={e}^{y}$ from $(1,0)$ to $(e,1).$

[T] Evaluate the integral ${\int }_{\gamma }^{}x{y}^{2}ds,$ where $\gamma$ is a triangle with vertices (0, 1, 2), (1, 0, 3), and $(0,-1,0).$