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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 $\vec{a}$.

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    @Qiaochu: You have raised two strong points: 1) The divergence de$f$ined by a limiting process in which a region shrinks down to a $f$ixed point requires the additional condition that $\mathbf{$F$}$ is *continuosly differentiable*, if we want the divergence theorem be applied. Without using this theorem I do not see how we could give a coordinate-free definition. $2$) Concerning the directional derivatives you are quite right. 3) So, the definition I took from a Calculus book has all the limitations you pointed out.2010-11-29

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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.

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    Small typo in my last comment: for $F$ as above, there are $m$ components, not $n$.2010-11-29