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Given that $y=x\varphi(z)+\psi(z)$ where $z$ is an implicit function of $x,y$, and $x\cdot\varphi'(z)+\psi'(z)\neq0$. Try to prove that $$\frac{\partial^2z}{\partial x^2}\cdot\left(\frac{\partial z}{\partial y}\right)^2-2\cdot\frac{\partial z}{\partial x}\cdot\frac{\partial z}{\partial y}\cdot\frac{\partial^2 z}{\partial x\partial y}+\frac{\partial^2 z}{\partial y^2}\cdot\left(\frac{\partial z}{\partial x}\right)^2=0$$

The outline of the proof from the book Григорий Михайлович Фихтенгольц:

Differentiating $y=x\varphi(z)+\psi(z)$ with $\dfrac{\partial^2z}{\partial x^2}$, $\dfrac{\partial^2z}{\partial x\partial y}$, $\dfrac{\partial^2z}{\partial y^2}$, and then multiplying special coefficients, we can get the answer.

I wonder whether there's some sensible ways to check these equations, or even more, some sysmatical ways to produce such equations.

Any help? Thanks a lot!

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up vote 3 down vote accepted

Let $\begin{pmatrix}z_{xx}&z_{xy}\\z_{xy}&z_{yy}\end{pmatrix}$ be the Hessian matrix of $z(x,y)$. For any unit vector $v$ the expression $v^THv$ is the second directional derivative of $z$ along $v$. The expression we are given is exactly of this form, with $v=\begin{pmatrix}-z_y \\ z_x\end{pmatrix}$, which we recognize as the gradient $\nabla z$ rotated by 90 degrees.

In other words, the formula we are asked to prove simply says that the second directional derivative of $z(x,y)$ vanishes in the direction tangent to the level curve of $z$. This sounds like the level curve is not allowed to have positive curvature, and indeed, a glance at the implicit equation tells us that the level curves of $z(x,y)$ are lines. Naturally, all directional derivatives of $z$ vanish along these lines, including the second one.

Additional comments:

  • The graph of $z$ is a special kind of a ruled surface: it is ruled by horizontal lines. Not sure if these have a name.
  • If we instead require that the second derivative vanishes in the direction of $\nabla z$ (without rotating the gradient by 90 degrees), we get $(\nabla z)^T\, H\, \nabla z=0$, the $\infty$-Laplace equation, a recently popular subject in PDE.
  • Григорий Михайлович Фихтенгольц died almost exactly 53 years ago, June 26th. I'm sure that he expected the students to actually carry out the computations, and that the students did.
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For your third comment: I've computed such partial derivative (such computation is valuable for me, although it's complicated), but I'm looking for a high way to know it. I think the author should have got a high way, otherwise he could not have figured out that equation. – Frank Science Jun 28 '12 at 1:17

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