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

  1. Is there a better notation for writing the above?

  2. Given $g$, can $f$ always be found?

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

  4. Is there a name given to this equation?

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    This equation is just a special case of the one you asked about in http://math.stackexchange.com/questions/49292/what-strategy-do-you-use-when-solving-vector-equations-involving-nabla2011-07-07
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    People are making differentiability/integrability assumptions about the functions in the answers. Usually for this sort of question it makes sense to state such assumptions yourself; otherwise the statements may be trivially false -- for instance, $f$ cannot be found if $g$ is nowhere continuous. That $\frac{\partial f}{\partial x} + \frac{\partial f}{\partial y}+\frac{\partial f}{\partial z}$ is the directional derivative along $(1,1,1)$ is only true under certain differentiability assumptions.2011-07-07
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    In fact this PDE can solve generally by using [Method of characteristics](http://en.wikipedia.org/wiki/Method_of_characteristics#Linear_and_quasilinear_cases).2012-09-06

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