Derivation of the identification relations

To derive (1.3), we let a positive number r, a positive integer <tex2html_comment_mark> m, and cube $\mathcal{C}\in \mathbf{C}_m$ be given, and we use the relation $h_m\,(\,bdy(\mathcal{C}_r(x)\cap \mathcal{C})\,)=\,\,bdy(h_m(% \mathcal{C}_r(x)\cap \mathcal{C})\,)$, the Divergence Theorem, and the Change of Variables Formula for volume integrals to write

$$\begin{eqnarray*}\int\limits_{h_m\,(\,bdy(\mathcal{C}_r(x)\cap \mathcal{C})\,)}\... ...\cap \mathcal{C}\,}\,(div\,w)(h_m(z))\,\det \nabla h_m(z)\,dV_z. \end{eqnarray*}$$

Summing over all cubes $\mathcal{C}\in \mathbf{C}_m$, dividing by r³, and letting $m\rightarrow \infty$ yields

$$\begin{eqnarray*}&&\lim\limits_{m\rightarrow \infty }r^{-3}\sum\limits_{\mathcal... ...z)\,\det K(z)\,dV_z \\ &\longrightarrow &(div\,w)(x)\,\det K(x) \end{eqnarray*}$$

as $r\rightarrow 0$, and this establishes the identification relation (1.3) for
$(div\,w)(x)\,\det K(x)$ as the volume density of the total flux of w.

To derive the second identification relation (1.4), we first use the Change of Variables Formula for surface integrals to write

$$\begin{eqnarray*}\int\limits_{h_m\,(\,(bdy\mathcal{C}_r(x))\,\cap \,\mathcal{C}%... ...mathcal{C}\,}\,\nabla h_m(z)^{*T}\,w(h_m(z))\cdot \nu (z)\,dA_z. \end{eqnarray*}$$

Summing over $\mathcal{C}\in \mathbf{C}_m$, letting $m\rightarrow \infty ,$and applying the Divergence Theorem, we conclude that

$$\begin{eqnarray*}&&\lim\limits_{m\rightarrow \infty }\sum\limits_{\mathcal{C}\in... ...\ &=&\int\limits_{\,\mathcal{C}_r(x)}\,div(K^{*T}\,w)(z)\,dV_z. \end{eqnarray*}$$

The identification relation (1.4) for the volume density of the flux of w without disarrangements follows upon division of the last relation by r³ and letting $r\rightarrow 0$.

The last identification relation (1.5) follows immediately from the first two identification relations, the decomposition of fluxes (1.2 ), and the relation:

$$\begin{eqnarray*}\int\limits_{h_m\,(\,\,bdy(\mathcal{C}_r(x)\cap \mathcal{C})\,\... ...al{C}_r(x))\,\cap \,\mathcal{C\,)}\,)}\,w(y)\cdot N(y)\,dA_y &=& \end{eqnarray*}$$

$$\int\limits_{h_m\,(\,\,bdy(\mathcal{C}_r(x)\cap \mathcal{C}% ... ...l{C}% _r(x))\,\cap \,\mathcal{C\,}\,)}\,w(y)\cdot N(y)\,dA_y.$$