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620ba102fd7
posted 5 days ago
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\begin
{aligned}
{\frac {\mathrm {d}
\Phi }
{\mathrm {d}
\varepsilon }}&=
{\frac {\mathrm {d}
}
{\mathrm {d}
\varepsilon }}\int _
{a}
^
{b}
L
(x,f(x)
+\varepsilon \eta
(x)
,
f'
(x)
+\varepsilon \eta '
(x)
)\,
\mathrm
{d}
x\\&=\int _
{a}
^
{b}
{\frac {\mathrm {d}
}
{\mathrm {d}
\varepsilon }}L
(x,f(x)
+\varepsilon \eta
(x)
,
f'
(x)
+\varepsilon \eta '
(x)
)\,
\mathrm
{d}
x\\&=\int _
{a}
^
{b}
\left
[\eta (x)
{\frac {\partial L}
{\partial {f}
}}
(x,f(x)
+\varepsilon \eta
(x)
,
f'
(x)
+\varepsilon \eta '
(x)
)+\eta .
.
.
