Necessary Condition for Integral Functional to have Extremum for given function/Dependent on N Functions

Theorem
Let $J[\mathbf{y}]=J[y_1,~y_2,~...,y_n]$ be a functional of the form

$\displaystyle J[\mathbf{y}]=\int_{a}^{b}F\left(x, \mathbf{y}\right)\mathrm{d}{x}=\int_{a}^{b}F\left(x,~y_1,~y_2,...,~y_n\right)\mathrm{d}{x}$