Let $P, Q$ be smooth functions on a domain $D \subseteq \mathbb{C}$, Find necessary and sufficient condition for the form $P dz + Q d\bar{z}$ to be closed.
ผมลองทำดู ได้เงื่อนไขตามนี้ครับ
Let $\omega = P dz + Q d \bar{z} = P \ (dx + i dy) + Q \ (dx - idy) = (P+Q) dx + (iP-iQ) dy$. Since $P, Q$ are smooth, $\omega$ is a $C^1$ differential form. So we have $$\omega \ \ \mbox{is closed iff} \ \ \frac{ \partial (P+Q)}{\partial y} = \frac{\partial (iP - iQ)}{\partial x} \ \ \mbox{iff} \ \ P_z = -Q_{\bar{z}}$$ where $P_z = \frac{1}{2}(P_x - i P_y)$ and $Q_{\bar{z}} = \frac{1}{2}(Q_x + i Q_y)$
ไม่แน่ใจว่าถูกมั้ยอ่ะครับ รบกวนช่วยดูให้หน่อย
