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?