Compute the first variation of

${\displaystyle J(y)=\int _{a}^{b}yy'dx.}$ 

From the definition above,

${\displaystyle {\begin{aligned}\delta J(y,h)&=\left.{\frac {d}{d\varepsilon }}J(y+\varepsilon h)\right|_{\varepsilon =0}\\&=\left.{\frac {d}{d\varepsilon }}\int _{a}^{b}(y+\varepsilon h)(y^{\prime }+\varepsilon h^{\prime })\ dx\right|_{\varepsilon =0}\\&=\left.{\frac {d}{d\varepsilon }}\int _{a}^{b}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\ dx\right|_{\varepsilon =0}\\&=\left.\int _{a}^{b}{\frac {d}{d\varepsilon }}(yy^{\prime }+y\varepsilon h^{\prime }+y^{\prime }\varepsilon h+\varepsilon ^{2}hh^{\prime })\ dx\right|_{\varepsilon =0}\\&=\left.\int _{a}^{b}(yh^{\prime }+y^{\prime }h+2\varepsilon hh^{\prime })\ dx\right|_{\varepsilon =0}\\&=\int _{a}^{b}(yh^{\prime }+y^{\prime }h)\ dx\end{aligned}}}$ 

• Calculus of variations
• Functional derivative