- #1
rdt2
- 124
- 2
Older textbooks on the Calculus of Variations seem to define the first variation of a functional \Pi as:
\delta \Pi = \Pi(f + \delta f) - \Pi (f)
which looks analogous to:
\delta f = \frac {df} {dx} \delta x = lim_{\delta x \rightarrow 0} (f(x+ \delta x) -f(x))
from differential calculus. However, newer books seem to define the first variation as the Gateaux derivative:
\left[ \frac {d} {d \epsilon} \Pi (f+ \epsilon h) \right]_{\epsilon = 0 }
which looks more like the gradient \frac {df} {dx} than the difference \delta x. Which is the better 'basic' definition?
\delta \Pi = \Pi(f + \delta f) - \Pi (f)
which looks analogous to:
\delta f = \frac {df} {dx} \delta x = lim_{\delta x \rightarrow 0} (f(x+ \delta x) -f(x))
from differential calculus. However, newer books seem to define the first variation as the Gateaux derivative:
\left[ \frac {d} {d \epsilon} \Pi (f+ \epsilon h) \right]_{\epsilon = 0 }
which looks more like the gradient \frac {df} {dx} than the difference \delta x. Which is the better 'basic' definition?