# Funciones de forma para elementos viga

## Elementos de dos nodos

\begin{equation*} v = N_{1} v_{1} + N_{2} \theta_{1} + N_{3} v_{2} + N_{4} \theta_{2} = \sum_{i = 0}^{3} \alpha_{i} x^{i} = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} + \alpha_{3} x^{3} \end{equation*}

en forma matricial

\begin{equation*} v = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} + \alpha_{3} x^{3} = \begin{bmatrix} 1 & x & x^{2} & x^{3} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} \end{equation*}

la deformación angular es

\begin{equation*} \theta = \frac{d v}{ d x} = \alpha_{1} + 2 \alpha_{2} x + 3 \alpha_{3} x^{2} \end{equation*}

reemplazando $x_{1}$ y $x_{2}$

\begin{align*} \alpha_{0} + \alpha_{1} x_{1} + \alpha_{2} x_{1}^{2} + \alpha_{3} x_{1}^{3} &= v_{1} \\ \alpha_{1} + 2 \alpha_{2} x_{1} + 3 \alpha_{3} x_{1}^{2} &= \theta_{1} \\ \alpha_{0} + \alpha_{1} x_{2} + \alpha_{2} x_{2}^{2} + \alpha_{3} x_{2}^{3} &= v_{2} \\ \alpha_{1} + 2 \alpha_{2} x_{2} + 3 \alpha_{3} x_{2}^{2} &= \theta_{2} \end{align*}

en forma matricial

\begin{equation*} \begin{bmatrix} 1 & x_{1} & x_{1}^{2} & x_{1}^{3} \\ 0 & 1 & 2 x_{1} & 3 x_{1}^{2} \\ 1 & x_{2} & x_{2}^{2} & x_{2}^{3} \\ 0 & 1 & 2 x_{2} & 3 x_{2}^{2} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} = \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{equation*}

resolviendo el sistema

\begin{equation*} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} = \begin{bmatrix} \frac{x_{2}^{2} (3 x_{1} - x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{x_{2}^{2} x_{1}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & \frac{x_{1}^{2} (-3 x_{2} + x_{1})}{(-x_{2} + x_{1})(x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{x_{1}^{2} x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \\ -\frac{6 x_{1} x_{2}}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{x_{2} (2 x_{1} + x_{2})}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & \frac{6 x_{1} x_{2}}{(-x_{2} + x_{1}) (x_{1}^{2} -2 x_{1} x_{2} + x_{2}^{2})} & \frac{x_{1} (x_{1} + 2 x_{2})}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \\ \frac{3 (x_{1} + x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{x_{1} + 2 x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & -\frac{3 (x_{1} + x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{2 x_{1} + x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \\ -\frac{2}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{1}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & \frac{2}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{1}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{equation*}

reemplazando las incógnitas

\begin{align*} v &= \begin{bmatrix} 1 & x & x^{2} & x^{3} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \end{bmatrix} \\ &= \begin{bmatrix} 1 & x & x^{2} & x^{3} \end{bmatrix} \begin{bmatrix} \frac{x_{2}^{2} (3 x_{1} - x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{x_{2}^{2} x_{1}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & \frac{x_{1}^{2} (-3 x_{2} + x_{1})}{(-x_{2} + x_{1})(x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{x_{1}^{2} x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \\ -\frac{6 x_{1} x_{2}}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{x_{2} (2 x_{1} + x_{2})}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & \frac{6 x_{1} x_{2}}{(-x_{2} + x_{1}) (x_{1}^{2} -2 x_{1} x_{2} + x_{2}^{2})} & \frac{x_{1} (x_{1} + 2 x_{2})}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \\ \frac{3 (x_{1} + x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{x_{1} + 2 x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & -\frac{3 (x_{1} + x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{2 x_{1} + x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \\ -\frac{2}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{1}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} & \frac{2}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{1}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \\ &= \begin{bmatrix} \frac{x_{2}^{2} (3 x_{1} - x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} - \frac{6 x_{1} x_{2}}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x + \frac{3 (x_{1} + x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x^{2} - \frac{2}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} {x}^{3} \\ - \frac{x_{2}^{2} x_{1}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} + \frac{x_{2} (2 x_{1} + x_{2})}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} x - \frac{x_{1} + 2 x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} x^{2} + \frac{1}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} {x}^{3} \\ \frac{x_{1}^{2} (-3 x_{2} + x_{1})}{(-x_{2} + x_{1}) ( x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} + \frac{6 x_{1} x_{2}}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x - \frac {3 (x_{1} + x_{2})}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} {x}^{2} + \frac{2}{(-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x^{3} \\ - \frac{x_{1}^{2} x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} + \frac{x_{1} (x_{1} + 2 x_{2})}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} x - \frac{2 x_{1} + x_{2}}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} x^{2} + \frac{1}{x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}} x^{3} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \\ &= \begin{bmatrix} N_{1} \\ N_{2} \\ N_{3} \\ N_{4} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \end{bmatrix} \end{align*}

## Elemento de tres nodos

\begin{equation*} v = N_{1} v_{1} + N_{2} \theta_{1} + N_{3} v_{2} + N_{4} \theta_{2} + N_{5} v_{3} + N_{6} \theta_{3} = \sum_{i = 0}^{5} \alpha_{i} x^{i} = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} + \alpha_{3} x^{3} + \alpha_{4} x^{4} + \alpha_{5} x^{5} \end{equation*}

en forma matricial

\begin{equation*} v = \alpha_{0} + \alpha_{1} x + \alpha_{2} x^{2} + \alpha_{3} x^{3} + \alpha_{4} x^{4} + \alpha_{5} x^{5} = \begin{bmatrix} 1 & x & x^{2} & x^{3} & x^{4} & x^{5} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \\ \alpha_{4} \\ \alpha_{5} \end{bmatrix} \end{equation*}

la deformación angular es

\begin{equation*} \theta = \frac{d v}{ d x} = \alpha_{1} + 2 \alpha_{2} x + 3 \alpha_{3} x^{2} + 4 \alpha_{4} x^{3} + 5 \alpha_{5} x^{4} \end{equation*}

reemplazando $x_{1}$, $x_{2}$ y $x_{3}$

\begin{align*} \alpha_{0} + \alpha_{1} x_{1} + \alpha_{2} x_{1}^{2} + \alpha_{3} x_{1}^{3} + \alpha_{4} x_{1}^{4} + \alpha_{5} x_{1}^{5} &= v_{1} \\ \alpha_{1} + 2 \alpha_{2} x_{1} + 3 \alpha_{3} x_{1}^{2} + 4 \alpha_{4} x_{1}^{3} + 5 \alpha_{5} x_{1}^{4} &= \theta_{1} \\ \alpha_{0} + \alpha_{1} x_{2} + \alpha_{2} x_{2}^{2} + \alpha_{3} x_{2}^{3} + \alpha_{4} x_{2}^{4} + \alpha_{5} x_{2}^{5} &= v_{2} \\ \alpha_{1} + 2 \alpha_{2} x_{2} + 3 \alpha_{3} x_{2}^{2} + 4 \alpha_{4} x_{2}^{3} + 5 \alpha_{5} x_{2}^{4} &= \theta_{2} \\ \alpha_{0} + \alpha_{1} x_{3} + \alpha_{2} x_{3}^{2} + \alpha_{3} x_{3}^{3} + \alpha_{4} x_{3}^{4} + \alpha_{5} x_{3}^{5} &= v_{3} \\ \alpha_{1} + 2 \alpha_{2} x_{3} + 3 \alpha_{3} x_{3}^{2} + 4 \alpha_{4} x_{3}^{3} + 5 \alpha_{5} x_{3}^{4} &= \theta_{3} \\ \end{align*}

en forma matricial

\begin{equation*} \begin{bmatrix} 1 & x_{1} & x_{1}^{2} & x_{1}^{3} & x_{1}^{4} & x_{1}^{5} \\ 0 & 1 & 2 x_{1} & 3 x_{1}^{2} & 4 x_{1}^{3} & 5 x_{1}^{4} \\ 1 & x_{2} & x_{2}^{2} & x_{2}^{3} & x_{2}^{4} & x_{2}^{5} \\ 0 & 1 & 2 x_{2} & 3 x_{2}^{2} & 4 x_{2}^{3} & 5 x_{2}^{4} \\ 1 & x_{3} & x_{3}^{2} & x_{3}^{3} & x_{3}^{4} & x_{3}^{5} \\ 0 & 1 & 2 x_{3} & 3 x_{3}^{2} & 4 x_{3}^{3} & 5 x_{3}^{4} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \\ \alpha_{4} \\ \alpha_{5} \end{bmatrix} = \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \\ v_{3} \\ \theta_{3} \end{bmatrix} \end{equation*}

resolviendo el sistema

\begin{equation*} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \\ \alpha_{4} \\ \alpha_{5} \end{bmatrix} = \begin{bmatrix} \frac{x_{2}^{2} x_{3}^{2} (5 x_{1}^{2} - 3 x_{1} x_{2} -3 x_{1} x_{3} + x_{2} x{3})}{(x_{1}^{2} -2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} & -\frac{x_{2}^{2} x_{1} x_{3}^{2}}{(x_{1}^{2} - 2 x_{1} x_{3}+ x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{x_{1}^{2} x_{3}^{2} (3 x_{1} x_{2} - x_{1} x_{3} - 5 x_{2}^{2} + 3 x_{2} x_{3})}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} & -\frac{x_{2} x_{1}^{2} x_{3}^{2}}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{x_{2}^{2} x_{1}^{2} (x_{1} x_{2} - 3 x_{1} x_{3} - 3 x_{2} x_{3} + 5 x_{3}^{2})}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} & -\frac{x_{2}^{2} x_{1}^{2} x_{3}}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} \\ -\frac{2 x_{1} x_{2} x_{3} ( 5 x_{1} x_ {2} + 5 x_{1} x_{3} - 3 x_{2}^{2} - 4 x_{2} x_{3} - 3 x_{3}^{2})}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} & \frac{x_{2} x_{3} (2 x_{1} x_{2} + 2 x_{1} x_{3} + x_{2} x_{3})}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{2 x_{1} x_{2} x_{3} (3 x_{1}^{2} - 5 x_{1} x_{2} + 4 x_{1} x_{3} - 5 x_{2} x_{3} + 3 x_{3}^{2})}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} & \frac{x_{1} x_{3} (2 x_{1} x_{2} + x_{1} x_{3} + 2 x_{2} x_{3})}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{2 x_{1} x_{2} x_{3} (3 x_{1}^{2} + 4 x_{1} x_{2} - 5 x_{1} x_{3} + 3 x_{2}^{2} - 5 x_{2} x_{3})}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} & \frac{x_{1} x_{2} (x_{1} x_{2} + 2 x_{1} x_{3} + 2 x_{2} x_{3})}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} \\ \frac{5 x_{2}^{2} x_{1}^{2} + 20 x_{1}^{2} x_{2} x_{3} + 5 x_{1}^{2} x_{3}^{2} - 3 x_{2}^{3} x_{1} - 7 x_{1} x_{2}^{2} x_{3} - 7 x_{1} x_{2} x_{3}^{2} - 3 x_{1} x_{3}^{3} - 3 x_{2}^{3} x_{3} - 4 x_{2}^{2} x_{3}^{2} - 3 x_{2} x_{3}^{3}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} & -\frac{x_{2}^{2} x_{1} + 4 x_{1} x_{2} x_{3} + x_{1} x_{3}^{2} + 2 x_{2}^{2} x_{3} + 2 x_{2} x_{3}^{2}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{3 x_{2} x_{1}^{3} + 3 x_{1}^{3} x_{3} - 5 x_{2}^{2} x_{1}^{2} + 7 x_{1}^{2} x_{ 2} x_{3} + 4 x_{1}^{2} x_{3}^{2} - 20 x_{1} x_{2}^{2} x_{3} + 7 x_{1} x_{2} x_{3}^{2} + 3 x_{1} x_{3}^{3} - 5 x_{2}^{2} x_{3}^{2} + 3 x_{2} x_{3}^{3}}{(x_{2} -x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} & -\frac{x_{2} x_{1}^{2} + 2 x_{1}^{2} x_{3} + 4 x_{1} x_{2} x_{3} + 2 x_{1} x_{3}^{2} + x_{2} x_{3}^{2}}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{3 x_{2} x_{1}^{3} + 3 x_{1}^{3} x_{3} + 4 x_{2}^{2} x_{1}^{2} + 7 x_{1}^{2} x_{2} x_{3} - 5 x_{1} ^{2} x_{3}^{2} + 3 x_{2}^{3} x_{1} + 7 x_{1} x_{2}^{2} x_{3} - 20 x_{1} x_{2} x_{3}^{2} + 3 x_{2}^{3} x_{3} - 5 x_{2}^{2} x_{3}^{2}}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{ 2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} & -\frac{2 x_{2} x_{1}^{2} + x_{1}^{2} x_{3} + 2 x_{2}^{2} x_{1} + 4 x_{1} x_{2} x_{3} + x_{2}^{2} x_{3}}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} \\ -\frac{10 x_{2} x_{1}^{2} + 10 x_{1}^{2} x_{3} - 2 x_{2}^{2} x_{1} + 4 x_{1} x_{2} x_{3} - 2 x_{1} x_{3}^{2} - 2 x_{2}^{3} - 8 x_{2}^{2} x_{3} - 8 x_{2} x_{3}^{2} - 2 x_{3}^{3}} {(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} & \frac{2 x_{1} x_{2} + 2 x_{1} x_{3} + x_{2}^{2} + 4 x_{2} x_{3} + x_{3}^{2}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{2 x_{1}^{3} + 2 x_{2} x_{1}^{2} + 8 x_{1}^{2} x_{3} - 10 x_{2}^{2} x_{1} - 4 x_{1} x_{2} x_{3} + 8 x_{1} x_{3}^{2} - 10 x_{2}^{2} x_{3} + 2 x_{2} x_{3}^{2} + 2 x_{3}^{3}}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} & \frac{x_{1}^{2} + 2 x_{1} x_{2} + 4 x_{1} x_{3} + 2 x_{2} x_{3} + x_{3}^{2}}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{2 x_{1}^{3} + 8 x_{2} x_{1}^{2} +2 x_{1}^{2} x_{3} + 8 x_{2}^{2} x_{1} - 4 x_{1} x_{2} x_{3} - 10 x_{1} x_{3}^{2} + 2 x_{2}^{3} + 2 x_{2}^{2} x_{3} - 10 x_{2} x_{3}^{2}} {(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} & \frac{x_{1}^{2} + 4 x_{1} x_{2} + 2 x_{1} x_{3} + x_{2}^{2} + 2 x_{2} x_{3}}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} \\ \frac{5 x_{1}^{2} + 5 x_{1} x_{2} + 5 x_{1} x_{3} - 4 x_{2}^{2} - 7 x_{2} x_{3} - 4 x_{3}^{2}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} & -\frac{x_{1} + 2 x_{2} + 2 x_{3}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{ 4 x_{1}^{2} - 5 x_{1} x_{2} + 7 x_{1} x_{3} - 5 x_{2}^{2} - 5 x_{2} x_{3} + 4 x_{3}^{2}}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} & -\frac{ 2 x_{1} + x_{2} + 2 x_{3}}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{4 x_{1}^{2} + 7 x_{1} x_{2} - 5 x_{1} x_{3} + 4 x_{2}^{2} - 5 x_{2} x_{3} - 5 x_{3}^{2}}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} & -\frac{2 x_{1} + 2 x_{2} + x_{3}}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} \\ -\frac{ 4 x_{1} - 2 x_{2} - 2 x_{3}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} & \frac{1}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{2 x_{1} - 4 x_{2} + 2 x_{3}} {(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} & \frac{1}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{2 x_{1} + 2 x_{2} - 4 x_{3}}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} & \frac{1}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \\ v_{3} \\ \theta_{3} \end{bmatrix} \end{equation*}

reemplazando las incógnitas

\begin{align*} v &= \begin{bmatrix} 1 & x & x^{2} & x^{3} & x^{4} & x^{5} \end{bmatrix} \begin{bmatrix} \alpha_{0} \\ \alpha_{1} \\ \alpha_{2} \\ \alpha_{3} \\ \alpha_{4} \\ \alpha_{5} \end{bmatrix} \\ &= \begin{bmatrix} 1 & x & x^{2} & x^{3} & x^{4} & x^{5} \end{bmatrix} \begin{bmatrix} \frac{x_{2}^{2} x_{3}^{2} (5 x_{1}^{2} - 3 x_{1} x_{2} -3 x_{1} x_{3} + x_{2} x{3})}{(x_{1}^{2} -2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} & -\frac{x_{2}^{2} x_{1} x_{3}^{2}}{(x_{1}^{2} - 2 x_{1} x_{3}+ x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{x_{1}^{2} x_{3}^{2} (3 x_{1} x_{2} - x_{1} x_{3} - 5 x_{2}^{2} + 3 x_{2} x_{3})}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} & -\frac{x_{2} x_{1}^{2} x_{3}^{2}}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{x_{2}^{2} x_{1}^{2} (x_{1} x_{2} - 3 x_{1} x_{3} - 3 x_{2} x_{3} + 5 x_{3}^{2})}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} & -\frac{x_{2}^{2} x_{1}^{2} x_{3}}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} \\ -\frac{2 x_{1} x_{2} x_{3} ( 5 x_{1} x_ {2} + 5 x_{1} x_{3} - 3 x_{2}^{2} - 4 x_{2} x_{3} - 3 x_{3}^{2})}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} & \frac{x_{2} x_{3} (2 x_{1} x_{2} + 2 x_{1} x_{3} + x_{2} x_{3})}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{2 x_{1} x_{2} x_{3} (3 x_{1}^{2} - 5 x_{1} x_{2} + 4 x_{1} x_{3} - 5 x_{2} x_{3} + 3 x_{3}^{2})}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} & \frac{x_{1} x_{3} (2 x_{1} x_{2} + x_{1} x_{3} + 2 x_{2} x_{3})}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{2 x_{1} x_{2} x_{3} (3 x_{1}^{2} + 4 x_{1} x_{2} - 5 x_{1} x_{3} + 3 x_{2}^{2} - 5 x_{2} x_{3})}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} & \frac{x_{1} x_{2} (x_{1} x_{2} + 2 x_{1} x_{3} + 2 x_{2} x_{3})}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} \\ \frac{5 x_{2}^{2} x_{1}^{2} + 20 x_{1}^{2} x_{2} x_{3} + 5 x_{1}^{2} x_{3}^{2} - 3 x_{2}^{3} x_{1} - 7 x_{1} x_{2}^{2} x_{3} - 7 x_{1} x_{2} x_{3}^{2} - 3 x_{1} x_{3}^{3} - 3 x_{2}^{3} x_{3} - 4 x_{2}^{2} x_{3}^{2} - 3 x_{2} x_{3}^{3}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} & -\frac{x_{2}^{2} x_{1} + 4 x_{1} x_{2} x_{3} + x_{1} x_{3}^{2} + 2 x_{2}^{2} x_{3} + 2 x_{2} x_{3}^{2}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{3 x_{2} x_{1}^{3} + 3 x_{1}^{3} x_{3} - 5 x_{2}^{2} x_{1}^{2} + 7 x_{1}^{2} x_{ 2} x_{3} + 4 x_{1}^{2} x_{3}^{2} - 20 x_{1} x_{2}^{2} x_{3} + 7 x_{1} x_{2} x_{3}^{2} + 3 x_{1} x_{3}^{3} - 5 x_{2}^{2} x_{3}^{2} + 3 x_{2} x_{3}^{3}}{(x_{2} -x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} & -\frac{x_{2} x_{1}^{2} + 2 x_{1}^{2} x_{3} + 4 x_{1} x_{2} x_{3} + 2 x_{1} x_{3}^{2} + x_{2} x_{3}^{2}}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{3 x_{2} x_{1}^{3} + 3 x_{1}^{3} x_{3} + 4 x_{2}^{2} x_{1}^{2} + 7 x_{1}^{2} x_{2} x_{3} - 5 x_{1} ^{2} x_{3}^{2} + 3 x_{2}^{3} x_{1} + 7 x_{1} x_{2}^{2} x_{3} - 20 x_{1} x_{2} x_{3}^{2} + 3 x_{2}^{3} x_{3} - 5 x_{2}^{2} x_{3}^{2}}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{ 2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} & -\frac{2 x_{2} x_{1}^{2} + x_{1}^{2} x_{3} + 2 x_{2}^{2} x_{1} + 4 x_{1} x_{2} x_{3} + x_{2}^{2} x_{3}}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} \\ -\frac{10 x_{2} x_{1}^{2} + 10 x_{1}^{2} x_{3} - 2 x_{2}^{2} x_{1} + 4 x_{1} x_{2} x_{3} - 2 x_{1} x_{3}^{2} - 2 x_{2}^{3} - 8 x_{2}^{2} x_{3} - 8 x_{2} x_{3}^{2} - 2 x_{3}^{3}} {(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} & \frac{2 x_{1} x_{2} + 2 x_{1} x_{3} + x_{2}^{2} + 4 x_{2} x_{3} + x_{3}^{2}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{2 x_{1}^{3} + 2 x_{2} x_{1}^{2} + 8 x_{1}^{2} x_{3} - 10 x_{2}^{2} x_{1} - 4 x_{1} x_{2} x_{3} + 8 x_{1} x_{3}^{2} - 10 x_{2}^{2} x_{3} + 2 x_{2} x_{3}^{2} + 2 x_{3}^{3}}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} & \frac{x_{1}^{2} + 2 x_{1} x_{2} + 4 x_{1} x_{3} + 2 x_{2} x_{3} + x_{3}^{2}}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{2 x_{1}^{3} + 8 x_{2} x_{1}^{2} +2 x_{1}^{2} x_{3} + 8 x_{2}^{2} x_{1} - 4 x_{1} x_{2} x_{3} - 10 x_{1} x_{3}^{2} + 2 x_{2}^{3} + 2 x_{2}^{2} x_{3} - 10 x_{2} x_{3}^{2}} {(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} & \frac{x_{1}^{2} + 4 x_{1} x_{2} + 2 x_{1} x_{3} + x_{2}^{2} + 2 x_{2} x_{3}}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} \\ \frac{5 x_{1}^{2} + 5 x_{1} x_{2} + 5 x_{1} x_{3} - 4 x_{2}^{2} - 7 x_{2} x_{3} - 4 x_{3}^{2}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} & -\frac{x_{1} + 2 x_{2} + 2 x_{3}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{ 4 x_{1}^{2} - 5 x_{1} x_{2} + 7 x_{1} x_{3} - 5 x_{2}^{2} - 5 x_{2} x_{3} + 4 x_{3}^{2}}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} & -\frac{ 2 x_{1} + x_{2} + 2 x_{3}}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{4 x_{1}^{2} + 7 x_{1} x_{2} - 5 x_{1} x_{3} + 4 x_{2}^{2} - 5 x_{2} x_{3} - 5 x_{3}^{2}}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} & -\frac{2 x_{1} + 2 x_{2} + x_{3}}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} \\ -\frac{ 4 x_{1} - 2 x_{2} - 2 x_{3}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} & \frac{1}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & -\frac{2 x_{1} - 4 x_{2} + 2 x_{3}} {(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} & \frac{1}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} & \frac{2 x_{1} + 2 x_{2} - 4 x_{3}}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} & \frac{1}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} \end{bmatrix} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \\ v_{3} \\ \theta_{3} \end{bmatrix} \\ & = \begin{bmatrix} \frac{x_{2}^{2} x_{3}^{2} (5 x_{1}^{2} - 3 x_{1} x_{2} - 3 x_{1} x_{3} + x_{2} x_{3})}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} -\frac{2 x_{1} x_{2} x_{3} (5 x_{1} x_{2} + 5 x_{1} x_{3} - 3 x_{2}^{2} - 4 x_{2} x_{3} - 3 x_{3}^{2})}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} x +\frac{5 x_{2}^{2} x_{1}^{2} + 20 x_{1}^{2} x_{2} x_{3} + 5 x_{1}^{2} x_{3}^{2} - 3 x_{2}^{3} x_{1} - 7 x_{1} x_{2}^{2} x_{3} - 7 x_{1} x_{2} x_{3}^{2} - 3 x_{1} x_{3}^{3} - 3 x_{2}^{3} x_{3} - 4 x_{2}^{2} x_{3}^{2} - 3 x_{2} x_{3}^{3}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} x^{2} -\frac{2(5 x_{2} x_{1}^{2} + 5 x_{1}^{2} x_{3} - x_{2}^{2} x_{1} + 2 x_{1} x_{2} x_{3} - x_{1} x_{3}^{2} - x_{2}^{3} - 4 x_{2}^{2} x_{3} - 4 x_{2} x_{3}^{2} - x_{3}^{3})}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} {x}^{3} +\frac{5 x_{1}^{2} + 5 x_{1} x_{2} + 5 x_{1} x_{3} - 4 x_{2}^{2} - 7 x_{2} x_{3} - 4 x_{3}^{2}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} {x}^{4} -\frac{2(2 x_{1} - x_{2} - x_{3})}{( x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (-x_{3} + x_{1})} {x}^{5} \\ -\frac{x_{1} x_{2}^{2} x_{3}^{2}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} +\frac{x_{2} x_{3} (2 x_{1} x_{2} + 2 x_{1} x_{3} + x_{2} x_{3})}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x -\frac{x_{2}^{2} x_{1} + 4 x_{1} x_{2} x_{3} + x_{1} x_{3}^{2} + 2 x_{2}^{2} x_{3} + 2 x_{2} x_{3}^{2}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x^{2} +\frac{2 x_{1} x_{2} + 2 x_{1} x_{3} + x_{2}^{2} + 4 x_{2} x_{3} + x_{3}^{2}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x^{3} -\frac{x_{1} + 2 x_{2} + 2 x_{3}}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x^{4} +\frac{1}{(x_{1}^{2} - 2 x_{1} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x^{5} \\ \frac{x_{1}^{2} x_{3}^{2} (3 x_{1} x_{2} - x_{1} x_{3} - 5 x_{2}^{2} + 3 x_{2} x_{3})}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} -\frac{2 x_{1} x_{2} x_{3} (3 x_{1}^{2} - 5 x_{1} x_{2} + 4 x_{1} x_{3} - 5 x_{2} x_{3} + 3 x_{3}^{2})}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} x +\frac{3 x_{2} x_{1}^{3} + 3 x_{1}^{3} x_{3} - 5 x_{2}^{2} x_{1}^{2} + 7 x_{1}^{2} x_{2} x_{3} + 4 x_{1}^{2} x_{3}^{2} - 20 x_{1} x_{2}^{2} x_{3} + 7 x_{1} x_{2} x_{3}^{2} + 3 x_{1} x_{3}^{3} - 5 x_{2}^{2} x_{3}^{2} + 3 x_{2} x_{3}^{3}}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} x^{2} -\frac{2 (x_{1}^{3} + x_{2} x_{1}^{2} + 4 x_{1}^{2} x_{3} - 5 x_{2}^{2} x_{1} - 2 x_{1} x_{2} x_{3} + 4 x_{1} x_{3}^{2} - 5 x_{2}^{2} x_{3} + x_{2} x_{3}^{2} + x_{3}^{3})}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} x^{3} +\frac{4 x_{1}^{2} - 5 x_{1} x_{2} + 7 x_{1} x_{3} - 5 x_{2}^{2} - 5 x_{2} x_{3} + 4 x_{3}^{2}}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} x^{4} -\frac{2 (x_{1} - 2 x_{2} + x_{3})}{(x_{2} - x_{3}) (-x_{2} + x_{1}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2}) (x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2})} x^{5} \\ -\frac{x_{1}^{2} x_{2} x_{3}^{2}}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} +\frac{x_{1} x_{3} (2 x_{1} x_{2} + x_{1} x_{3} + 2 x_{2} x_{3})}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x -\frac{x_{2} x_{1}^{2} + 2 x_{1}^{2} x_{3} + 4 x_{1} x_{2} x_{3} + 2 x_{1} x_{3}^{2} + x_{2} x_{3}^{2}}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x^{2} +\frac{x_{1}^{2} + 2 x_{1} x_{2} + 4 x_{1} x_{3} + 2 x_{2} x_{3} + x_{3}^{2}}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x^{3} -\frac{2 x_{1} + x_{2} + 2 x_{3}}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x^{4} +\frac{1}{(x_{2}^{2} - 2 x_{2} x_{3} + x_{3}^{2}) (x_{1}^{2} - 2 x_{1} x_{2} + x_{2}^{2})} x^{5} \\ \frac{x_{1}^{2} x_{2}^{2} (x_{1} x_{2} - 3 x_{1} x_{3} - 3 x_{2} x_{3} + 5 x_{3}^{2})}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} +\frac{2 x_{1} x_{2} x_{3} (3 x_{1}^{2} + 4 x_{1} x_{2} - 5 x_{1} x_{3} + 3 x_{2}^{2} - 5 x_{2} x_{3})}{(x_{1} x_{2} - x_{1} x_{3} - x_{ 2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} x -\frac{3 x_{2} x_{1}^{3} + 3 x_{1}^{3} x_{3} + 4 x_{2}^{2} x_{1}^{2} + 7 x_{1}^{2} x_{2} x_{3} - 5 x_{1}^{2} x_{3}^{2} + 3 x_{2}^{3} x_{1} + 7 x_{1} x_{2}^{2} x_{3} - 20 x_{1} x_{2} x_{3}^{2} + 3 x_{2}^{3} x_{3} - 5 x_{2}^{2} x_{3}^{2}}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} x^{2} +\frac{2 (x_{1}^{3} + 4 x_{2} x_{1}^{2} + x_{1}^{2} x_{3} + 4 x_{2}^{2} x_{1} - 2 x_{1} x_{2} x_{3} - 5 x_{1} x_{3}^{2} + x_{2}^{3} + x_{2}^{2} x_{3} - 5 x_{2} x_{3}^{2})}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} x^{3} -\frac{4 x_{1}^{2} + 7 x_{1} x_{2} - 5 x_{1} x_{3} + 4 x_{2}^{2} - 5 x_{2} x_{3} - 5 x_{3}^{2}}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} x^{4} +\frac{2 (x_{1} + x_{2} - 2 x_{3})}{(x_{1} x_{2} - x_{1} x_{3} - x_{2} x_{3} + x_{3}^{2}) (x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4})} x^{5} \\ -\frac{x_{1}^{2} x_{2}^{2} x_{3}}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} +\frac{x_{1} x_{2} (x_{1} x_{2} + 2 x_{1} x_{3} + 2 x_{2} x_{3})}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} x -\frac {2 x_{2} x_{1}^{2} + x_{1}^{2} x_{3} + 2 x_{2}^{2} x_{1} + 4 x_{1} x_{2} x_{3} + x_{2}^{2} x_{3}}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} x^{2} +\frac{x_{1}^{2} + 4 x_{1} x_{2} + 2 x_{1} x_{3} + x_{2}^{2} + 2 x_{2} x_{3}}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} x^{3} -\frac{2 x_{1} + 2 x_{2} + x_{3}}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} x^{4} +\frac{1}{x_{2}^{2} x_{1}^{2} - 2 x_{1}^{2} x_{2} x_{3} + x_{1}^{2} x_{3}^{2} - 2 x_{1} x_{2}^{2} x_{3} + 4 x_{1} x_{2} x_{3}^{2} - 2 x_{1} x_{3}^{3} + x_{2}^{2} x_{3}^{2} - 2 x_{2} x_{3}^{3} + x_{3}^{4}} x^{5} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \\ v_{3} \\ \theta_{3} \end{bmatrix} \\ &= \begin{bmatrix} N_{1} \\ N_{2} \\ N_{3} \\ N_{4} \\ N_{5} \\ N_{6} \end{bmatrix}^{\mathrm{T}} \begin{bmatrix} v_{1} \\ \theta_{1} \\ v_{2} \\ \theta_{2} \\ v_{3} \\ \theta_{3} \end{bmatrix} \\ \end{align*}