### 3. Thermodynamic equilibrium, reversible and irreversible processes

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### 2. The first law of thermodynamics (the law of conservation of energy) and the equation of heat influx

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### 1. The living forces theorem and the work of internal surface forces

One of the most important general consequences of the dynamic equations of motion of a continuous medium is the living forces theorem.

Let $V$ – be an arbitrary finite volume moving together with particles of the material medium, $\Sigma$ – the surface bounding it. Suppose that inside the volume $V$ the components of the stress tensor $P = p^{ij}\vec{э}_{i}\vec{э}_{j}$ and components the speed vector $\vec{v} = v^i\vec{э}_i = v_i\vec{э}^i$ – are continuous differentiable functions of spatial coordinates and time.

Take the vector $d\vec{r} = \vec{v}dt$ – the displacement vector of an infinitely small volume of a continuous medium $d\tau$ during the time $dt$; scalarly multiply the equation of impulses by $d\vec{r}$ and integrate by volume $V$. We get

 $\int\limits_V \rho\vec{a} \cdot \vec{v} dt d\tau = \int\limits_V \rho\vec{F}\cdot d\vec{r}d\tau + \int\limits_V\left( \nabla_i p^{ij} \right) v_i dt d\tau$ (1.1)

Lets transform the integrals involved in this relation.

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