선적분

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

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

Evaluate the following integrals.

$\int_{C}{x}^{2}dy+(2x-3xy)dx,$ along $C:y=\frac{1}{2}x$ from (0, 0) to (4, 2)

Evaluate the following integrals.

$\int_{C}ydx+x{y}^{2}dy,$ where $C:x=\sqrt{t},y=t-1,0\le t\le 1$

Evaluate the following integrals.

${\iint }_{S}x{y}^{2}dS,$ where S is surface $z={x}^{2}-y,0\le x\le 1,0\le y\le 4$

Find the divergence and curl for the following vector fields.

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

Find the divergence and curl for the following vector fields.

$\text{F}(x,y,z)={e}^{x}\text{i}+{e}^{xy}\text{j}+{e}^{xyz}\text{k}$

$\int_{C}3xydx+2x{y}^{2}dy,$ where C is a square with vertices (0, 0), (0, 2), (2, 2) and (2, 0) oriented counterclockwise.

${\int }_{C}3ydx+(x+{e}^{y})dy,$ where C is a circle centered at the origin with radius 3

$\text{F}(x,y,z)=y\text{i}-x\text{j}+z\text{k},$ where $S$ is the upper half of the unit sphere

$\text{F}(x,y,z)=y\text{i}+xyz\text{j}-2zx\text{k},$ where $S$ is the upward-facing paraboloid $z={x}^{2}+{y}^{2}$ lying in cylinder ${x}^{2}+{y}^{2}=1$

Use the divergence theorem to evaluate ${\iint }_{S}\text{F}\cdot dS.$

$\text{F}(x,y,z)=({x}^{3}y)\text{i}+(3y-{e}^{x})\text{j}+(z+x)\text{k},$ over cube $S$ defined by $-1\le x\le 1,$ $0\le y\le 2,$ $0\le z\le 2$

Use the divergence theorem to evaluate ${\iint }_{S}\text{F}\cdot dS.$

$\text{F}(x,y,z)=(2xy)\text{i}+(-{y}^{2})\text{j}+(2{z}^{3})\text{k},$ where $S$ is bounded by paraboloid $z={x}^{2}+{y}^{2}$ and plane $z=2$

Find the total mass of a thin wire in the shape of an upper semicircle with radius $\sqrt{2,}$ and a density function of $\rho (x,y)=y+{x}^{2}.$

Find the total mass of a thin sheet in the shape of a hemisphere with radius 2 for $z\ge 0$ with a density function $\rho (x,y,z)=x+y+z.$

Use the divergence theorem to compute the value of the flux integral over the unit sphere with $\text{F}(x,y,z)=3z\text{i}+2y\text{j}+2x\text{k}.$

In Exercises 5–10, a planar curve $C$ is given along with a surface $f$ that is defined over $C$. Evaluate the line integral $\displaystyle \int_Cf(s)\ ds.$

$C$ is the line segment joining the points $(-2,-1)$ and $(1,2)$; the surface is $f(x,y)=x^2+y^2+2$.

In Exercises 5–10, a planar curve $C$ is given along with a surface $f$ that is defined over $C$. Evaluate the line integral $\displaystyle \int_Cf(s)\ ds.$

$C$ is the segment of $y=3x+2$ on $[1,2]$; the surface is $f(x,y)=5x+2y$.

In Exercises 5–10, a planar curve $C$ is given along with a surface $f$ that is defined over $C$. Evaluate the line integral $\displaystyle \int_Cf(s)\ ds.$

$C$ is the circle with radius 2 centered at the point $(4,2)$; the surface is $f(x,y)=3x-y$.

In Exercises 5–10, a planar curve $C$ is given along with a surface $f$ that is defined over $C$. Evaluate the line integral $\displaystyle \int_Cf(s)\ ds.$

$C$ is the curve given by $\vec r(t) = \langle \cos t+t\sin t, \sin t-t\cos t\rangle$ on $[0,2\pi]$; the surface is $f(x,y)=5$.

In Exercises 5–10, a planar curve $C$ is given along with a surface $f$ that is defined over $C$. Evaluate the line integral $\displaystyle \int_Cf(s)\ ds.$

$C$ is the piecewise curve composed of the line segments that connect $(0,1)$ to $(1,1)$, then connect $(1,1)$ to $(1,0)$; the surface is $f(x,y)=x+y^2$.