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.

  • 1
    $\begingroup$ 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

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).

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.