What does the geodesic equation for a surface involve?

[kleinwolf]

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 ?

 

[dextercioby]

Where did you get that equation...?It should come from the tensor one involving Christoffel symbols.

Daniel.

 

[kleinwolf]

This is the equation in the special case where z=f(x,y)...the geodesics being given in the direct form : y=y(x)...I got this in Bronstein Taschenbuch der Mathematik.

 

[dextercioby]

I'm sorry,i can't get that book.Could u please indicate other source (it would be sizzling,if online) ?

Daniel.

 

[kleinwolf]

Here is a scan :

 

[dextercioby]

It's a typo,i'm sure the German dude meant the derivative of the first order

$$\frac{dy}{dx}$$.

Daniel.