In Calculus of Variations by Gelfand and Fomin the authors defined the variation $\delta J[h]$ of a functional $J[y]$ as the principle linear part of the increment $\Delta J[h]$:
$$\Delta J[h] = J[y+h] - J[y] = \delta J[h] + \epsilon(h) ||h||$$
where $\delta J[h]$ is a linear functional and $\epsilon(h) \rightarrow 0$ as $||h|| \rightarrow 0$.
I can now introduce an auxiliary variable $\alpha$ to calculate the variation as follows
\begin{align} \Delta J[\alpha h] &= J[y+\alpha h] - J[y] \\ &= \delta J[\alpha h] + \epsilon(\alpha h) ||\alpha h|| \\ &= \alpha\delta J[h] + |\alpha|\epsilon(ah)||h|| \end{align}
Therefore,
$$ \frac{\partial J[y+\alpha h]}{\partial\alpha} = \delta J[h] + \operatorname{sgn}(\alpha) \epsilon(\alpha h)||h|| + |\alpha|\frac{\partial\epsilon(\alpha h)}{\partial\alpha} ||h||$$
The second and third term vanishe when $\alpha \rightarrow 0$, hence the variation
$$ \delta J[h] = \frac{\partial J[y+\alpha h]}{\partial\alpha} \bigg|_{\alpha \rightarrow 0} $$
However, in Classical Mechenics by Goldstein and Classical Dynamics by Thornton and Marion the variation is defined respectfully as
$$ \delta J = \frac{\partial J}{\partial\alpha} \bigg|_{\alpha = 0} d\alpha $$
and
$$ \delta J = \frac{\partial J}{\partial\alpha} d\alpha$$
So what is this $d\alpha$ that Goldstein, Thornton and Marion added to the partial derivative and does it make their definition of the variation any different from the one from Gelfand and Fomin?