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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