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.
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Example
Compute the first variation of
J ( y ) = ∫ a b y y ′ d x . {\displaystyle J(y)=\int _{a}^{b}yy'\mathrm {d} x.}From the definition above,
δ J ( y , h ) = d d ε J ( y + ε h ) | ε = 0 = d d ε ∫ a b ( y + ε h ) ( y ′ + ε h ′ ) d x | ε = 0 = d d ε ∫ a b ( y y ′ + y ε h ′ + y ′ ε h + ε 2 h h ′ ) d x | ε = 0 = ∫ a b d d ε ( y y ′ + y ε h ′ + y ′ ε h + ε 2 h h ′ ) d x | ε = 0 = ∫ a b ( y h ′ + y ′ h + 2 ε h h ′ ) d x | ε = 0 = ∫ a b ( y h ′ + y ′ h ) d x {\displaystyle {\begin{aligned}\delta J(y,h)&=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0}\\&=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}(y+\varepsilon h)(y^{\prime }+\varepsilon h^{\prime })\ \mathrm {d} x\right|_{\varepsilon =0}\\&=\left.{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}\int _{a}^{b}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\ \mathrm {d} x\right|_{\varepsilon =0}\\&=\left.\int _{a}^{b}{\frac {\mathrm {d} }{\mathrm {d} \varepsilon }}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\ \mathrm {d} x\right|_{\varepsilon =0}\\&=\left.\int _{a}^{b}(yh^{\prime }+y^{\prime }h+2\varepsilon hh^{\prime })\ \mathrm {d} x\right|_{\varepsilon =0}\\&=\int _{a}^{b}(yh^{\prime }+y^{\prime }h)\ \mathrm {d} x\\\end{aligned}}}