I put this question in the 'Calculus' forum but didn't really get a response. Maybe it's a silly question but I thought I'd try here anyway:(adsbygoogle = window.adsbygoogle || []).push({});

Older textbooks on the Calculus of Variations seem to define the first variation of a functional [tex] \Pi [/tex] as:

[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?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# First variation

**Physics Forums | Science Articles, Homework Help, Discussion**