Per this post on math.stackexchange.com, I am told it is impossible to find an elementary formula for a coordinate patch covering of the orientable surface of genus 2 (or higher genus) (or the non-orientable ones?). Per this post on mathematica.stackexchange.com, there appear to be elementary formulas for the above-mentioned surfaces.

Is it possible to 1) write elementary formula coordinate patches for the above-mentioned orientable surfaces and elementary formulas for their volume 2-forms and integrate their volume forms over them, as in the Mathematics notebooks using the DifferentialForms.m package here and here? I would like to have Mathematica scripts for all of these phenomena (of course, there are infinitely many such surfaces, but there should be a pattern?). Thanks in advance.



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