Distribution functions: differentials in the numerator or denominator One paper I'm looking at says,


  $n(M, z) \, dM \, dz$, the number of sources with mass $M$ at a
  redshift $z$, in the mass interval $dM$ occurring in the redshift
  interval $dz$.


While another says that,


  $\frac{d^2n}{dM dz}$, the number of sources with mass and redshift in the range $[M, M+dM]$ and $[z, z+dz]$ respectively.


What is the difference (conceptually and technically, please) between these formalisms -- and the (different?) meaning(s) of the distribution $n$?

Perhaps unrelated: sometimes when integrating numerically (e.g. to find the total number of sources over all parameters) we have to include the differential itself, i.e. multiply by $\sum n_i \Delta z_i = n_i (z_{i+1} - z_i)$, whereas other times we just sum the distribution $\sum n_i$ --- is this because the latter version implicitly includes the differential-elements within it?
Both representations are of the same thing, depending upon the choice and preference of the author.

These two similar representation, account to the fact that in the function, $n(M, z)$ is may well be possible, that, $M := M(z)$, that is, it could be in itself, a function of $z$, please note that this fact is implicit and may not be always be true. This is the supposed technicality behind these representations, otherwise these two are the different visuals pointing to the same thing.

Let's consider an example which illustrates this point:
consider the integral, $$\int_{\pi/4}^{\pi/2}(1 + e^{tan(\theta)}) sec^{2}\theta d\theta.$$

Please note that this integral can also be written as $$\int_{\pi/4}^{\pi/2}(1 + e^{tan(\theta)}) d({tan(\theta)}),$$
since $$\frac{d(tan\theta)}{d\theta} = sec^{2}\theta$$
so, $$d(tan\theta) = sec^{2}\theta d\theta.$$

So, whether you say $d(tan\theta)$ or $sec^{2}\theta d\theta$, it represents the same thing.
Both representations are of the same thing, depending upon the choice and preference of the author.

These two similar representation, account to the fact that in the function, $n(M, z)$ is may well be possible, that, $M := M(z)$, that is, it could be in itself, a function of $z$, please note that this fact is implicit and may not be always be true. This is the supposed technicality behind these representations, otherwise these two are the different visuals pointing to the same thing.

Let's consider an example which illustrates this point:
consider the integral, $$\int_{\pi/4}^{\pi/2}(1 + e^{tan(\theta)}) sec^{2}\theta d\theta.$$

Please note that this integral can also be written as $$\int_{\pi/4}^{\pi/2}(1 + e^{tan(\theta)}) d({tan(\theta)}),$$
since $$\frac{d(tan\theta)}{d\theta} = sec^{2}\theta$$
so, $$d(tan\theta) = sec^{2}\theta d\theta.$$

So, whether you say $d(tan\theta)$ or $sec^{2}\theta d\theta$, it represents the same thing.