\begin{definition}[Definition:Geodetic Distance]
Let $y_i$, $F$ be real functions.
Let $\mathbf y = \sequence {y_i}_ {1 \mathop \le i \mathop \le n}$ be a vector. 
Let:
:$\ds J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$
be a functional with only one extremal passing any two points:
:$A = \map A {x_0, \mathbf y^0}$
:$B = \map B {x_1, \mathbf y^1}$
Suppose a curve $\gamma$ is an extremal of $J$.
{{explain|Extremum or extremal?}}
Then:
:$\ds \map S {x_0, x_1, \mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \big \rvert_{\gamma}\rd x$
is called a '''geodetic distance''' between $A$ and $B$.
\end{definition}