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This PDE came up in my research: $$\frac{\partial f(x, y)}{\partial x} + \frac{\partial g(x, y)}{\partial y} = 0$$ where $f, g: \mathbb{R}^{2} \to \mathbb{R}$.

Does anyone know what this equation is called? WolframAlpha was of no help.

Is there a method to identify solutions for $f$ and $g$?

Ameya
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  • $f$ and $g$ are scalar or vector fields? – Daniel Muñoz Feb 28 '23 at 00:56
  • Let's assume $f, g: \mathbb{R}^2 \to \mathbb{R}$ for now. But I am also interested in the case where $f, g: \mathbb{R}^{2n} \to \mathbb{R}^n$ for $n > 1$ as well. – Ameya Feb 28 '23 at 01:03
  • Never mind. I misread the question. – Jap88 Feb 28 '23 at 01:15
  • Suppose that we draw a tangent vector with direction $x$ in the $f\text{-field}$ and a tangent vector with direction $y$ in the $g\text{-field}$. If ${\partial f\over\partial x}=-{\partial g\over\partial y}$, then $g$ is $f$ is rotated around $z\text{-axis}$ by $90°$ clockwise I'm presume. – Daniel Muñoz Feb 28 '23 at 01:20
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    This says the divergence vanishes: $\nabla\cdot(f,g)=0$. – anon Feb 28 '23 at 02:03

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This looks almost exactly like one of the Cauchy-Riemann equations. If we have real-valued functions $f(x,y),g(x,y)$ then $$\frac{\partial f}{\partial y}=-\frac{\partial g}{\partial x}$$Is a Cauchy-Riemann equation. We could just switch the variables $y$ and $x$ around to get your equation.

Kamal Saleh
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    I also think that can be – Daniel Muñoz Feb 28 '23 at 01:28
  • Your comment reminded me of it though, since I haven't used these equations for a few months. – Kamal Saleh Feb 28 '23 at 01:42
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    It's one of the CR equations. It's pretty easy to construct examples of $f$ and $g$ that aren't harmonic but satisfy this equation, e.g. by taking harmonic examples and perturbing them by the variable being differentiated. – Brevan Ellefsen Feb 28 '23 at 01:45
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    The CR equations basically say the flow induced by the (Cartesian) product of the functions (up to a sign, representing a rotation) is irrotational and solenoidal respectively. I'd thus refer the OP's equation as a generalization to $n$ dimensions of Laplace's Equation for Irrational Flow, though I don't know of any non-obvious results off the top of my head regarding stability or the moduli space of solutions. – Brevan Ellefsen Feb 28 '23 at 01:55
  • Interesting! Aren't the Cauchy-Riemann equations a pair of two equations though? – Ameya Feb 28 '23 at 02:13
  • @Ameya Yes, one of the equations is what you wanted. Remember to accept, it helps both of us! – Kamal Saleh Feb 28 '23 at 02:35
  • Yes, but there could be solutions that satisfy only one of the CR equations but not the other? Sorry if I am missing something. – Ameya Feb 28 '23 at 02:46
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    Yes, that is precisely what I said in my first above comment @Ameya. – Brevan Ellefsen Feb 28 '23 at 02:52
  • @BrevanEllefsen You might have, but in complicated notation :) Your comment could be an answer by the way. – Kamal Saleh Feb 28 '23 at 02:58
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    @KamalSaleh If you're unfamiliar with the notation, let me expand my comment. It's easy to prove that if $u$ and $v$ satisfy the CR equations (both of them) so are the real and imaginary parts of some holomorphic function then $u$ and $v$ are harmonic, i.e. $u_{xx} + u_{yy} = 0$ and $v_{xx} + v_{yy} = 0$. Thus satisfying one CR equation but not the other is equivalent to satisfying one but not being harmonic, hence my use of that notation. As I mentioned, we can use this idea to find $u,v$ satisfying one but not the other by perturbation, i.e. let $u$ and $v$ be the real and imaginary... – Brevan Ellefsen Feb 28 '23 at 03:25
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    ... parts of your favorite holomorphic function so that $u_y = -v_x$, and define $f = u + p_1(x,y)$ and $g' = v + p_2(x,y)$, where $p_1, p_2$ aren't harmonic (e.g. you could just take a third order polynomial in $x$). Then $f_y = -g_x$ so $f,g$ satisfy the first CR equation, but clearly $f$ and $g$ are not harmonic so cannot satisfy the second CR equation. – Brevan Ellefsen Feb 28 '23 at 03:29
  • I keep getting downvotes for some reason, and when I flag the answer/question for targeted downvote the moderator says "needs evidence" even though my post is fine- and has nothing wrong about it. Oh well, it is just a downvote I guess. – Kamal Saleh Feb 28 '23 at 16:49