A function is harmonic if its non-mixed second partial derivatives with respect to each input sum to $0$. Is there a similar notion for the sum of a function's first partial derivatives, or for any other derivatives? If not, why is this attribute only useful in conjunction with the second partial derivative?
2026-03-25 12:45:50.1774442750
Harmonic Function Analog
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You could define such a notion, but it probably wouldn't be particularly useful. One thing that's special about the sum of the second derivatives is that it is invariant under orthogonal transformations.