Weierstrass's Necessary Condition

Theorem
Let $\mathbf y$ be an $ n $-dimensional vector such that $\map {\mathbf y} a=A$ and $\map {\mathbf y} b=B$.

Let $J$ be a functional such that:


 * $\displaystyle J\sqbrk{\mathbf y}=\int_a^b \map F {x,\mathbf y,\mathbf y'}\rd x$

Let $\mathbf w$ be a finite vector.

Let $\gamma$ be a strong minimum of $J$.

Then:
 * $\map E {x,\mathbf y,\mathbf y',\mathbf w}\ge 0$

along $\gamma$ and for every $\mathbf w$, and $E$ stands for Weierstrass E-Function.