91影视

Already we have the vector field:

$\mathbf{F}(x,y,z)=\mathbf{i}+\mathbf{j}+\mathbf{k}$.

The goal is to determine how this vector field circulates about $C$, where $C$is the curve of intersection of the plane $胃=\frac{蟺}{4}$or $胃=\frac{5蟺}{4}$and the unit sphere.

The following is the description of the circulating of a vector field $F$around $C$:

${鈭�}_C鈥�\mathbf{F}(x,y,z)脳d\mathbf{r}$

To evaluate this integral, use Stokes' Theorem. "Let $S$be an orientated, smooth or piecewise-smooth surface bounded by a curve $C$," says Stokes. 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 $n$.

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

Equation $1$

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

Calculate the vector field's curl first.$\mathbf{F}(x,y,z)=\mathbf{i}+\mathbf{j}+\mathbf{k}$

A vector field's curvature $\mathbf{F}(x,y,z)=F_1(x,y,z)\mathbf{i}+F_2(x,y,z)\mathbf{j}+F_3(x,y,z)\mathbf{k}$is created as:

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

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

The vector field will then have a curl of $\mathbf{F}(x,y,z)=\mathbf{i}+\mathbf{j}+\mathbf{k}$.

$\mathrm{curl}鈦�\mathbf{F}(x,y,z)=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ \frac{\mathrm{鈭�}}{\mathrm{鈭�}x}& \frac{\mathrm{鈭�}}{\mathrm{鈭�}y}& \frac{\mathrm{鈭�}}{\mathrm{鈭�}z}\\ 1& 1& 1\end{array}\right|$

$=\left[\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}\left(1\right)鈭�\frac{\mathrm{鈭�}}{\mathrm{鈭�}z}\left(1\right)\right]\mathbf{i}鈭�\left[\frac{\mathrm{鈭�}}{\mathrm{鈭�}x}\left(1\right)鈭�\frac{\mathrm{鈭�}}{\mathrm{鈭�}z}\left(1\right)\right]\mathbf{j}+\left[\frac{\mathrm{鈭�}}{\mathrm{鈭�}x}\left(1\right)鈭�\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}\left(1\right)\right]\mathbf{k}$

$=0i+0j+0k$

$=0$

Evaluate the equation$1$${鈭�}_C鈥�\mathbf{F}(x,y,z)脳d\mathbf{r}$as below:

$={鈭�}_S鈥�\mathbf{0}\mathbf{脳}\mathbf{n}dS$

$={鈭�}_S鈥�\mathbf{0}dS$

$=0$