91影视

Consider the given vector field$\mathbf{F}(x,y)=x^2{e}^y\mathbf{i}+\cos 鈦�x\sin 鈦�y\mathbf{j}.$

The objective is to write the line integral ${鈭�}_C鈥�\mathbf{F}鈰�d\mathbf{r}$as a double integral by using the divergence form of Green's Theorem, where C is the unit circle traversed counterclockwise.

"Let R be a region in the plane with smooth boundary curve C oriented counterclockwise by

$\mathbf{r}\left(t\right)=鉄�\left(x\right(t),y(t\left)\right)鉄�\text{for}a鈮�t鈮�b$

If a vector field is defined on R, then

${鈭�}_C鈥�\mathbf{F}鈰�d\mathbf{r}={鈭�}_R鈥�\left(\frac{\mathrm{鈭�}{F}_2}{\mathrm{鈭�}x}鈭�\frac{\mathrm{鈭�}{F}_1}{\mathrm{鈭�}y}\right)dA鈰�"$

If a unit vector n is perpendicular to the curve C, then Green鈥檚 Theorem is equivalent to the following statement:

${鈭�}_C鈥�\mathbf{F}(x,y)鈰�\mathbf{n}ds={鈭�}_R鈥�\mathrm{div}鈦�\mathbf{F}dA."鈥�鈥�\left(1\right)$

The divergence of a vector field $F(x,y)=\left(F_1(x,y)\mathbf{i}+F_2(x,y)\mathbf{j}\right)$in ${\mathrm{鈩�}}^2$is defined as follows:

$\mathrm{div}鈦�\mathbf{F}=\left(\frac{\mathrm{鈭�}}{\mathrm{鈭�}x}\mathbf{i}+\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}\mathbf{j}\right)鈰�\left(F_1(x,y)\mathbf{i}+F_2(x,y)\mathbf{j}\right)$

For the vector field $\mathbf{F}(x,y)=x^2{e}^y\mathbf{i}+\cos 鈦�x\sin 鈦�y\mathbf{j}$, the divergence of F will be,

localid="1654152262260" $\begin{array}{rcl}\begin{array}{r}\mathrm{div}鈦�\mathbf{F}\end{array}& =& \left(\frac{\mathrm{鈭�}}{\mathrm{鈭�}x}\mathbf{i}+\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}\mathbf{j}\right)鈰�\left(x^2{e}^y\mathbf{i}+\cos 鈦�x\sin 鈦�y\mathbf{j}\right)\\ & =& \begin{array}{r}\frac{\mathrm{鈭�}}{\mathrm{鈭�}x}\left(x^2{e}^y\right)+\frac{\mathrm{鈭�}}{\mathrm{鈭�}y}(\cos 鈦�x\sin 鈦�y)\end{array}\\ & =& 2x{e}^y+\cos 鈦�x\cos 鈦�y\end{array}$

Use the Divergence form of Green's Theorem (1) to evaluate the line integral as follows:

$\begin{array}{rcl}\begin{array}{r}{鈭�}_C鈥�\mathbf{F}(x,y)鈰�\mathbf{n}ds\end{array}& =& {鈭�}_R鈥�\mathrm{div}鈦�\mathbf{F}dA\\ & =& {鈭�}_R鈥�\left(2x{e}^y+\cos 鈦�x\cos 鈦�y\right)dA\end{array}$

Where R is the unit circle traversed counterclockwise.

Therefore, the line integral as double integral is ${鈭�}_R鈥�\left(2x{e}^y+\cos 鈦�x\cos 鈦�y\right)dA$