Question about derivation of Differential Vector Operators in Orthogonal Curvilinear Coordinates on Griffiths' Introduction to Electrodynamics I'm working through Introduction to Electrodynamics by Griffiths, and there's a derivation of gradient, divergence, and curl in (orthogonal) curvilinear coordinates (Appendix A) which I don't entirely follow.
The derivation for divergence starts as follows. Consider a vector function $\mathbf{A}=A_u\mathbf{\hat{u}}+A_v\mathbf{\hat{v}}+A_w\mathbf{\hat{w}}$ and the solid found by starting at $(u,v,w)$ and changing each of the coordinates by $\text{d}u$, $\text{d}v,$ and $\text{d}w$ respectively.
The volume of this solid is $\text{d}V = fgh \ \text{d}u \ \text{d}v \ \text{d}w$ where $f$, $g$, and $h$ are the functions that determine the infinitesimal displacement vector, i.e., $\text{d}\mathbf{l}=f\mathbf{\hat{u}} + g \mathbf{\hat{v}} + h \mathbf{\hat{w}}$. The flux out of the front and back surfaces respectively are $-A_u g  h|_{u} \text{d}v\text{d}w$ and $-(A_ugh)|_{u+\text{d}u}\text{d}v\text{d}w$ so the total flux contributed by these two surfaces is $\frac{\partial A_ugh}{\partial u}\text{d}u\text{d}v\text{d}w$
There's a couple of questions I have with this part of the derivation.

*

*Firstly, how is it known that the normal vector from the back surface will be $\mathbf{\hat{u}}$ if the solid can 'curve' in an arbitrary manner? I understand that the unit vectors may be functions of position, but I don't see how that can ensure that the faces will always be orthogonal to $\mathbf{\hat{u}}$ (it would be true if all surfaces were planes, but there's no reason for that to be true in general coordinates).

*Furthermore, how can $A_u$ be considered constant on each of the surfaces, this seems to imply it can be considered constant over distances $\text{d}v$ and $\text{d}w$ (along the face) but can vary significantly over the distance $\text{d}u$ (between the faces).

*Also, how can the difference in length for each of the sides be neglected when calculating the volume, but matter when finding the flux out of the surfaces? These two calculations seem inconsistent with each other. In general, is there a criterion for how accurate these approximations have to be when deriving formulas like this?

Any insight would be appreciated.
 A: Griffiths' calculations are a formal manipulation that can be done more rigorously through other approaches.
For the first issue, he seems to be defining the solid in this way. The vectors are orthogonal at every point, but, as you mentioned, they changed from point to point. The surface he has constructed has sides of constant values of $u$, $v$, and $w$, and as a consequence one of the unit vectors is always constant to each surface. Think in spherical coordinates, for an example: the vector $\mathbf{\hat{r}}$ is always orthogonal to a sphere.
For the second issue, he is considering the difference because in here the difference is the leading order term. In other words, if he did not consider it, the expression would vanish. For an example, suppose you are considering the expression $f(x + \epsilon) + f(x)$ for $\epsilon$ quite small. Taylor expanding, we have
\begin{align}
f(x + \epsilon) + f(x) &= f(x) + f(x) + \epsilon f'(x) + \mathcal{O}(\epsilon^2), \\
\approx 2f(x),
\end{align}
for the corrections $\epsilon f'(x) + \mathcal{O}(\epsilon^2)$ are being assumed to be much smaller than $f(x)$. However, consider now $f(x + \epsilon) - f(x)$. This time, we have
\begin{align}
f(x + \epsilon) - f(x) &= f(x) - f(x) + \epsilon f'(x) + \mathcal{O}(\epsilon^2), \\
\approx 0 + \epsilon f'(x),
\end{align}
and we keep the $\epsilon$ term because it is now the leading term. In both computations we are keeping the most important term, while dropping along the way those terms which are just small corrections (rigorously, they would vanish when taking the appropriate limits).
The third issue is, deep down, just the same as the second one, but written in a slightly different manner.
In all cases, the weirdness in these computations comes from the formal manipulation. See also this answer I've written a while ago to a similar issue in a similar topic. To obtain the expressions with less mathematical awkwardness, one could use the differential geometric expressions for the differential operators and work their way around the exterior derivatives, Christoffel symbols, and so on. This is actually a cool exercise in Differential Geometry, but it does require a bunch of more complicated Math. Griffiths' approach is a bit rough and depends more on the awkward formal manipulations, but gets to the same conclusions through an easier route.
