La transformación de un punto $(x_1,x_2,x_3)$ en el espacio del sistema cartesiano hacia el sistema de coordenadas generalizadas $(x'_1,x'_2,x'_3)$, se define por medio de:\begin{align*}x_1&=x_1(x'_1,x'_2,x'_3)\\x_2&=x_2(x'_1,x'_2,x'_3)\\x_3&=x_3(x'_1,x'_2,x'_3)\end{align*}La transformación inverza entre estas coordenadas es dado por:\begin{align*}x'_1&=x'_1(x_1,x_2,x_3)\\x'_2&=x'_2(x_1,x_2,x_3)\\x'_3&=x'_3(x_1,x_2,x_3)\end{align*}
Tensor Métrico\begin{align}ds^2=dx_1^2+dx_2^2+dx_3^2\end{align} teniendo en cuenta la notación de índices repetidos\begin{align}
dx_1&=\frac{\partial x_1}{\partial x'_1}dx'_1+\frac{\partial x_1}{\partial x'_2}dx'_2+\frac{\partial x_1}{\partial x'_3}dx'_3=\frac{\partial x_1}{\partial x'_i}dx'_i\\
dx_2&=\frac{\partial x_2}{\partial x'_1}dx'_1+\frac{\partial x_2}{\partial x'_2}dx'_2+\frac{\partial x_2}{\partial x'_3}dx'_3=\frac{\partial x_2}{\partial x'_i}dx'_i\\
dx_3&=\frac{\partial x_3}{\partial x'_1}dx'_1+\frac{\partial x_3}{\partial x'_2}dx'_2+\frac{\partial x_3}{\partial x'_3}dx'_3=\frac{\partial x_3}{\partial x'_i}dx'_i
\end{align}$\Longrightarrow$\begin{align}
ds^2&=\left(\frac{\partial x_1}{\partial x'_i}dx'_i\right)\left(\frac{\partial x_1}{\partial x'_j}dx'_j\right)+\left(\frac{\partial x_2}{\partial x'_i}dx'_i\right)\left(\frac{\partial x_2}{\partial x'_j}dx'_j\right)+\left(\frac{\partial x_3}{\partial x'_i}dx'_i\right)\left(\frac{\partial x_3}{\partial x'_j}dx'_j\right)\\
&=\left(\frac{\partial x_1}{\partial x'_i}\frac{\partial x_1}{\partial x'_j}+\frac{\partial x_2}{\partial x'_i}\frac{\partial x_2}{\partial x'_j}+\frac{\partial x_3}{\partial x'_i}\frac{\partial x_3}{\partial x'_j}\right)dx'_idx'_j=\sum_{ij}g_{ij}dx'_idx'_j\end{align}Teniendo en cuenta que el vector posición es\begin{align}
\vec{r}=(x_1(x'_1,x'_2,x'_3),x_2(x'_1,x'_2,x'_3),x_3(x'_1,x'_2,x'_3))\end{align}$\Longrightarrow$\begin{align}g_{ij}&=\frac{\partial x_1}{\partial x'_i}\frac{\partial x_1}{\partial x'_j}+\frac{\partial x_2}{\partial x'_i}\frac{\partial x_2}{\partial x'_j}+\frac{\partial x_3}{\partial x'_i}\frac{\partial x_3}{\partial x'_j}\\&=\left(\frac{\partial x_1}{\partial x'_i},\frac{\partial x_2}{\partial x'_i},\frac{\partial x_3}{\partial x'_i}\right)\cdot\left(\frac{\partial x_1}{\partial x'_j},\frac{\partial x_2}{\partial x'_j},\frac{\partial x_3}{\partial x'_j}\right)\\
&=\left(\frac{\partial\vec{r}}{\partial x'_i}\right)\cdot\left(\frac{\partial\vec{r}}{\partial x'_j}\right)=\left(h_i\hat{e}_i\right)\cdot\left(h_j\hat{e}_j\right)=h_ih_j\delta_{ij}
\end{align}$\Longrightarrow$
\begin{align}
g_{ij}=\left(\begin{array}{ccc}
g_{11} & g_{12} & g_{13} \\
g_{21} & g_{22} & g_{23} \\
g_{31} & g_{32} & g_{33} \\
\end{array}
\right)
=
\left(\begin{array}{ccc}
h^2_1 & 0 & 0 \\
0 & h^2_2 & 0 \\
0 & 0 & h^2_3 \\
\end{array}
\right)
\end{align}\begin{align}ds^2=\left(h_1dx'_1\right)^2+\left(h_2dx'_2\right)^2+\left(h_3dx'_3\right)^2\hspace{0.3cm}\text{donde}\hspace{0.3cm}h_i=\left\|\frac{\partial \vec{r}}{\partial x'_i}\right\|\end{align}
Matriz de transformación de vectores unitarios. Teniendo en cuenta el vector posición en coordenadas generalizadas, podemos transformar de $e_i$ hacia $e'_i$ de la siguiente forma:\begin{align}
&\vec{r}=x_1\hat{e}_1+x_2\hat{e}_2+x_3\hat{e}_3\hspace{0.5cm}\text{donde}\hspace{0.5cm} x_i=x_i(x'_1,x'_2,x'_3)\\
&e'_i=\frac{1}{h_i}\frac{\partial\vec{r}}{\partial x'_i}
\end{align}\begin{align}
\left(\begin{array}{c}e'_1 \\
\\e'_2 \\
\\e'_3 \\
\end{array}
\right)&=\left(\begin{array}{ccc}
\frac{1}{h_1}\frac{\partial x_1}{\partial x'_1} & \frac{1}{h_1}\frac{\partial x_2}{\partial x'_1} & \frac{1}{h_1}\frac{\partial x_3}{\partial x'_1} \\
\\
\frac{1}{h_2}\frac{\partial x_1}{\partial x'_2} & \frac{1}{h_2}\frac{\partial x_2}{\partial x'_2} & \frac{1}{h_2}\frac{\partial x_3}{\partial x'_2} \\
\\
\frac{1}{h_3}\frac{\partial x_1}{\partial x'_3} & \frac{1}{h_3}\frac{\partial x_2}{\partial x'_3} & \frac{1}{h_3}\frac{\partial x_3}{\partial x'_3} \\
\end{array}
\right)
\left(\begin{array}{c}e_1 \\
\\e_2 \\
\\e_3 \\
\end{array}
\right)\end{align}
dx_1&=\frac{\partial x_1}{\partial x'_1}dx'_1+\frac{\partial x_1}{\partial x'_2}dx'_2+\frac{\partial x_1}{\partial x'_3}dx'_3=\frac{\partial x_1}{\partial x'_i}dx'_i\\
dx_2&=\frac{\partial x_2}{\partial x'_1}dx'_1+\frac{\partial x_2}{\partial x'_2}dx'_2+\frac{\partial x_2}{\partial x'_3}dx'_3=\frac{\partial x_2}{\partial x'_i}dx'_i\\
dx_3&=\frac{\partial x_3}{\partial x'_1}dx'_1+\frac{\partial x_3}{\partial x'_2}dx'_2+\frac{\partial x_3}{\partial x'_3}dx'_3=\frac{\partial x_3}{\partial x'_i}dx'_i
\end{align}$\Longrightarrow$\begin{align}
ds^2&=\left(\frac{\partial x_1}{\partial x'_i}dx'_i\right)\left(\frac{\partial x_1}{\partial x'_j}dx'_j\right)+\left(\frac{\partial x_2}{\partial x'_i}dx'_i\right)\left(\frac{\partial x_2}{\partial x'_j}dx'_j\right)+\left(\frac{\partial x_3}{\partial x'_i}dx'_i\right)\left(\frac{\partial x_3}{\partial x'_j}dx'_j\right)\\
&=\left(\frac{\partial x_1}{\partial x'_i}\frac{\partial x_1}{\partial x'_j}+\frac{\partial x_2}{\partial x'_i}\frac{\partial x_2}{\partial x'_j}+\frac{\partial x_3}{\partial x'_i}\frac{\partial x_3}{\partial x'_j}\right)dx'_idx'_j=\sum_{ij}g_{ij}dx'_idx'_j\end{align}Teniendo en cuenta que el vector posición es\begin{align}
\vec{r}=(x_1(x'_1,x'_2,x'_3),x_2(x'_1,x'_2,x'_3),x_3(x'_1,x'_2,x'_3))\end{align}$\Longrightarrow$\begin{align}g_{ij}&=\frac{\partial x_1}{\partial x'_i}\frac{\partial x_1}{\partial x'_j}+\frac{\partial x_2}{\partial x'_i}\frac{\partial x_2}{\partial x'_j}+\frac{\partial x_3}{\partial x'_i}\frac{\partial x_3}{\partial x'_j}\\&=\left(\frac{\partial x_1}{\partial x'_i},\frac{\partial x_2}{\partial x'_i},\frac{\partial x_3}{\partial x'_i}\right)\cdot\left(\frac{\partial x_1}{\partial x'_j},\frac{\partial x_2}{\partial x'_j},\frac{\partial x_3}{\partial x'_j}\right)\\
&=\left(\frac{\partial\vec{r}}{\partial x'_i}\right)\cdot\left(\frac{\partial\vec{r}}{\partial x'_j}\right)=\left(h_i\hat{e}_i\right)\cdot\left(h_j\hat{e}_j\right)=h_ih_j\delta_{ij}
\end{align}$\Longrightarrow$
\begin{align}
g_{ij}=\left(\begin{array}{ccc}
g_{11} & g_{12} & g_{13} \\
g_{21} & g_{22} & g_{23} \\
g_{31} & g_{32} & g_{33} \\
\end{array}
\right)
=
\left(\begin{array}{ccc}
h^2_1 & 0 & 0 \\
0 & h^2_2 & 0 \\
0 & 0 & h^2_3 \\
\end{array}
\right)
\end{align}\begin{align}ds^2=\left(h_1dx'_1\right)^2+\left(h_2dx'_2\right)^2+\left(h_3dx'_3\right)^2\hspace{0.3cm}\text{donde}\hspace{0.3cm}h_i=\left\|\frac{\partial \vec{r}}{\partial x'_i}\right\|\end{align}
Matriz de transformación de vectores unitarios. Teniendo en cuenta el vector posición en coordenadas generalizadas, podemos transformar de $e_i$ hacia $e'_i$ de la siguiente forma:\begin{align}
&\vec{r}=x_1\hat{e}_1+x_2\hat{e}_2+x_3\hat{e}_3\hspace{0.5cm}\text{donde}\hspace{0.5cm} x_i=x_i(x'_1,x'_2,x'_3)\\
&e'_i=\frac{1}{h_i}\frac{\partial\vec{r}}{\partial x'_i}
\end{align}\begin{align}
\left(\begin{array}{c}e'_1 \\
\\e'_2 \\
\\e'_3 \\
\end{array}
\right)&=\left(\begin{array}{ccc}
\frac{1}{h_1}\frac{\partial x_1}{\partial x'_1} & \frac{1}{h_1}\frac{\partial x_2}{\partial x'_1} & \frac{1}{h_1}\frac{\partial x_3}{\partial x'_1} \\
\\
\frac{1}{h_2}\frac{\partial x_1}{\partial x'_2} & \frac{1}{h_2}\frac{\partial x_2}{\partial x'_2} & \frac{1}{h_2}\frac{\partial x_3}{\partial x'_2} \\
\\
\frac{1}{h_3}\frac{\partial x_1}{\partial x'_3} & \frac{1}{h_3}\frac{\partial x_2}{\partial x'_3} & \frac{1}{h_3}\frac{\partial x_3}{\partial x'_3} \\
\end{array}
\right)
\left(\begin{array}{c}e_1 \\
\\e_2 \\
\\e_3 \\
\end{array}
\right)\end{align}
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