Calculus of Variations: The Gateaux vs First Variation Debate

[rdt2]

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?

 

[EnumaElish]

The first definition is similar to the discrete $\Delta$ operator in real analysis. The 2nd def. is similar to the continuous d operator.