Some questions on $\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}+\frac{\partial f}{\partial z} = g(x,y,z)$ Let  $\quad\displaystyle \frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}+\frac{\partial f}{\partial z} = g(x,y,z)\;$.


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*Is there a better notation for writing the above?

*Given $g$, can $f$ always be found?

*Given $g$, is there a unique $f$ that satisfies the above equation?

*Is there a name given to this equation?

Your condition is that the directional derivative of $f$ in the $(1,1,1)$ direction is given by $g(x,y,z)$. It's a rotated version of the simple equation ${\partial f \over \partial x} = h(x,y,z)$. 
Just as you can solve this by $f(x,y,z) = \int_0^x h(t,y,z) \,dt + H(y,z)$, where $H(y,z)$ can be anything, you can always solve your equation by integrating $g(x,y,z)$ in the $(1,1,1)$ direction, and you have nonuniqueness because you can add any function of $x - y$ and $y - z$ to your solution and obtain another solution; the directional derivative of any function $H(x-y,y-z)$ in the $(1,1,1)$ direction is zero. 
