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Let $ \Omega_1 $ and $ \Omega_2 $ be domains (open and connected) in $ \mathbb{R}^2 $. $ \psi:\Omega_1\to\mathbb{R} $ and $ \phi:\Omega_1\to\mathbb{R} $ are $ C^1 $ functions with two variables. Moreover, we assume that map $ (x,y)\to (\phi(x,y),\psi(x,y)) $ is homeomorphism from $ \Omega_1 $ to $ \Omega_2 $, i.e. the map $ (x,y)\to (\phi(x,y),\psi(x,y)) $ is continuous from $ \Omega_1 $ to $ \Omega_2 $ and has continuous inverse map. I want to ask that if I can obtain that the Jacobi determinant of the map, denoted as $ \frac{\partial(\phi,\psi)}{\partial(x,y)} $ is either non-positive or non-negative in $ \Omega_1 $, i.e. either $ \frac{\partial(\phi,\psi)}{\partial(x,y)}\geq 0 $ for all $ (x,y)\in\Omega_1 $, or $ \frac{\partial(\phi,\psi)}{\partial(x,y)}\leq 0 $ for all $ (x,y)\in\Omega_1 $. I have tried by considering the image sets of a curve in $ \Omega_1 $ and by using the connectedness of $ \Omega_1 $ but failed. Can you give me some references or hints?

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1 Answer 1

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Let $f=(\phi,\psi):\Omega_1\to\Omega_2$.

For every point $p\in\Omega_1$ consider the curve $\gamma:t\mapsto p+\varepsilon e^{it}$, for $\varepsilon$ so small that the curve is contained in $\Omega_1$. Let $n(p)$ be the winding number of $f\circ \gamma$ around $f(p)$. As $f$ is a homeomorphism, $n(p)$ is well defined (and an integer) and depends continuously on $p$, so as $\Omega_1$ is connected, it is constant. So your statement is a direct consequence of the fact that $n(p)=1$ in points $p$ where $\frac{\partial(\phi,\psi)}{\partial(x,y)}(p)>0$ and $n(q)=-1$ in points $q$ where $\frac{\partial(\phi,\psi)}{\partial(x,y)}(q)<0$.

This can be generalized to higher dimensions using the local degree of $f$ instead of winding numbers.

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