# Definition of extrema in calculus of variations

I am reading Gelfand and Fomin about Calculus of Variations and in page 12 they say:

' Analogously, we say that the functional $J[y]$ has a (relative) extremum for $y=\hat{y}$ if $J[y]-J[\hat{y}]$ does not change its sign in some neighborhood of the curve $y=\hat{y}(x)$.'

Now the content of this definition is clear for me. As they say, its analogous with analysis. What troubles me is the notion '$y=\hat{y}$'. Why they express it in that way?

• It is the same as "extremum at $\hat y$". Notation $y=\hat y$ is used, I guess, to stress that it is $y$ that should be substituted by $\hat y$ (not $x$ or anything else).
– A.Γ.
Jun 26 '16 at 12:12
• $y$ is an arbitrary function for which the functional $J$ is being evaluated at, the $\hat{y}$ is a specific function (which is the location of the local extrema of $J$). Jun 26 '16 at 13:49