# Is it possible to label particles of a continuum body?

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)?

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

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