Calculus of Variations: First Variation Definition?

[rdt2]

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 $$\Pi$$ 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?

 

[leon1127]

If you know vector calculus, of course you should go on with Gateaux derivative.

 

[DeadWolfe]

I've never seen anyone call the first one called the first variation. Everyone calls the Gateaux derivative the first variation.