# Metric Tensor Grid

Let, we are in a 2d metric where $$g_{xx}=1, g_{yy}=x^2$$, therefore $$|e_x|=1$$, $$|e_y|=x$$. If we try to draw the metric in a grid - it looks something like the image I uploaded. Note that, along the X axis the $$e_x$$ basis is always 1 unit long, but $$e_y$$ grows as we move along the X axis. And that causes a problem. $$g_{xx}=1$$, therefore a constant, and should not grow as we move along the Y axis. But as we can see, $$g_{xx}$$, which is the squared length of $$e_x$$ grows, $$CD >AB$$. What is the reason for this inconsistency? What am I missing? Use Christoffel Symbols in answer if needed.

First of all note, that the metric you wrote actually is the metric for polar coordinates, where x=r and y=$$\varphi$$. The problem in your picture is, that you drew the lines from A to C and from B to D as straight lines, which they are not, as they go along circles around the origin (constant r). Although this answer is very short I think it contains all the info you need, I hope it helped you.