First variation

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In applied mathematics and the calculus of variations, the first variation of a functional J(y)\, is defined as \delta J(y, h)= \frac{d}{d\varepsilon} J(y + \varepsilon h)\left.\right|_{\varepsilon = 0}.

[edit] Example

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

[edit] External links

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