Geodesic Definition and 256 Threads

I don't understand the equation of the geodesic y=y(x) for the surface given by z=f(x,y) : a(x)y''(x)=b(x)y'(x)^3+c(x)y'(x)^2+d(x)dxdy-e(x) the functions a,b,c,d,e are here not very important, what I don't understand, is that there is terms in \frac{dy}{dx} and dxdy...What does this mean ?

can't delete post ? w/e. i'm trying to follow the path numerically. in ase of light geodesic, given gij, xi, and very small deltas dxi on step k, what would be dxi on step k+1?

I'm studying for my math physics final tomorrow and I'm going through a derivation done in our book, but I'm stuck on this one step. The derivation is of the geodesic equation using variational calculus (this is done in the Arfken and Weber book, on page 156 if you have it). Anyways, I follow...

Show that x1=asecx2 is a geodesic for the Euclidean metric in polar coordinates. So I tried taking all the derivatives and plugging into polar geodesic equations. Obviously, bad idea. Now I'm thinking I need to use Dgab/du=gab;cx'c and prove that the lengths of some vectors and their dot...

Fields of singular probabilities are inherent to quantum mechanics, but what method determines the statistics of curve segments like random geodesics bounded by definite black hole singularities, horizons or observers? Have Feynman path integrals been of use there, and if so, how?

Can someone take a look at http://wps.aw.com/wps/media/objects/500/512494/supplements/Ch21.pdf and tell me how they go from Eq. (7) to Eq. (8)? I've tried this and keep getting additional terms.