Conditions for existence of divergence of a vector field What are the necessary and sufficient conditions on a vector field $F$ for the divergence $\nabla\cdot F$ to exist at a given point.
EDIT
In Divergence in the second line under the heading "Application in Cartesian coordinates", why is it assumed that $\vec{F}$ to be a continuously differentiable vector field ?
EDIT 2
Ideally one would expect each component of $F$ to be differentiable at a given point $\vec{a}$ no matter through which continuous contour you traverse the point $\vec{a}$. 
EDIT 3
Or is it that each component of $F$ to be differentiable and the derivative being continuous at a given point $\vec{a}$ no matter through which continuous contour you traverse the point $\vec{a}$. 
 A: Perhaps the point is that existence of partial derivatives is not enough for a function of several variables to be differentiable in the multivariable sense, or even necessarily continuous: see this MO question for a standard example of a function of two variables all of whose partial (and even all directional) derivatives exist but is not even continuous at the origin.  
[Edit: the following paragraph was added after reading Qiaochu's comment.]
It is also possible for all the partial derivatives in a given coordinate system to exist -- e.g. $\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}$ -- but for directional derivatives in other directions not to exist.  (For instance, let $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ be the function which is $0$ at $(x,y,z)$ if at least two of the variables are zero or if all of $x$,$y$,$z$ have rational coordinates and which is $1$ at all other points.)  Here the divergence will be defined with respect to the standard coordinate system but not with respect to a different coordinate system.
Anyway, in general a function which has a continuous derivative (or partial derivatives) is much better behaved than a merely differentiable function.  Inserting the hypothesis that a function is $C^1$ -- unless you really need to be considering weaker hypotheses than that for a specific application -- is generally a prudent practice.
