Let $\mathcal V$ be a real [vector space](/page/Vector%20Space) of variations, let $\mathcal A$ be an admissible class for which expressions of the form $y+\varepsilon h$ are defined, and let $J:\mathcal A\to\mathbb R$ be a functional. Fix $y\in\mathcal A$. Assume that, for every admissible two-sided variation direction $h\in\mathcal V$ at $y$, there exists $\rho_h>0$ such that $y+\varepsilon h\in\mathcal A$ for all $\varepsilon\in(-\rho_h,\rho_h)$, and the first variation

\begin{align*} \delta J[y;h] \end{align*}

exists and is given by

\begin{align*} \delta J[y;h]=\frac{d}{d\varepsilon}\bigg|_{\varepsilon=0}J[y+\varepsilon h]. \end{align*}

If $y$ is a local minimizer or a local maximizer of $J$ relative to these admissible two-sided variations, then

\begin{align*} \delta J[y;h]=0 \end{align*}

for every admissible two-sided variation direction $h$ at $y$.