Let's say same point in two co-ordinate system has the following relation from partial derivatives,
$$dx'=\frac{\partial x'}{\partial x} dx + \frac{\partial x'}{\partial y} dy$$ and $$dy'=\frac{\partial y'}{\partial x} dx + \frac{\partial y'}{\partial y} dy.$$
I have to derive the conditions for $$ dx'^2 + dy'^2 \propto dx^2 + dy^2. $$
Now my question is how do define $dx'^2$? My memory and hunch says, it should be $$ dx'^2 = \left( \frac{\partial x'}{\partial x} \right)^2 dx^2 + 2 \frac{\partial x'^2}{\partial x \partial y} dxdy + \left( \frac{\partial x'}{\partial y} \right)^2 dy^2$$
However, I'm not sure. Is it correct?
Also, could anyone tell me a good text to revise these stuffs from. I did them a long time ago and forgot.