Differential Geometry: Coordinate Patches

by SNOOTCHIEBOOCHEE
Tags: coordinate, differential, geometry, patches
 P: 145 Sorry i wasnt able to get help in the hw department. figured id try here. 1. The problem statement, all variables and given/known data For a coordinate patch x: U--->$$\Re^{3}$$show that$$u^{1}$$is arc length on the $$u^{1}$$ curves iff $$g_{11} \equiv 1$$ 3. The attempt at a solution So i know arc legth of a curve $$\alpha (t) = \frac{ds}{dt} = \sum g_{ij} \frac {d\alpha^{i}}{dt} \frac {d\alpha^{j}}{dt}$$ (well thats actually arclength squared but whatever). But im not sure how to write this for just a $$u^{1}$$ curve. A $$u^{1}$$ curve throught the point P= x(a,b) is $$\alpha(u^{1})= x(u^{1},b)$$ But i have no idea how to find this arclength applies to u^1 curves. Furthermore i know some stuff about our metric $$g_{ij}(u^{1}, u^{2})=$$ We know that $$x_{1}= (1,0)$$ and that is as far as i got :/ Any help appreciated.
 Sci Advisor P: 1,716 A curve tangent to a coordinate direction only has one metric tensor component that is not zero - I think.

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