I have positive function $f$ of two variables, $u$ and $v$, and a surface $M$ such that the metric takes on the form $$ds^2 = f^2 \ du^2 + f^{-2} \ dv^2.$$ I want to compute the area of the square given by $0 \leq u,v \leq 1$. Is this done by just integrating the metric over the square?
Computing Area of a Square on a Surface given a Metric
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geometry
differential-geometry
surfaces
riemann-surfaces
1 Answers
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Note the first fundamental form is $ds^2=Edu^2+2Fdudv+Gdv^2$. Comparing this with $ds^2$, one has $$ E=f^2,F=0,G=f^{-2}. $$ Also note the area element is $dA=\sqrt{EG-F^2}dudv$. Then the area of the square given by $0≤u,v≤1$ is $$ A=\int_0^1\int_0^1\sqrt{EG-F^2}dudv=1.$$