Suppose I am given a function $f(x, y, z)$ that is such that

$3 f_x + xf_y + 2yf_z = 0$

I want to know how to write down a general representation for functions with such a property.

Proceeding by method of characteristics, I obtained

$\phi_1 = y - \dfrac{x^2}{6}$, and $\phi_2 = z + \dfrac{2}{27}x^3 - \dfrac{2}{3} xy$

and f satisfies the orthogonality condition if it is of the form

$f = g(\phi_1, \phi_2)$

Q1) Is this a necessary and sufficient condition for the function f to have the stated property ?

Q2) Is the representation unique ?

  • $\begingroup$ If the solution $f$ is smooth, this is necessary and sufficient. Now, you can multiply $\phi_{1,2}$ by arbitrary constants without loosing any generality of the result. $\endgroup$
    – EditPiAf
    Feb 19, 2019 at 11:56

1 Answer 1


Follow the method in http://en.wikipedia.org/wiki/Method_of_characteristics#Example:

$\dfrac{dx}{dt}=3$ , letting $x(0)=0$ , we have $x=3t$

$\dfrac{dy}{dt}=x=3t$ , letting $y(0)=y_0$ , we have $y=\dfrac{3t^2}{2}+y_0=\dfrac{x^2}{6}+y_0$

$\dfrac{dz}{dt}=2y=3t^2+2y_0$ , letting $z(0)=z_0$ , we have $z=t^3+2y_0t+z_0=\dfrac{2xy}{3}-\dfrac{2x^3}{27}+z_0$

$\dfrac{df}{dt}=0$ , letting $f(0)=F(y_0,z_0)$ , we have $f(x,y,z)=F(y_0,z_0)=F\left(y-\dfrac{x^2}{6},z+\dfrac{2x^3}{27}-\dfrac{2xy}{3}\right)$

  • $\begingroup$ In my question, I already mentioned that I used the method of characteristics to get the result. My question was something else. $\endgroup$
    – me10240
    Jul 15, 2015 at 0:57

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