# What is meant when we say that a differential takes on a certain value?

In all cases in which I met with the expression $\mathrm{d}A=0$ it meant that the map associated to it was identically null on its codomain. Then it's only an "abuse of notation", an abbreviation to mean that the differential maps all the points in which is evaluated to the null element of the codomain.

$\mathrm{d}A=0 \equiv$ *$\mathrm{d}A(\vec{v})=0$ $\forall$ $\vec{v} \in D(\mathrm{d}A)$*

where $D(\mathrm{d}A)$ is the domain of $\mathrm{d}A$ (a set in which its application is *well defined*).

[EDIT] - Answer to the second part of the question

With the differential, saying that it's positive means that $\mathrm{d}A$ is such that the direction of growth of the function that has $\mathrm{d}A$ as differential is coherent whit the direction of growth of the coordinates. For example for $f: \mathbb{R} \to \mathbb{R}$ if $\mathrm{d}f$ is said positive then the value of $f$ increases while we increase the value of the input variable. In particular the case $\mathbb{R} \to \mathbb{R}$ is the one in which I met with the expression "*the differential is positive*" more often, because there is only one variable. Trying to extend the expression in the case of multivariable functions, $f: \mathbb{R^n} \to \mathbb{R}$, the *total* differential takes in input a vector and gives in output a real number, and if it's positive in that point means that the function grows up in the direction coherent whit the one of the vector. So the direction intended should to be specified to be more precise, otherwise I agree it could sound so ambiguous, saying just that the total differential is positive.

An expression like $$ dA = 0 $$ means that $dA$ is the zero map, i.e. the map that returns $0$ for all inputs. It is not hard to see that this is also the zero vector of the space of linear maps to which $dA$ belongs when treated as a vector space in its own right.