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 $delta\; J(y,\; h)=\; frac\{d\}\{dvarepsilon\}\; J(y\; +\; varepsilon\; h)left.\; ight|\_\{varepsilon\; =\; 0\}$.

Example

Compute the first variation of $J(y)=int\_a^b\; yy\text{'}\; dx,$

$delta\; J(y,\; h),$	$=\; frac\{d\}\{dvarepsilon\}\; J(y\; +\; varepsilon\; h)left.\; ight|\_\{varepsilon\; =\; 0\}$
	$=\; frac\{d\}\{dvarepsilon\}\; int\_a^b\; (y\; +\; varepsilon\; h)(y^prime\; +\; varepsilon\; h^prime)\; dxleft.\; ight|\_\{varepsilon\; =\; 0\}$
	$=\; frac\{d\}\{dvarepsilon\}\; int\_a^b\; (yy^prime\; +\; yvarepsilon\; h^prime\; +\; y^primevarepsilon\; h\; +\; varepsilon^2\; hh^prime)\; dxleft.\; ight|\_\{varepsilon\; =\; 0\}$
	$=\; int\_a^b\; frac\{d\}\{dvarepsilon\}\; (yy^prime\; +\; yvarepsilon\; h^prime\; +\; y^primevarepsilon\; h\; +\; varepsilon^2\; hh^prime)\; dxleft.\; ight|\_\{varepsilon\; =\; 0\}$
	$=\; int\_a^b\; (yh^prime\; +\; y^prime\; h\; +\; 2varepsilon\; hh^prime)\; dxleft.\; ight|\_\{varepsilon\; =\; 0\}$
	$=\; int\_a^b\; (yh^prime\; +\; y^prime\; h)\; dx$

External links

* [http://www.exampleproblems.com Exampleproblems.com] has more examples of computing the first variation of functionals.