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Given a sufficiently smooth scalar field $\phi$, how to get another scalar field $\psi$ so that their gradients are orthogonal. In 2D case, this is equivalent to:$$\nabla \phi \cdot \nabla \psi=\frac{\partial\phi}{\partial x}\frac{\partial\psi}{\partial x}+\frac{\partial\phi}{\partial y}\frac{\partial\psi}{\partial y}=0$$.

Notice the solution of $\psi$ may be up to a constant. This may be similar to the problem of finding a orthogonal curvlinear coordinate. I just want to know when does the solution exist and how to calculate it.

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  • $\begingroup$ You can see a scalar field as a surface. This surface is parallel to the gradient at each point. You are looking for a surface whose gradient is then perpendicular to your given surface, at each point. So will be the solution. This helps? $\endgroup$ Commented Oct 20, 2017 at 10:13
  • $\begingroup$ You are solving a homogeneous, linear, partial differential equation in two variables to find your field. Look in an introductory book on PDE's maybe? $\endgroup$
    – Paul
    Commented Oct 20, 2017 at 10:31
  • $\begingroup$ @DaniloGregorin I know that this is equivalent to finding two scalars whose isosurfaces are orthogonal at each point. But I don’t know how to find one given another. $\endgroup$
    – MiniUFO
    Commented Oct 20, 2017 at 14:39
  • $\begingroup$ @Paul is there a clear simple expression of solution to this problem?or standard numerical procedure to get he solution? A link from the web will be much appreciated. $\endgroup$
    – MiniUFO
    Commented Oct 20, 2017 at 14:44
  • $\begingroup$ Sorry, away for the weekend. Don't know a website sorry. $\endgroup$
    – Paul
    Commented Oct 23, 2017 at 10:17

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