一次變分

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在應用數學變分法裏,一個泛函 J(y)\,一次變分定義為 \delta J(y, h)= \frac{d}{d\varepsilon} J(y + \varepsilon h)\left.\right|_{\varepsilon = 0}\,

實例[编辑]

計算 J(y)=\int_a^b yy' dx\,的一次變分?

\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

外連[编辑]