벡터 미적분 전체 문제

What is vector field $\text{F}(x,y)$ with a value at $(x,y)$ that is of unit length and points toward $(1,0)?$

All vectors are parallel to the x-axis and all vectors on a vertical line have the same magnitude.

All vectors point toward the origin and have constant length.

All vectors are of unit length and are perpendicular to the position vector at that point.

Give a formula $\text{F}(x,y)=M(x,y)\text{i}+N(x,y)\text{j}$ for the vector field in a plane that has the properties that $\text{F}=0$ at $(0,0)$ and that at any other point $(a,b),$ F is tangent to circle ${x}^{2}+{y}^{2}={a}^{2}+{b}^{2}$ and points in the clockwise direction with magnitude $||F||=\sqrt{{a}^{2}+{b}^{2}}.$

For the following exercises, write formulas for the vector fields with the given properties.

Is vector field $\text{F}(x,y)=<P(x,y),Q(x,y)>=(\sin \,x+y)\text{i}+(\cos \,y+x)\text{j}$ a gradient field?

Find a formula for vector field $\text{F}(x,y)=M(x,y)\text{i}+N(x,y)\text{j}$ given the fact that for all points $(x,y),$ F points toward the origin and $||F||=\frac{10}{{x}^{2}+{y}^{2}}.$

Find the components of the electric field in the x- and y-directions, where $\text{E}(x,y)=-\nabla V(x,y).$

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.

[T] ${\int }_{C}^{}(x+y)ds$

$C:\,x=t,y=(1-t),\,z=0$ from (0, 1, 0) to (1, 0, 0)

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.

[T] ${\int }_{C}^{}(x-y)ds$

$C:\,\text{r}(t)=4t\text{i}+3t\text{j}$ when $0\le t\le 2$

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.

[T] ${\int }_{C}^{}({x}^{2}+{y}^{2}+{z}^{2})ds$

$C:\,\text{r}(t)=\sin \,t\text{i}+\cos \,t\text{j}+8t\text{k}$ when $0\le t\le \frac{\pi }{2}$

[T] Evaluate ${\int }_{C}^{}x{y}^{4}ds,$ where C is the right half of circle ${x}^{2}+{y}^{2}=16$ and is traversed in the clockwise direction.

[T] Evaluate ${\int }_{C}^{}4{x}^{3}ds,$ where C is the line segment from $(-2,-1)$ to (1, 2).

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 $(0,5,1).$

Find the work done by a person weighing 150 lb walking exactly one revolution up a circular, spiral staircase of radius 3 ft if the person rises 10 ft.

Find the work done by force field $\text{F}(x,y,z)=-\frac{1}{2}x\text{i}-\frac{1}{2}y\text{j}+\frac{1}{4}\text{k}$ on a particle as it moves along the helix $\text{r}(t)=\cos \,t\text{i}+\sin \,t\text{j}+t\text{k}$ from point $(1,0,0)$ to point $(-1,0,3\pi ).$

Find the work done by vector field $\text{F}(x,y)=y\text{i}+2x\text{j}$ in moving an object along path C, the straight line which joins points (1, 0) and (0, 1).

Find the work done by force $\text{F}(x,y)=2y\text{i}+3x\text{j}+(x+y)\text{k}$ in moving an object along curve $\text{r}(t)=\cos (t)\text{i}+\sin (t)\text{j}+\frac{1}{6}\text{k},$ where $0\le t\le 2\pi .$

Find the mass of a wire in the shape of a circle of radius 2 centered at (3, 4) with linear mass density $\rho (x,y)={y}^{2}.$

For the following exercises, evaluate the line integrals.

Evaluate ${\int }_{\gamma }^{}{({x}^{2}+{y}^{2}+{z}^{2})}^{-1}ds,$ where $\gamma$ is the helix $x=\cos \,t,y=\sin \,t,z=t(0\le t\le T).$