Today I was working through a handout my professor gave me on geodesics and stumbled upon the section on semigeodesic parameterization...and then I got lost. I was able to follow the material through the point where the text points out that the first fundemental form will consist of: I= E(du)^2 + G*(dv)^2, since the parametrization is orthogonal. The text then moves into an example of how if v were constant the differential equation for the geodesic u''v'-u'v''+Av'-Bu' = 0 (where A and B's values can be derived from here: http://mathworld.wolfram.com/GeodesicCurvature.html ) becomes -1/2*[tex]E_v[/tex]/G = 0. Which makes sense, but then it gives an alternative statement later on that the above equation of the geodesic could also be expressed as da/dv = - partial((G)^(1/2))/partial(u). Where a is defined as the angle the geodesics intersect the curves v=constant. Whats worse is this alternative to the above was offered in the "theorems" section of the chapter and doesn't have a proof with it. So two questions: 1) Wouldn't the angle be pi/2, since in a semigeodesic parameterization any geodesic would be orthogonal to the to the coordinate curves (in this case v=constant)? 2) Where are they getting the alternative expression...is is from Gauss-Bonnet? Or am I just reading the theorem incorrectly. In any case, where the heck is the text coming up with said "theorem." I don't want a proof, I just want a reasonable feel for where the heck they are getting it from. Because from their definations I just don't see the correlation.