I am currently reading "The Variational Principles of Mechanics - Cornelius Lanczos", in which the author talks about the variation of a function $F(q_1, q_2, \dots q_n)$ where $q_1, q_2, \dots q_n$ are the generalized coordinates
$$F=F(q_1, q_2, \dots q_n)$$
$$ \delta F=\frac{\partial F}{\partial q_1}\delta q_1+\frac{\partial F}{\partial q_n}\delta q_n+\dots+\frac{\partial F}{\partial q_n}\delta q_n \tag{1} $$ next he writes, $\delta q_1=\epsilon a_1, \delta q_2=\epsilon a_2, \dots ,\delta q_n=\epsilon a_n$
where $a_1, a_2 \dots , a_n$ are the direction cosines, and $\epsilon$ is a small variation. I find it wrong to use the same $\epsilon$ for all $\delta q_i $'s as it seems to be inconsistent with dimensions
For example if we are dealing with spherical coordinates $(r, \theta, \phi)$, according to the above the individual variations become
$\delta r=\epsilon \hat{r}$, So I expect $\epsilon$ to have a dimension of $r$ (unit vectors are dimensionless)
$ \delta \theta = \epsilon \hat{\theta},$ now $\epsilon$ has a dimension of $\theta$?
substituting these in $\text{eq}(1)$ makes it worse,
Further I like to think (I may be wrong) of the $\delta F$ as $\nabla F$, since both seem to have the same form, but as we know $\nabla$ is different for different coordinates, but $\text{eq}(1)$ seems to use $\delta$ as we use for Cartesian coordinates
Am I missing out something?
P.S
This still isn't a very great problem since the main goal is to find the stationary value which anyways leads to a conclusion $\frac{\partial F}{\partial q_k}=0$