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First variation

  1. Sep 30, 2007 #1
    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:

    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 to define the 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?
  2. jcsd
  3. Sep 30, 2007 #2
    If you know vector calculus, of course you should go on with Gateaux derivative.
  4. Sep 30, 2007 #3
    I've never seen anyone call the first one called the first variation. Everyone calls the Gateaux derivative the first variation.
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