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Let $f(x,y,z) = \log(x^3+y^3+z^3-3xyz)$ then to find the value of $\displaystyle \left(\frac{\partial}{\partial x} + \frac{\partial}{\partial y}+\frac{\partial}{\partial z}\right)^2 f$.

We can do it by brute force and lengthy process ...is there any elegant method?

Will Euler's Theorem for homogeneous equation come to help us!

Michael Hardy
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User8976
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    Standard notation for partial derivatives is $\displaystyle\frac{\partial}{\partial x} f$, not $\displaystyle\frac{\delta}{\delta x} f$. I changed it. $\qquad$ – Michael Hardy Feb 18 '16 at 17:14

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Omitting the point of application, we have $e^{f} = x^3+y^2+z^3-3xyz$. Applying $(\partial_x+\partial_y+\partial_z)$ on both sides we get: $$e^f(\partial_x+\partial_y+\partial_z)f = 3(x^2+y^2+z^2 - xz-yz-xy)$$Repeating, we get: $$e^f((\partial_x+\partial_y+\partial_z)f)^2 + e^f(\partial_x+\partial_y+\partial_z)^2f = 0,$$so: $$(\partial_x+\partial_y+\partial_z)^2f = - ((\partial_x+\partial_y+\partial_z)f)^2 = -\left(\frac{3(x^2+y^2+z^2 - xz-yz-xy)}{x^3+y^2+z^3-3xyz}\right)^2$$

Ivo Terek
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