91影视

Already we have the vector field,

The goal is to show that if the curve $C$is formed as follows, the line incremental ${鈭�}_C鈥�\mathbf{F}\mathbf{脳}d\mathbf{r}$does not dependent on $F_3$.

The curve $C$is a smooth curve in the plane $z=r$, where $r$ is an arbitrary number in the$xy$ plane.

Applying stokes theorem,

"Assume $S$is an orientated, smooth or piecewise-smooth surface with a curve $C$enclosed by it. Assume that $n$is an orientated unit normal vector of $S$, and that $C$has a modeling method that traverses $C$ circular in relation to $n$.

The vector field$\mathbf{F}(x,y,z)=F_1(x,y,z)\mathbf{i}+F_2(x,y,z)\mathbf{j}+F_3(x,y,z)\mathbf{k}$is created on $S$.

Then,

Equation$1$

${鈭�}_C鈥�\mathbf{F}(x,y,z)脳d\mathbf{r}={鈭�}_S鈥�\mathrm{curl}鈦�\mathbf{F}(x,y,z)脳\mathbf{n}dS$

The vector field of curlis explained below

$\mathrm{curl}鈦�\mathbf{F}(x,y,z)=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\mathrm{鈭�}}{\mathrm{鈭�}x}& \frac{\mathrm{鈭�}}{\mathrm{鈭�}y}& \frac{\mathrm{鈭�}}{\mathrm{鈭�}z}\\ F_1(x,y,z)& F_2(x,y,z)& F_3(x,y,z)\end{array}\right|$

$=\left[\frac{\mathrm{鈭�}{F}_3}{\mathrm{鈭�}y}鈭�\frac{\mathrm{鈭�}{F}_2}{\mathrm{鈭�}z}\right]\mathbf{i}鈭�\left[\frac{\mathrm{鈭�}{F}_3}{\mathrm{鈭�}x}鈭�\frac{\mathrm{鈭�}{F}_1}{\mathrm{鈭�}z}\right]\mathbf{j}+\left[\frac{\mathrm{鈭�}{F}_2}{\mathrm{鈭�}x}鈭�\frac{\mathrm{鈭�}{F}_1}{\mathrm{鈭�}y}\right]\mathbf{k}$

The value of$\mathrm{curl}鈦�\mathbf{F}(x,y,z)脳\mathbf{n}$ explained,

$=\left(\frac{\mathrm{鈭�}{F}_3}{\mathrm{鈭�}y}鈭�\frac{\mathrm{鈭�}{F}_2}{\mathrm{鈭�}z}\right)脳0+\left(\frac{\mathrm{鈭�}{F}_1}{\mathrm{鈭�}z}鈭�\frac{\mathrm{鈭�}{F}_3}{\mathrm{鈭�}x}\right)脳0+\left(\frac{\mathrm{鈭�}{F}_2}{\mathrm{鈭�}x}鈭�\frac{\mathrm{鈭�}{F}_1}{\mathrm{鈭�}y}\right)r$

$=\left(\frac{\mathrm{鈭�}{F}_2}{\mathrm{鈭�}x}鈭�\frac{\mathrm{鈭�}{F}_1}{\mathrm{鈭�}y}\right)r$

Using stokes theorem, substitute in equation$1$

${鈭�}_{\mathrm{C}}鈥�\mathbf{F}(x,y,z)脳d\mathbf{r}={鈭�}_S鈥�\mathrm{curl}鈦�\mathbf{F}(x,y,z)脳\mathbf{n}dS$

$={鈭�}_D鈥�\left(\frac{\mathrm{鈭�}{F}_2}{\mathrm{鈭�}x}鈭�\frac{\mathrm{鈭�}{F}_1}{\mathrm{鈭�}y}\right)rdA$