Higher orders of divergence and curl In the standard definition of div and curl, a limit is taken.
If one instead expands the integral out into a series(Taylor?) of the vol/area then there are higher order terms that vanish. Do these higher order terms have any usefulness or commonly defined in mathematics or physics?
For 2D, we can define the div and curl as
$$div(F) = \lim_{C\rightarrow 0}\frac{1}{A}\int_C F \cdot \hat{r} dC$$
$$curl_z(F) = \lim_{C\rightarrow 0}\frac{1}{A}\int_C F \cdot \hat{v} dC$$
Where $C$ is a circle around the point under consideration, A its area, r is a normal vector to the circle, and v is a tangent vector.
But what if we look at simply the integral
$$\int_C F \cdot \hat{r} dC$$
and expand it out in a Taylor series about C? The the first non-constant term is the divergence of F, the term after that is what? Similarly we could do the same for the curl above.
 A: This is an interesting question.  I can't tell you if any of this has any well-known use in mathematics or physics, but I can provide an answer.  I'm going to focus on the divergence, though I think similar remarks will apply to the curl.
To begin, let's define the divergence not as the limit that you wrote about but rather as the differential operator $div(P,Q) = P_X + Q_y$.  Applying the divergence theorem to the circle $C_R$ of radius $R$ centered at the origin, we have:
$$\int_{C_R} F \cdot \hat{r}\, dC = \iint_{D_R} div(F)\, dA$$
Here $D_R$ is the disk of radius $R$.  Your limit follows by arguing that for every $\epsilon > 0$ there exists $R$ such that $div(F)(p)$ is within $\epsilon$ of $div(F)(0)$ for $p$ in $D_R$.
To get the higher order terms, we should look at the Taylor expansion of $div(F)(p)$ near $0$.  Up to the quadratic term, we get:
$$div(F)(p) \sim (P_x(0) + Q_y(0)) + (P_{xx}(0) + Q_{xy}(0))x + (P_{xy}(0) + Q_{yy}(0))y +\\
\frac{1}{2}(P_{xxx}(0) + Q_{xxy}(0))x^2 + (P_{xxy}(0) + Q_{xyy}(0))xy + (P_{xyy}(0) + Q_{yyy}(0))y^2$$
Now let's integrate this over $D_R$.  


*

*The integral of the constant term leaves $div(F)(0) \cdot Area(D_R)$ as before.

*The integral of the linear term vanishes by symmetry (the integral of $x$ or $y$ over any disk centered at the origin is $0$).

*The integral of the quadratic cross term $xy$ over the disk is again $0$ by symmetry, so we are left with the second order term:
$$\frac{1}{2}(P_{xxx}(0) + Q_{xxy}(0)) \iint_{D_R} x^2\, dA + \frac{1}{2}(P_{xyy}(0) + Q_{yyy}(0)) \iint_{D_R} y^2\, dA$$


You can certainly calculate these integrals explicitly, but you should recognize them as the moments of inertia of the disk times its area.
Here's how you can express the general answer.  Form the vector fields
$$F_{\alpha_1, \alpha_2} = \left(\frac{\partial^{\alpha_1}}{\partial x^{\alpha_1}}\frac{\partial^{\alpha_2}}{\partial y^{\alpha_2}}P,\frac{\partial^{\alpha_1}}{\partial x^{\alpha_1}}\frac{\partial^{\alpha_2}}{\partial y^{\alpha_2}}Q\right)$$
and let $I_{\alpha_1,\alpha_2}$ denote the higher moments of $D_R$:
$$I_{\alpha_1,\alpha_2} = \frac{1}{Area(D_R)}\iint_{D_R} x^{\alpha_1}y^{\alpha_2}\, dA$$
(By the symmetry argument above, these are $0$ when either $\alpha_1$ or $\alpha_2$ is odd.)  The $n$th order part of $\frac{1}{Area(D_R)}\int_{C_R} F \cdot \hat{r}\, dC$ is given by:
$$\sum_{\alpha_1 + \alpha_2 = n} B_{\alpha_1, \alpha_2} \cdot div(F_{\alpha_1,\alpha_2})(0) \cdot I_{\alpha_1,\alpha_2}$$
where $B_{\alpha_1, \alpha_2}$ is the appropriate factorial term.  The higher moments are certainly standard mathematical objects, but I don't really know what to make of the vector field $F_{\alpha_1,\alpha_2}$.
A: The integral $\int_C F\cdot \hat rd
C$ is the flux of the vector field $F$ across the surface $C$. We take the limit to eliminate higher terms because we are interested in an instantaneous behavior of a vector field at a point in space. It is analogous to a directional derivative of a function. In this case, instead of asking how a vector field changes as we move in a certain direction we ask how the field changes if we move away from the point radially.
Without the limit, we are getting an average change in the vector field in the surface surrounding a point. It is like getting the secant line instead of the tangent line for a point on a differentiable curve.
