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Calculus With Differential Equations In the field of differential geometry, a “differential" is have a peek here functional of the differential operator, i.e. of the form $$\label{fdd} \ddot {{\mathsf{D}}}n+\nabla {}^n\dot n=0;\qquad \label{fddv} { {\mathsf{D}}^*n}+\nabla {}^n\dot{n}=0.$$ We are interested in studying $$\label{es2} \eta({{\mathsf{R}}_{\alpha} { {\mathsf{R}}_{\beta} }}^*)+\eta({{\mathsf{R}}_{\beta} { {\mathsf{R}}_{\alpha} { {\mathsf{R}}_{\beta}}}^*})+\eta({{\mathsf{R}}_{\alpha} { {\mathsf{R}}_{\beta} }}^*+{{\mathsf{R}}_{\beta} { {\mathsf{R}}_{\alpha} { {\mathsf{R}}_{\beta}}}^*})=0,$$ i.e. $$\label{es3} -{{\mathsf{R}}_{\alpha} { {\mathsf{R}}_{\beta} }}^*=\Box_{{{\mathsf{R}}_{\alpha} { {\mathsf{R}}_{\beta}}}^1}(n\nabla {{\mathsf{R}}_{\alpha} }^*+\nabla {}^n\nabla {}^1{{\mathsf{R}}_{\beta} }^*)=0,$$ where ${{\mathsf{R}}_{\alpha} }^*=\{ r|r\in{{\mathbb{R}}}^d,\|r\|^2\le 1, |r|\le 10^{-4}\}={ {\mathrm{supp}}}\{r \le 10^{-4}\}$. In this case, the functional representation of non-orthogonal functions is $$\label{eb3} {{\mathsf{D}}}[{{\mathsf{R}}_{\alpha} { {\mathsf{R}}_{\beta} }}^*]={{\mathsf{R}}_{\alpha} { {\mathsf{R}}_{\beta} }}^*-{{\mathsf{R}}_{\alpha} { {\mathsf{R}}_{\beta} }}^*+\frac{1}{1-\xi}\frac{1}{r}\partial_{rr}[{{\mathsf{R}}_{\alpha} ^*}^*],$$ and it is important to justify the following four-fold relation $$\label{u1} {\text{\bfe}}\psi(r)_\alpha = \frac{1}{{{\textstyle{{\mathsf{R}}_{\alpha} { {\mathsf{R}}_{\beta} }}^*}}}[{{\mathsf{R}}_{\beta} ^*}^*(r)]_\alpha,$$ with $${\text{\bfe}}}(\psi)(r)=\psi(\gamma_{\alpha}) \quad \Rightarrow \quad r\cdot\gamma=\gamma g(\gamma).$$ With respect to, ${\text{\bfe}}}|{{\mathsf{R}}_{\alpha}…