We all know from school that density is defined as mass over volume $$\rho = \frac{m}{V}$$
I'm wondering what the mathematically correct definition of density is. I'm considering two options.
OPTION 1. Is density defined by the means of multiple integral? $$m = \iiint \rho \, \mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$ This way density is $\rho = \cfrac{\partial^3 m}{\partial x \partial y \partial z}$.
OPTION 2. Is density defined by the means of volume integral? $$m = \int \rho \, \mathrm{d}V$$ This way density is $\rho = \cfrac{dm}{dV}$.
If Option 2 is correct, then what is volume integral? In my engineering calculus course, I learnt about multiple and curvilinear integrals. Curvilinear integrals include line integrals (of two types) and surface integrals (of two types). I assume that volume integral is a curvilinear integral, but I'm not sure.
If volume integral is a curviulinear integral, then what is the way of calculating it in Decart (Cartesian) coordinates? I mean that there are formulas to calculate line integrals in Cartesian coordinates (one formula for each type of the line integral) and there are formulas to calculate surface integrals in Cartesian coordinates (again, each type of the surface integral has a formula to calculate it in Cartesian coordinates); is there an analogous formula for the volume integral?
If both options 1 and 2 are not correct, then what is the mathematically rigorous way to define density?
P.S. I did read this post. But I do still have my question unanswered. Is density a mass derivative over volume or a third mass derivative over three Cartesian coordinates? In other words, I'm trying to figure out is there such a thing as a coordinate along a volume (I know about coordinate along a line - hence, line integral; I know about coordinate along a surface - hence, surface integral; but is there volume integral in the same sense ...)
ANSWERS
I found an answer to a half of my question here. I.e., the answer to the post I've cited basically validates that there's such a thing as volume integral and we can map it to Cartesian coordinates using Jacobian (the same way as we do for surface integrals).
There's a very interesting answer by @CyclotomicField. See two CyclotomicField's comments bellow.
The answer by @Othing calrifies it. Basically, the definition of density is $\rho = m / V$. Then I need to pick a MEASURE I want. The notion of measure is explained in the wikipedia article cited by @CyclotomicField.
The comment by herb steinberg is very interesting as well.
@Peek-a-boo comment is very good! Definitely, check the link provided in the comment.
Thank you very much for all your inputs!
The definition of (mass) density is precisely the mass per unit volume. The mathematical formulation of this is the very first formula you wrote: $\rho=m/V$. If you take the mass of any object and divide by its volume, you end up with the density. The formula $$ m=\int_{V}\rho(\mathbf{r}) d^3x $$ is actually used to measure $\rho$, not to define it, so this is actually more of a physical discussion. This last formula implies that the density is a function of your coordinates, and its value at a point $\mathbf{r}$ is given by $\rho=dm/dV$, where $dV$ is the volume element at $\mathbf{r}$. In an experiment you usually can't measure $\rho$ directly. But you can measure mass and volume pretty accurately. So it is possible to measure the mass of a substance/object over a region with a very small (and known) volume. In the limit of an infinitesimal element of volume you would find the function $\rho(x,y,z)$. You'll never reach that limit of course, but you can make progressively better measurements.
Edit: To be clear the integral formula above is just a consequence of the definition of density, and it holds in an euclidean space. I wrote it that way because it is used for measurements, which are usually made in that space. If you change your volume form, the formula changes and there might be extreme cases where you can't define such a formula.