Let, p = x – y, q = y – z and r = z – x. ∴ u =f(p, q, r)

$ \therefore \dfrac{\partial p}{\partial x}=1 \; \; and \; \; \dfrac{\partial p}{\partial y}=-1 $

Similarly,

$ \dfrac{\partial q}{\partial y}=1 \; , \; \dfrac{\partial q}{\partial z}=-1 \; , \; \dfrac{\partial r}{\partial z}=1 \; and \; \dfrac{\partial r}{\partial x}=-1 \; \; \\ \; \\ \; \\ Now , \; \dfrac{\partial u}{\partial x} \;=\; \dfrac{\partial u}{\partial p} . \dfrac{\partial p}{\partial x} + \dfrac{\partial u}{\partial r} . \dfrac{\partial r}{\partial x} \;=\; \dfrac{\partial u}{\partial p} (1)+ \dfrac{\partial u}{\partial r} (-1) \;=\; \dfrac{\partial u}{\partial p} - \dfrac{\partial u}{\partial r} \; \; \; \ldots (i) $

Similarly, $ \dfrac{\partial u}{\partial y} \;=\; \dfrac{\partial u}{\partial p} . \dfrac{\partial p}{\partial y} + \dfrac{\partial u}{\partial q} . \dfrac{\partial q}{\partial y} \;=\; \dfrac{\partial u}{\partial p} (-1) + \dfrac{\partial u}{\partial q} (1) \;=\; \dfrac{\partial u}{\partial q} - \dfrac{\partial u}{\partial p} \; \; \; \ldots (ii) $

and $ \dfrac{\partial u}{\partial z} \;=\; \dfrac{\partial u}{\partial q} . \dfrac{\partial q}{\partial z} + \dfrac{\partial u}{\partial r} . \dfrac{\partial r}{\partial z} \;=\; \dfrac{\partial u}{\partial q} (-1)+ \dfrac{\partial u}{\partial r} (1) \;=\; \dfrac{\partial u}{\partial r} - \dfrac{\partial u}{\partial q} \; \; \; \; \ldots (iii) \\ \; \\ \; \\ \; \\ \; \\ \therefore \dfrac{\partial u}{\partial x} + \dfrac{\partial u}{\partial y} + \dfrac{\partial u}{\partial z} = \dfrac{\partial u}{\partial p} - \dfrac{\partial u}{\partial r} + \dfrac{\partial u}{\partial p} - \dfrac{\partial u}{\partial r} + \dfrac{\partial u}{\partial q} - \dfrac{\partial u}{\partial p} \; = \; 0 $

Hence Proved.