\begin{definition}[Definition:Euler's Equation for Vanishing Variation]
Let $\map y x$ be a real function.
Let $\map F {x, y, z}$ be a real function belonging to $C^2$ {{WRT}} all its variables.
Let $J \sqbrk y$ be a functional of the form:
:$\ds \int_a^b \map F {x, y, y'} \rd x$
Then '''Euler's equation for vanishing variation''' is defined a differential equation, resulting from condition:
:$\ds \delta \int_a^b \map F {x, y, y'} \rd x = 0$
In other words:
:$F_y - \dfrac \d {\d x} F_{y'} = 0$
{{Stub|Check if conditions can be stricter, add special cases, examples, multidimensional and multiderivative forms}}
{{NamedforDef|Leonhard Paul Euler|cat = Euler}}
\end{definition}