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I very often see the symbol "$\partial$" is used to define the boundary of a body in three-dimensional space. For example, if the body is called $\Omega$ the boundaries is labeled as $\partial\Omega$.

  • Is it any reason using this symbol?
  • What can one call it orally?
  • Is it appropriate to call the boundaries of $\partial\Omega$ (for example the path surrounding the surface $\partial\Omega$) $\partial^2\Omega$?

Thanks for sharing your thoughts.

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    $\begingroup$ One of the responses here: mathoverflow.net/questions/46252/… claims that $\partial$ comes from German {\it Rand}. Something like $\partial \Omega$ is usually just read out as "d-omega." $\endgroup$ – anomaly Dec 15 '17 at 1:02
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Consider the ball $B_r$ of radius $r$. This is a manifold with boundary whose boundary is the sphere of radius $r$. Now consider specifically the difference between this ball and a slightly larger ball $B_{r + dr}$. The difference is a thin spherical shell which is $dr$ thick, and as $dr \to 0$ it approaches, in some sense, the boundary sphere of $B_r$.

So the boundary is something like a "derivative." This is made somewhat more precise by Stokes' theorem, and is also related to ideas like cobordism. A simple example of the way in which the boundary behaves like a derivative is that if $M, N$ are manifolds with boundary, then loosely speaking we have the "product rule"

$$\partial(M \times N) = \left( \partial(M) \times N \right) \cup \left( M \times \partial(N) \right).$$

There's some subtlety to making this precise because $M \times N$ is generally a manifold with corners (consider for example $M = N = I$, which is perhaps the easiest case to visualize).

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  • $\begingroup$ Have you happened to see this symbol for showing the boundary of a surface? which is, in fact, the boundary of the boundary of a body. In my problem (in applied mechanics), I have three kinds of domains to integrate over. One is the volume of a 3D body which I called $\Gamma$, one is a subset of its boundary which I called $\partial\Gamma_t$ but I do not know how to call the boundary of $\partial \Gamma_t$. $\endgroup$ – Reza Dec 15 '17 at 2:07
  • $\begingroup$ I think $\partial^2$ is fine but it would be worth explaining explicitly. I have not seen it but it makes sense. $\endgroup$ – Qiaochu Yuan Dec 15 '17 at 4:13

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