First variation
In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional δ J ( y ) {\displaystyle \delta J(y)} mapping the function h to δ J ( y , h ) = lim ε → 0 J ( y + ε h ) − J ( y ) ε = d d ε J ( y + ε h ) | ε = 0 , {\displaystyle \delta J(y,h)=\lim _{\varepsilon \to 0}{\frac {J(y+\varepsilon h)-J(y)}{\varepsilon }}=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0},} where y and h are functions, and ε is a scalar. This is recognizable as the Gateaux derivative of the functional.