Legendre's Condition/Lemma 1/Dependent on N Functions

Lemma
Let $\mathbf y = \paren {\sequence {\map {y_i} x}_{1 \mathop \le i \mathop \le N} }$ be a vector real function, such that:


 * $\map {\mathbf y} a = A$
 * $\map {\mathbf y} b = B$

Let $J \sqbrk {\mathbf y}$ be a functional, such that:


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

where:


 * $F \in C^2 \closedint a b$

all its variables.

Then:


 * $\displaystyle \delta^2 J \sqbrk {\mathbf y; \mathbf h} = \int_a^b \paren {\mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$

where:


 * $\mathbf P = \dfrac 1 2 F_{y_i'y_j'}$
 * $\mathbf Q = \dfrac 1 2 \paren {F_{ y_i y_j} - \dfrac \d {\d x} F_{y_i y_j'} }$