Without prethought, I mentioned in class once that the reason the symbol $\partial$
is used to represent the boundary operator in topology is
that its behavior is akin to a derivative.
But after reflection and some research,
I find little support for my unpremeditated claim.
Just sticking to the topological boundary (as opposed to the boundary
of a manifold or of a simplicial chain),
$\partial^3 S = \partial^2 S$
for any set $S$.
So there seems to be no possible analogy to
Taylor series. Nor can I see an analogy with the
fundamental theorem of calculus.
The only tenuous sense in which I can see the boundary
as a derivative is that $\partial S$ is a transition between $S$
and the "background" complement $\overline{S}$.


I've looked for the origin of the use of the symbol $\partial$ in topology without luck. 
I have only found [references](http://en.wikipedia.org/wiki/%E2%88%82)
for its use in calculus.
I've searched through *History of Topology* (Ioan Mackenzie James)
online without success (but this may be my poor searching).
Just visually scanning the 1935 *Topologie* von Alexandroff und Hopf, I do not
see $\partial$ employed.


I have two questions:



> 
> **Q1**. Is there a sense in which the boundary operator $\partial$ is
>  analogous to a derivative?
> 
> 
> **Q2**. What is the historical origin for the use of the symbol
>  $\partial$ in topology?
> 
> 
> 


Thanks!


**Addendum** (*2010*). Although **Q2** has not been addressed
[was subsequently addressed by @FrancoisZiegler], it seems appropriate to accept one
among the wealth of insightful responses to **Q1**. Thanks to all!



Is the boundary $\partial S$ analogous to a derivative?
The surface area $|\partial S|$ of a (bounded, smooth) body $S$ is the derivative of the volume $|S\_r|$ of the $r$-neighbourhoods $S\_r$ of $S$ at $r=0$:


$$ |\partial S| = \frac{d}{dr} |S\_r| |\_{r=0}.$$


Thus, for instance, the boundary $\partial D\_r$ of the disk $D\_r$ of radius $r$ has circumference $\frac{d}{dr} (\pi r^2) = 2\pi r$.


More generally, one intuitively has the Newton quotient-like formula


$$ \partial S = \lim\_{h \to 0^+} \frac{S\_h \backslash S}{h};$$


the right-hand side does not really make formal sense, but certainly one can view $S\_h \backslash S$ as a $[0,h]$-bundle over $\partial S$ for $h$ sufficiently small (in particular, smaller than the radius of curvature of $S$).


In a similar spirit, one informally has the "chain rule"


$$ {\mathcal L}\_X S "=" (X \cdot n) \partial S $$


for the "Lie derivative" of $S$ along a vector field $X$, where $n$ is the outward normal. (There may also be a divergence term, depending on whether one is viewing $S$ as a set, a measure, or a volume form.) Again, this does not really make formal sense, although Stokes' theorem already captures most of the above intuition rigorously (and, as noted in the comments, Stokes' theorem is probably the clearest way to link boundaries and derivatives together).


EDIT: A more rigorous way to link boundaries with derivatives proceeds via the theory of distributions. The weak derivative $\nabla 1\_S$ of the indicator function of a smooth body $S$ is equal to $-n d\sigma$, where $n$ is the outward normal and $d\sigma$ is the surface measure on $\partial S$. (This is really just a fancy way of restating Stokes' theorem, after one unpacks all the definitions.) This can be used, for instance, to link the Sobolev inequality with the isoperimetric inequality.


In a similar spirit, $\frac{1\_{S\_h} - 1\_S}{h}$ converges in the sense of distributions as $h \to 0$ to surface measure $d\sigma$ on $\partial S$, thus providing a rigorous version of the intuitive difference formula given previously.


