One says that balance of energy holds if, for every nice open set $\mathcal{U} \subset \mathcal{B}$,
$$ \frac{d}{\mathrm{~d} t} \int_{\varphi_t(\mathcal{U})} \rho\left(e+\frac{1}{2}\langle\langle\mathbf{v}, \mathbf{v}\rangle\rangle\right) \mathrm{d} v=\int_{\varphi_t(\mathcal{U})} \rho(\langle\langle\mathbf{b}, \mathbf{v}\rangle\rangle+r) \mathrm{d} v+\int_{\partial \varphi_l(\mathcal{U})}(\langle\langle\mathbf{t}, \mathbf{v}\rangle\rangle+h) \mathrm{d} a, $$where $e=e(\mathbf{x}, t), r=r(\mathbf{x}, t)$ and $h=h(\mathbf{x}, \hat{\mathbf{n}}, t)$ are internal energy per unit mass, heat supply per unit mass and heat flux, respectively.
Balance of energy. For an elastic material one assumes that there exists a strain energy function $e$ per unit mass whose change represents the change in the internal energy due to mechanical deformations. Balance of energy may be written as
$$ \int_V \rho\langle\mathbf{v}, \mathbf{b}\rangle \mu+\int_S\langle\mathbf{v}, \boldsymbol{\mathcal { T }}\rangle=\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{1}{2} \int_V \rho\langle\langle\mathbf{v}, \mathbf{v}\rangle\rangle \mu+\int_V e \rho \mu\right), $$or, equivalently, on $\mathcal{R}_0$ as
$$ \int_{V_0} \rho_0\langle\mathbf{V}, \mathbf{B}\rangle \mu_0+\int_{S_0}\langle\mathbf{V}, \mathcal{P}\rangle=\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{1}{2} \int_{V_0} \rho_0\langle\langle\mathbf{V}, \mathbf{V}\rangle\rangle \mu_0+\int_{V_0} e \rho_0 \mu_0\right) . $$Here, the $\langle\langle\rangle$,$\rangle denotes the inner products both on T \mathcal{R}$ and $T \mathcal{R}_0$, respectively. The volume integrals are taken over an arbitrary subset $V \subseteq \mathcal{R}$ while $V_0=\varphi^{-1}(V) \subseteq$ $\mathcal{R}_0$ and the area integrals are taken over the bounding surfaces $S=\partial V$ and $S_0=\partial V_0$.