First variation

In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional \delta J(y) mapping the function h to : \delta J(y,h) = \lim_{\varepsilon\to 0} \frac{J(y + \varepsilon h)-J(y)}{\varepsilon} = \left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon} J(y + \varepsilon h)\right|_{\varepsilon = 0}, where y and h are functions, and ε is a scalar. This is recognizable as the Gateaux derivative of the functional.

In applied mathematics and the calculus of variations, the first variation of a functional J(y) is defined as the linear functional \delta J(y) mapping the function h to \delta J(y,h) = \lim_{\varepsilon\to 0} \frac{J(y + \varepsilon h)-J(y)}{\varepsilon} = \left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon} J(y + \varepsilon h)\right|_{\varepsilon = 0}, where y and h are functions, and ε is a scalar. This is recognizable as the Gateaux derivative of the functional.

Example

Compute the first variation of J(y)=\int_a^b yy' \mathrm{d}x. From the definition above: \begin{align} \delta J(y,h)&=\left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon} J(y + \varepsilon h)\right|_{\varepsilon = 0}\\ &= \left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon} \int_a^b (y + \varepsilon h)(y^\prime + \varepsilon h^\prime) \ \mathrm{d}x\right|_{\varepsilon = 0}\\ &= \left.\frac{\mathrm{d}}{\mathrm{d}\varepsilon} \int_a^b (yy^\prime + y\varepsilon h^\prime + y^\prime\varepsilon h + \varepsilon^2 hh^\prime) \ \mathrm{d}x\right|_{\varepsilon = 0}\\ &= \left.\int_a^b \frac{\mathrm{d}}{\mathrm{d}\varepsilon} (yy^\prime + y\varepsilon h^\prime + y^\prime\varepsilon h + \varepsilon^2 hh^\prime) \ \mathrm{d}x \right|_{\varepsilon = 0}\\ &= \left.\int_a^b (yh^\prime + y^\prime h + 2\varepsilon hh^\prime) \ \mathrm{d}x\right|_{\varepsilon = 0}\\ &= \int_a^b (yh^\prime + y^\prime h) \ \mathrm{d}x \\ \end{align}

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