proof of Bendixson’s negative criterion

Suppose that there exists a periodic solution called $\mathrm{\Gamma }$ which has a period of $T$ and lies in $E$. Let the interior of $\mathrm{\Gamma }$ be denoted by $D$. Then by Green’s Theorem we can observe that

${\displaystyle \int {\int }_D\nabla \cdot \text{𝐟}dxdy}$$=$${\displaystyle \int {\int }_D\frac{\partial \text{𝐗}}{\partial x}}+{\displaystyle \frac{\partial \text{𝐘}}{\partial y}}\mathrm{d}x\mathrm{d}y$

$=$${\displaystyle {\oint }_{\mathrm{\Gamma }}}(\text{𝐗}\mathrm{d}y-\text{𝐘}\mathrm{d}x)$

$=$${\displaystyle {\int }_0^T}(\text{𝐗}\dot{y}-\text{𝐘}\dot{x})dt$

$=$${\displaystyle {\int }_0^T}(\text{𝐗𝐘}-\text{𝐘𝐗})dt$

$=$$0$

Since $\nabla \cdot \text{𝐟}$ is not identically zero by hypothesis and is of one sign, the double integral on the left must be non zero and of that sign. This leads to a contradiction since the right hand side is equal to zero. Therefore there does not exists a periodic solution in the simply connected region $E$.