{{Short description|Mathematical theory}} {{Use dmy dates|date=March 2026}} {{More citations needed|date=March 2026}} In applied mathematics and the calculus of variations, the '''first variation''' of a functional ''J''(''y'') is defined as the linear functional <math> \delta J(y) </math> mapping the function ''h'' to
:<math>\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},</math>
where ''y'' and ''h'' are functions, and ''ε'' is a scalar.<ref name=":0">{{Cite web |title=1.3.2 First variation and first-order necessary condition |url=https://liberzon.csl.illinois.edu/teaching/cvoc/node15.html |access-date=2026-03-03 |website=liberzon.csl.illinois.edu}}</ref> This is recognizable as the Gateaux derivative of the functional.<ref name=":0" />
==Example==
Compute the first variation of
:<math>J(y)=\int_a^b yy' \mathrm{d}x.</math>
From the definition above:
:<math> \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} </math>
== See also == *Calculus of variations *Functional derivative *Second variation
== References == {{Reflist}}
Category:Calculus of variations
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