When, usually in text of physics or concerning thermodynamical aspects of chemistry, I find notations such as$$\mathrm{d}f=g\,\mathrm{d}t$$ I always interpretate it as $\frac{\mathrm{d}f(t)}{\mathrm{d}t}=g(t)$ in the sense that the derivative of $f$ at $t$ is $f'(t)=g(t)$ and always try to keep that in mind when I see steps of calculations involving such differentials because of my love for the spirit of mathematical proofs.
I have been told by friend of mine, who is a professor of mathematics, that, at a more or less elementary didactic level, the notations such as $\mathrm{d}A$ and $\delta A$ found in thermodynamics, are really intended to be "small quantities" with no pretence of mathematical rigour. Parenthetically, I notice that these differentials, as they are usually manipulated in elementary physics textbooks, cannot be the differential forms defined in differential geometry because the differential forms of differential geometry do not constitute an algebraic field, while these differentials usually are multiplied and divided by each other as if they were real numbers.
Nevertheless, I notice that integration is usually operated on such differentials and my love for the spirit of mathematical proof leads me to try to find a rigourous way to read such forms.
Can we then read a form such as, for example, $\delta A$, which is of the kind of forms that books integrate getting $\int \delta A=\int A_1\mathrm{d}x_1+\ldots+ A_n\mathrm{d}x_n$ , as $$A_1(x_1,\ldots,x_n)x_1'(s)\mathrm{d}s+A_2(x_1,\ldots,x_n)x_2'(s)\mathrm{d}s+\ldots +A_n(x_1,\ldots,x_n)x_n'(s)\mathrm{d}s$$where $s$ is the variable of a parametrisation of a path we are going to integrate on?
To give a practical example, since $p$, $V$ and $T$ are variables often used in thermodynamics, I would tend to read $\delta Q$ as $$Q_1(p,V,T)p'(s)\mathrm{d}s+Q_2(p,V,T)V'(s)\mathrm{d}s+Q_3(p,V,T)T'(s)\mathrm{d}s$$where the $Q_i$ are opportune functions of $(p,V,T)$ . Therefore, if $(p,V,T)=:\mathbf{r}:[a,b]\to\mathbb{R}^3$ parametrises the path of the transformation and we define $\mathbf{Q}:=(Q_1,Q_2,Q_3)$, I would read $\int\delta Q$ just as $\int_{\gamma}\mathbf{Q}\cdot\mathrm{d}\mathbf{r}$, i.e. as $$\int_a^bQ_1(p,V,T)p'(s)+Q_2(p,V,T)V'(s)+Q_3(p,V,T)T'(s)\,\mathrm{d}s$$where it is worth of note that one of the $Q_i$ usually is constantly $0$ because of the interdependence of the variables and where simplifications such as $\int_a^bQ_1(p,V,T)p'(s)\mathrm{d}s=\int_{p(a)}^{p(b)}Q_1(p,V,T)\mathrm{d}p$ are usually introduced when calculating .
I $\infty$-ly thank any answerer
