| \begin{eqnarray} \frac{dV}{dr}=\frac{1}{L}\left[\begin{matrix}-1&1\\\end{matrix}\right]\left\{\begin{matrix}V_1\\V_2\\\end{matrix}\right\}=\left[B\right]\left\{V\right\} \end{eqnarray} | \begin{eqnarray} \frac{d\delta V}{dr}=\frac{1}{L}\left[\begin{matrix}-1&1\\\end{matrix}\right]\left\{\begin{matrix}{\delta V}_1\\{\delta V}_2\\\end{matrix}\right\}=\left[B\right]\left\{\delta V\right\} \end{eqnarray} |
| \begin{eqnarray} \int_{r_1}^{r_2}\left(r\frac{dV}{dr}\delta V\right)dr=\left\{\delta V\right\}^T\int_{r_1}^{r_2}{r\left[N\right]^T}\left[B\right]dr\left\{V\right\} \end{eqnarray} |
| \begin{eqnarray} \int_{r_1}^{r_2}\left(r\frac{dV}{dr}\delta V\right)dr=\left\{\delta V\right\}^T\frac{1}{L}\int_{r_1}^{r_2}\left[\begin{matrix}-rN_1\\-rN_2\\\end{matrix}\right]\left[\begin{matrix}-1&1\\\end{matrix}\right]dr\left\{V\right\}=\left\{\delta V\right\}^T\frac{1}{L}\int_{r_1}^{r_2}\left[\begin{matrix}-rN_1&rN_1\\-rN_2&rN_2\\\end{matrix}\right]dr\left\{V\right\} \end{eqnarray} |
| \begin{eqnarray} \int_{r_1}^{r_2}\left(r\frac{dV}{dr}\delta V\right)dr=\left\{\delta V\right\}^T\frac{1}{L^2}\left[\begin{matrix}\left(\frac{R^3}{3}-\frac{r_2R^2}{2}\right)&\left(\frac{r_2R^2}{2}-\frac{R^3}{3}\right)\\\left(\frac{r_1R^2}{2}-\frac{R^3}{3}\right)&\left(\frac{R^3}{3}-\frac{r_1R^2}{2}\right)\\\end{matrix}\right]\left\{V\right\}=\left\{\delta V\right\}^T\left[C\right]\left\{V\right\} \end{eqnarray} |
| \begin{eqnarray} R^2=r_2^2-r_1^2 \end{eqnarray} | \begin{eqnarray} R^3=r_2^3-r_1^3 \end{eqnarray} |
| \begin{eqnarray} \int_{r_1}^{r_2}\left(r^2\frac{dV}{dr}\frac{d\delta V}{dr}\right)dr=\left\{\delta V\right\}^T\frac{R^3}{3L^2}\left[\begin{matrix}\ \ \ 1&-1\\-1&\ \ \ 1\\\end{matrix}\right]\left\{V\right\} \end{eqnarray} |