CoV25

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Let J:A\to {\mathbb  {R}}\, be a functional on a subset A\, of a normed linear space V\,.

(a) Define precisely the first variation \delta J(y_{0},h)\, of J\, at y_{0}\, and admissible h(x)\,.

\delta J(y_{0},h)={\frac  {d}{d\varepsilon }}J(y_{0}+\varepsilon h)\left.\right|_{{\varepsilon =0}}

Alternatively,

\delta J(y_{0},h)=\lim _{{\varepsilon \to 0}}{\frac  {J(y_{0}+\varepsilon h)-J(y_{0})}{\varepsilon }}

The admissible h\, are \{h\in V:y_{0}+\varepsilon h\in A\}

(b) Show that if \delta J(y_{0},h)\, exists for a certain admissible h\in V\,, then \delta J(y_{0},\alpha h)\, also exists for every real number \alpha \,, and \delta J(y_{0},\alpha h)=\alpha \delta J(y_{0},h)\,.

The solution is here.


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