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So if I have a surface in R^3 z(x,y)=sin(x)+sin(y). I'm interested in finding surface coordinates S_1 and S_2 that lie along the lines of constant x and y. I don't REALLY get how to find the forms S_1(x,y,z) and S_2(x,y,z) because the distance along either S_1 or S_2 is the arclength of a curve that depends on x, y, and z.
Basically I don't know how I should go about finding the functions that will give me the coordinate transformations. I'm interested in certain aspects of the differential geometry of this surface but alas I'm too dumb to figure out the relations that will let me calculate the metric tensor on this surface, mean curvature, etc.