In basic continuum mechanics (e.g. fluid dynamics), we label particles of the continuum, i.e., each particle can be identified by a label, e.g., $p$. Then other quantities are defined accordingly, e.g., velocity: $V=\frac{\partial r(p,t)}{\partial t}|_p$ where $r$ is the location vector of $p$ and $t$ is time.

Now, I was wondering what kind of number $p$ is? It cannot be a natural number e.g. $1,2,3,...$, because the set of particles of continuum is uncountably infinite, while the set of natural numbers is countably infinite, so there cannot be a one-to-one correspondence. So, is it true to say $p\in\mathbb{R}$ (real numbers)?

  • 1
    $\begingroup$ can you give a link for the formula? AFAIK velocity of the fluid is defined for a small volume dV, not for a particle . $\endgroup$
    – anna v
    Commented Jan 29 at 10:10
  • $\begingroup$ @annav, Sure, see here, at the end of p.6. $\endgroup$
    – Naghi
    Commented Jan 29 at 12:05

1 Answer 1


Is it true to say $p \in R$ (real numbers) ?

Yes, or more generally $p \in R^n$ i.e. $p$ is a tuple of real numbers. For example, one way to label infinitesimal fluid parcels in a continuum model is to label them by their location at some fixed time $t_0$, so that

$p = (x,y,z) \iff \vec r(p, t_0) = (x,y,z)$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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