Planar Space

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

$$ds^2={d\xi_1}^2+{d\xi_2}^2.$$ (1)

Now plug in

$$d\xi_1={\partial\xi_1\over\partial x_1} dx_1+{\partial\xi_1\over\partial x_2}dx_2$$ (2)

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

to obtain

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

$$\phantom{=} + \left[{\left({\partial\xi_1\over \partial x_2}\right)^2+\left({\partial\xi_2\over\partial x_2}\right)^2}\right]{dx_2}^2.$$ (4)

Reading off the Coefficients from

$$ds^2=g_{11}\,{dx_1}^2+2g_{12}\,dx_1\,dx_2+g_{22}\,(dx_2)^2$$ (5)

gives

$$g_{11} = \left({\partial\xi_1\over \partial x_1}\right)^2+\left({\partial\xi_2\over \partial x_1}\right)^2$$ (6)

$$g_{12} = {\partial\xi_1\over\partial x_1}{\partial\xi_1\over\partial x_2}+{\partial\xi_2\over\partial x_1} {\partial\xi_2\over\partial x_2}$$ (7)

$$g_{22} = \left({\partial\xi_1\over \partial x_2}\right)^2+\left({\partial\xi_2\over \partial x_2}\right)^2.$$ (8)

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

$$g_{11}' = \left({\partial\xi_1\over\partial x_1'}\right)^2+\left({\partial\xi_2\over\partial x_1'}\right)^2$$

$$= \left({{\partial\xi_1\over\partial x_1}{\partial x_1\over\partial... ...1'}+{\partial\xi_2\over\partial x_2} {\partial x_2\over\partial x_1'}}\right)^2$$

$$= g_{11}\left({\partial x_1\over\partial x_1'}\right)^2+2g_{12}{\pa... ...al x_2\over\partial x_1'}+g_{22}\left({\partial x_2\over\partial x_1'}\right)^2$$ (9)

$$g_{12}' = {\partial\xi_1\over\partial x_1}{\partial x_1\over\partial x_1'} ... ...artial x_1'}{\partial \xi_2\over \partial x_2} {\partial x_2\over\partial x_2'}$$

$$= g_{12}{\partial x_1\over\partial x_1'}{\partial x_2\over\partial x_2'}$$ (10)

$$g_{22}' = g_{11}\left({\partial x_1\over\partial x_1'}\right)^2+2g_{12}{\pa... ...l x_2\over\partial x_2'}+g_{22}\left({\partial x_2\over\partial x_2'}\right)^2.$$ (11)