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Planar Space

Let $(\xi_1,\xi_2)$ be a locally Euclidean coordinate system. Then

\begin{displaymath}
ds^2={d\xi_1}^2+{d\xi_2}^2.
\end{displaymath} (1)

Now plug in
\begin{displaymath}
d\xi_1={\partial\xi_1\over\partial x_1} dx_1+{\partial\xi_1\over\partial x_2}dx_2
\end{displaymath} (2)


\begin{displaymath}
d\xi_2={\partial\xi_2\over\partial x_1} dx_1+{\partial\xi_2\over\partial x_2}dx_2
\end{displaymath} (3)

to obtain


$\displaystyle ds^2$ $\textstyle =$ $\displaystyle \left[{\left({\partial\xi_1\over \partial x_1}\right)^2+\left({\p...
...ial\xi_2\over\partial x_1} {\partial\xi_2\over\partial x_2}}\right]\,dx_1\,dx_2$  
  $\textstyle \phantom{=}$ $\displaystyle + \left[{\left({\partial\xi_1\over \partial x_2}\right)^2+\left({\partial\xi_2\over\partial x_2}\right)^2}\right]{dx_2}^2.$ (4)

Reading off the Coefficients from
\begin{displaymath}
ds^2=g_{11}\,{dx_1}^2+2g_{12}\,dx_1\,dx_2+g_{22}\,(dx_2)^2
\end{displaymath} (5)

gives
$\displaystyle g_{11}$ $\textstyle =$ $\displaystyle \left({\partial\xi_1\over \partial x_1}\right)^2+\left({\partial\xi_2\over \partial x_1}\right)^2$ (6)
$\displaystyle g_{12}$ $\textstyle =$ $\displaystyle {\partial\xi_1\over\partial x_1}{\partial\xi_1\over\partial x_2}+{\partial\xi_2\over\partial x_1}
{\partial\xi_2\over\partial x_2}$ (7)
$\displaystyle g_{22}$ $\textstyle =$ $\displaystyle \left({\partial\xi_1\over \partial x_2}\right)^2+\left({\partial\xi_2\over \partial x_2}\right)^2.$ (8)

Making a change of coordinates $(x_1,x_2)\to(x_1',x_2')$ gives


$\displaystyle g_{11}'$ $\textstyle =$ $\displaystyle \left({\partial\xi_1\over\partial x_1'}\right)^2+\left({\partial\xi_2\over\partial x_1'}\right)^2$  
  $\textstyle =$ $\displaystyle \left({{\partial\xi_1\over\partial x_1}{\partial x_1\over\partial...
...1'}+{\partial\xi_2\over\partial x_2} {\partial x_2\over\partial x_1'}}\right)^2$  
  $\textstyle =$ $\displaystyle g_{11}\left({\partial x_1\over\partial x_1'}\right)^2+2g_{12}{\pa...
...al x_2\over\partial x_1'}+g_{22}\left({\partial x_2\over\partial x_1'}\right)^2$ (9)
$\displaystyle g_{12}'$ $\textstyle =$ $\displaystyle {\partial\xi_1\over\partial x_1}{\partial x_1\over\partial x_1'}
...
...artial x_1'}{\partial \xi_2\over \partial x_2}
{\partial x_2\over\partial x_2'}$  
  $\textstyle =$ $\displaystyle g_{12}{\partial x_1\over\partial x_1'}{\partial x_2\over\partial x_2'}$ (10)
$\displaystyle g_{22}'$ $\textstyle =$ $\displaystyle g_{11}\left({\partial x_1\over\partial x_1'}\right)^2+2g_{12}{\pa...
...l x_2\over\partial x_2'}+g_{22}\left({\partial x_2\over\partial x_2'}\right)^2.$ (11)




© 1996-9 Eric W. Weisstein
1999-05-25