# First variation

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{{Short description|Mathematical theory}}
{{Use dmy dates|date=March 2026}}
{{More citations needed|date=March 2026}}
In applied [mathematics](/source/mathematics) and the [calculus of variations](/source/calculus_of_variations), the '''first variation''' of a [functional](/source/Functional_(mathematics)) ''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](/source/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](/source/Calculus_of_variations)
*[Functional derivative](/source/Functional_derivative)
*[Second variation](/source/Second_variation)

== References ==
{{Reflist}}

Category:Calculus of variations

{{mathanalysis-stub}}

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