Older textbooks on the Calculus of Variations seem to define the first variation of a functional [tex] \Pi [/tex] as:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \delta \Pi = \Pi(f + \delta f) - \Pi (f) [/tex]

which looks analogous to:

[tex] \delta f = \frac {df} {dx} \delta x = lim_{\delta x \rightarrow 0} (f(x+ \delta x) -f(x)) [/tex]

from differential calculus. However, newer books seem todefinethe first variation as the Gateaux derivative:

[tex] \left[ \frac {d} {d \epsilon} \Pi (f+ \epsilon h) \right]_{\epsilon = 0 } [/tex]

which looks more like the gradient [tex]\frac {df} {dx} [/tex] than the difference [tex]\delta x [/tex]. Which is the better 'basic' definition?

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# First variation

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