Use the same vector field as in Exercise 13 together with the divergence form of Green’s Theorem to write the line integral of F(x, y) about the unit circle as a double integral. Do not evaluate the integral.

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

The objective is to write the line integral ${\int }_C \mathbf{F}\cdot d\mathbf{r}$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

$\mathbf{r}\left(t\right)=⟨\left(x\right(t),y(t\left)\right)⟩\text{for}a\le t\le b$

If a vector field is defined on R, then

${\int }_C \mathbf{F}\cdot d\mathbf{r}={\iint }_R \left(\frac{\mathrm{\partial }{F}_2}{\mathrm{\partial }x}-\frac{\mathrm{\partial }{F}_1}{\mathrm{\partial }y}\right)dA\cdot "$

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

${\int }_C \mathbf{F}(x,y)\cdot \mathbf{n}ds={\iint }_R \mathrm{div}\mathbf{F}dA."\dots \dots \left(1\right)$

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

$\mathrm{div}\mathbf{F}=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }x}\mathbf{i}+\frac{\mathrm{\partial }}{\mathrm{\partial }y}\mathbf{j}\right)\cdot \left(F_1(x,y)\mathbf{i}+F_2(x,y)\mathbf{j}\right)$

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

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

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

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

Where R is the unit circle traversed counterclockwise.

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