In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional ${\displaystyle \delta J(y)}$ mapping the function h to

${\displaystyle \delta J(y,h)=\lim _{\varepsilon \to 0}{\frac {J(y+\varepsilon h)-J(y)}{\varepsilon }}=\left.{\frac {d}{d\varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0},}$

where y and h are functions, and ε is a scalar. This is recognizable as the Gateaux derivative of the functional.