CoV13

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Compute the first variation of J(y)=\int _{a}^{b}yy'dx\,

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

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