선적분

Find the line integral of ${\int }_{C}^{}(1+{x}^{2}y)ds,$ where C is ellipse $\text{r}(t)=2\,\cos \,t\text{i}+3\,\sin \,t\text{j}$ from $0\le t\le \pi .$

Compute the flux of $\text{F}={x}^{2}\text{i}+y\text{j}$ across a line segment from (0, 0) to (1, 2).

Let $\text{F}=5\text{i}$ and let C be curve $y=0,0\le x\le 4.$ Find the flux across C.

Let $\text{F}=5\text{j}$ and let C be curve $y=0,0\le x\le 4.$ Find the flux across C.

For the following exercises, find the flux.

Let $\text{F}=-y\text{i}+x\text{j}$ and let C: $\text{r}(t)=\cos \,t\text{i}+\sin \,t\text{j}$ $(0\le t\le 2\pi ).$ Calculate the flux across C.

For the following exercises, find the flux.

Let $\text{F}=({x}^{2}+{y}^{3})\text{i}+(2xy)\text{j}.$ Calculate flux F orientated counterclockwise across curve C: ${x}^{2}+{y}^{2}=9.$

Find the line integral of ${\int }_{C}^{}{z}^{2}dx+ydy+2ydz,$ where C consists of two parts: ${C}_{1}$ and ${C}_{2}.$ ${C}_{1}$ is the intersection of cylinder ${x}^{2}+{y}^{2}=16$ and plane $z=3$ from (0, 4, 3) to $(-4,0,3).$ ${C}_{2}$ is a line segment from $(-4,0,3)$ to (0, 1, 5).

A spring is made of a thin wire twisted into the shape of a circular helix $x=2\,\cos \,t,\,y=2\,\sin \,t,\,z=t.$ Find the mass of two turns of the spring if the wire has constant mass density.

A thin wire is bent into the shape of a semicircle of radius a. If the linear mass density at point P is directly proportional to its distance from the line through the endpoints, find the mass of the wire.

An object moves in force field $\text{F}(x,y,z)={y}^{2}\text{i}+2(x+1)y\text{j}$ counterclockwise from point (2, 0) along elliptical path ${x}^{2}+4{y}^{2}=4$ to $(-2,0),$ and back to point (2, 0) along the x-axis. How much work is done by the force field on the object?

Find the work done when an object moves in force field $\text{F}(x,y,z)=2x\text{i}-(x+z)\text{j}+(y-x)\text{k}$ along the path given by $\text{r}(t)={t}^{2}\text{i}+({t}^{2}-t)\text{j}+3\text{k},$ $0\le t\le 1.$

If an inverse force field F is given by $\text{F}(x,y,z)=\frac{k}{{\| \text{r}\| }^{3}}\text{r},$ where k is a constant, find the work done by F as its point of application moves along the x-axis from $A(1,0,0)\,\text{to}\,B(2,0,0).$

David and Sandra plan to evaluate line integral ${\int }_{C}^{}\text{F}\cdot d\text{r}$ along a path in the xy-plane from (0, 0) to (1, 1). The force field is $\text{F}(x,y)=(x+2y)\text{i}+(-x+{y}^{2})\text{j}.$ David chooses the path that runs along the x-axis from (0, 0) to (1, 0) and then runs along the vertical line $x=1$ from (1, 0) to the final point (1, 1). Sandra chooses the direct path along the diagonal line $y=x$ from (0, 0) to (1, 1). Whose line integral is larger and by how much?

For vector field $\text{F}(x,y)=P(x,y)\text{i}+Q(x,y)\text{j},$ if ${P}_{y}(x,y)={Q}_{x}(x,y)$ in open region $D,$ then ${\int }_{C}Pdx+Qdy=0.$

The divergence of a vector field is a vector field.

Draw the following vector fields.

$\text{F}(x,y)=\frac{1}{2}\text{i}+2x\text{j}$

Draw the following vector fields.

$\text{F}(x,y)=\frac{<y,3x>}{\sqrt{{x}^{2}+{y}^{2}}}$

Are the following the vector fields conservative? If so, find the potential function $f$ such that $\text{F}=\nabla f.$

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

Are the following the vector fields conservative? If so, find the potential function $f$ such that $\text{F}=\nabla f.$

$\text{F}(x,y)=(6xy)\text{i}+(3{x}^{2}-y{e}^{y})\text{j}$

Are the following the vector fields conservative? If so, find the potential function $f$ such that $\text{F}=\nabla f.$

$\text{F}(x,y,z)=(2xy+{z}^{2})\text{i}+({x}^{2}+2yz)\text{j}+(2xz+{y}^{2})\text{k}$