보존 벡터장

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

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

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

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

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

$\text{F}(x,y,\,z)=\left(-\frac{1}{y}\right)\text{i}+\left(\frac{x}{{y}^{2}}\right)\text{j}+(2z-1)\text{k}$

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

$\text{F}(x,y,z)=3{z}^{2}\text{i}-\cos y\text{j}+2xz\text{k}$

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

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

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

$\text{F}(x,y)=(4{e}^{x}\cos y)\text{i}-(4{e}^{x}\sin y)\text{j}$

For the following exercises, determine whether the vector field is conservative and, if so, find a potential function.

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

Evaluate ${\int }_{C}\nabla f\cdot d\text{r},$ where $f(x,y,\,z)=\cos (\pi x)+\sin (\pi y)-xyz$ and C is any path that starts at $\left(1,\frac{1}{2},2\right)$ and ends at $(2,1,-1).$

[T] Evaluate ${\int }_{C}\nabla f\cdot d\text{r},$ where $f(x,y)=xy+{e}^{x}$ and C is a straight line from $(0,0)$ to $(2,1).$

[T] Evaluate ${\int }_{C}\nabla f\cdot d\text{r},$ where $f(x,y)={x}^{2}y-x$ and C is any path in a plane from (1, 2) to (3, 2).

Evaluate ${\int }_{C}\nabla f\cdot d\text{r},$ where $f(x,y,\,z)=xy{z}^{2}-yz$ and C has initial point (1, 2, 3) and terminal point (3, 5, 1).

Calculate the line integral of F over C₁.

Calculate the line integral of G over C₁.

Calculate the line integral of F over C₂.

Calculate the line integral of G over C₂.

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 $B=(3,1,2).$

[T] Find line integral ${\int }_{C}\text{F}\cdot dr$ of vector field $\text{F}(x,y,z)=3{x}^{2}z\text{i}+{z}^{2}\text{j}+({x}^{3}+2yz)\text{k}$ along curve C parameterized by $r(t)=\left(\frac{\ln t}{\ln \,2}\right)\text{i}+{t}^{3/2}\text{j}+t\,\cos (\pi t)k,\,1\le t\le 4.$

$\text{F}=(x{y}^{2}+3{x}^{2}y)\text{i}+(x+y){x}^{2}\text{j};$ C is the curve consisting of line segments from $(1,1)$ to $(0,2)$ to $(3,0).$

For the following exercises, show that the following vector fields are conservative by using a computer. Calculate ${\int }_{C}\text{F}\cdot d\text{r}$ for the given curve.

$\text{F}=\frac{2x}{{y}^{2}+1}\text{i}-\frac{2y({x}^{2}+1)}{{({y}^{2}+1)}^{2}}\text{j};$ C is parameterized by $x={t}^{3}-1,y={t}^{6}-t,0\le t\le 1.$

For the following exercises, show that the following vector fields are conservative by using a computer. Calculate ${\int }_{C}\text{F}\cdot d\text{r}$ for the given curve.

[T] $\text{F}=[\cos (x{y}^{2})-x{y}^{2}\sin (x{y}^{2})]\text{i}-2{x}^{2}y\,\sin (x{y}^{2})\text{j};$ C is curve $r(t)=<{e}^{t},{e}^{t+1}>,-1\le t\le 0.$