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In Potter's Thermodynamics: Demystified (page 68), the author wrote:

Since only two independent variables are necessary to establish the state of a simple system, the specific internal energy can be expressed as a function of temperature and specific volume; that is, $$u=u(T,v)$$ $$du = \frac{\partial u}{\partial T}dT+\frac{\partial u}{\partial v}dv \tag{1}$$ Joule’s classical experiment showed that u = u(T) for an ideal gas, so that $\frac{\partial u}{\partial v}=0$. We define the constant-volume specific heat $C_v$ as, $$C_v =: \frac{\partial u}{\partial T}$$ so that $(1)$ becomes, $$du=C_{v}dT \tag{2}$$ This can be integrated to give,

$$u_2-u_1 = \int_{T_1}^{T_2} C_{v}dT \tag{3} $$

So, I know that the process of differentiation and integration is used as a "tool", and it's quite simple to convince yourself (with basic integral calculus) that we can integrate $(2)$ or even talk about differentials in $(1)$. But with this theorem here (Stokes theorem):

$$\int_{\partial \Omega} \omega = \int_{\Omega} \mathrm{d}\omega \tag{4}$$

the whole concept of differentials changes because the $\mathrm{d}\omega$ isn't a differential but rather another object called a differential form; for a 1-form version, we have

$$ \mathrm{d}\omega =: \langle \nabla \omega, \cdot \rangle \tag{5}$$

Now, I know that $(3)$ isn't the Stokes theorem, and even in the most advanced books the concept of $(1)$ is take for granted as simply the basic notion of differentials from calculus. But, again, with $(5)$ we know that the concept of differentials is extended and different from calculus, and with $(4)$ integration and differential forms are quite related (maybe we can't talk about integration without dealing with differential forms in a general concept).

So, how can I formalize the concepts of differentials, inexact differentials and integration, used in thermodynamics, in terms of differential forms theory?

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    $\begingroup$ see BERNARD F. SCHUTZ: "Geometrical methods of mathematical physics" pp 163-167 $\endgroup$ – hyportnex Dec 11 '18 at 16:41

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