CoV27

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Compute the first variation \delta J(y,h)\, for y\in C[0,1]\,: J(y)=\int _{0}^{1}\int _{0}^{1}\sin(xt)y(x)y(t)dxdt\,

\delta J(y,h)\, ={\frac  {d}{d\varepsilon }}J(y+\varepsilon h)\left.\right|_{{\varepsilon =0}}
={\frac  {d}{d\varepsilon }}\int _{0}^{1}\int _{0}^{1}\sin(xt)[(y+\varepsilon h)(x)][(y+\varepsilon h)(t)]\ dxdt\left.\right|_{{\varepsilon =0}}
=\int _{0}^{1}\int _{0}^{1}{\frac  {d}{d\varepsilon }}\left[\sin(xt)(y(x)+\varepsilon h(x))(y(t)+\varepsilon h(t))\right]\ dxdt\left.\right|_{{\varepsilon =0}}
=\int _{0}^{1}\int _{0}^{1}{\frac  {d}{d\varepsilon }}\left[\sin(xt)(y(x)y(t)+\varepsilon h(x)y(t)+\varepsilon h(t)y(x)+\varepsilon ^{2}h(x)h(t))\right]\ dxdt\left.\right|_{{\varepsilon =0}}
=\int _{0}^{1}\int _{0}^{1}\sin(xt)[0+h(x)y(t)+h(t)y(x)+2\varepsilon h(x)h(t)]\ dxdt\left.\right|_{{\varepsilon =0}}
=\int _{0}^{1}\int _{0}^{1}\sin(xt)(h(x)y(t)+h(t)y(x))\ dxdt


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