Geodesic Equation/2d Surface Embedded in 3d Euclidean Space

Theorem
Let $\sigma: U \subset \R^2 \to V \subset \R^3$ be a smooth surface specified by a vector-valued function:


 * $\mathbf r = \map {\mathbf r} {u, v}$

Then a geodesic of $\sigma$ satisfies the following system of differential equations:


 * $\dfrac {E_u u'^2 + 2 F_u u' v' + G_u v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \dfrac \d {\d t} \dfrac {2 \paren {E u' + F v'} } {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$


 * $\dfrac {E_v u'^2 + 2 F_v u' v' + G_v v'^2} {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } - \dfrac \d {\d t} \dfrac {2 \paren {F u' + G v'} } {\sqrt{E u'^2 + 2 F u' v' + G v'^2} } = 0$

where $E, F, G$ are the functions of the first fundamental form:


 * $\displaystyle E = {\mathbf r}_u \cdot {\mathbf r}_u, F = {\mathbf r}_u \cdot {\mathbf r}_v, G = {\mathbf r}_v \cdot {\mathbf r}_v$

Proof
A curve on the surface $\mathbf r$ can be specified as $u = \map u t$, $v = \map v t$

The arc length between the points corresponding to $t_0$ and $t_1$ equals:


 * $\displaystyle J \sqbrk {u, v} = \int_{t_0}^{t_1} \sqrt {E u'^2 + 2 F u'v' + G v'^2} \rd t$

The following derivatives will appear in Euler's Equations:

Analogous relations hold for derivatives $v$ and $v'$.

Euler's Equation together with the results above yield the geodesic equations.