Q:

What does δA mean in differentiation?

To be more specific, I met this when doing analytical mechanics involving the principle of least action:

A:

The easiest way to understand this notation is as follows:

A variation is a transformation of the independent variables in a problem; it says that for each  $\epsilon$ we can transform $x_i$ to $x_i=x_i(\epsilon)=x_i + \epsilon\,\delta x_i + O(\epsilon^2)$ for some $\delta x_i$. You can think of it as a path in $\mathbf x$ space parametrized by $\epsilon$.
Notice $$\left[\frac{\mathrm d x_i}{\mathrm d \epsilon}\right]_{\epsilon=0} = \delta x_i$$Thus the vector $\mathbf{\delta x}$ is a tangent to that path.
In general, we define $$\delta F(x_1,\cdots,x_n) = \delta F(x_1(\epsilon),\cdots,x_n(\epsilon)) = \left[\frac{\mathrm d F}{\mathrm d \epsilon}\right]_{\epsilon=0}$$Hence $\epsilon\times \delta F$ is the small change in $F$ under this variation.
Using the chain rule $$\delta F = \left[\frac{\mathrm d F}{\mathrm d \epsilon}\right]_{\epsilon=0} = \sum_i\frac{\partial F}{\partial x_i}\left[\frac{\mathrm d x_i}{\mathrm d \epsilon}\right]_0 = \sum_i\frac{\partial F}{\partial x_i}\delta x_i$$

This can be formalized using differential forms, but this above is key to a more intuitive understanding. The message is that $\delta F$ is the small change in $F$ due to the small change in $\delta \mathbf x$ in $\mathbf x$.

For completeness, we can very quickly sketch how one moves towards differential forms.
Idea: What derivatives $F'(x)$ are for is telling you how to figure out what a small change in $F$ is given a small change in $x$. Define $\mathrm d F$ to be something mapping a change in $x$ to the change in $F$ to first order in the Taylor expansion. $\mathrm d F(U) = U \times \partial_x F$.
In more arbitrary settings, you are allowed to change $\mathbf x$, thinking of it as a set of coordinates, by moving along any path in $\mathbf x$ space. (Here, we're thinking of allowing any path. In general, we might be constrained to manifolds like the unit sphere.) Then a small change along a path is determined by derivatives along that path; this is simply the tangent vector dotted with the gradient, $\mathbf U \cdot \nabla F$. In analogy with the above, we define
$$\mathrm d F(\mathbf U) = \text{derivative of $F$ along curve tangent to $\mathbf U$} = U_i \cdot \partial F/\partial x_i$$
using the chain rule for the last step.
This is essentially the idea of (exact) differential 1-forms: they take vectors to the derivatives of functions.

A:

In this case, the author is using $\delta$ for the exterior derivative, which is usually denoted $\operatorname{d}$.
The function $A$ is a function of several variables, say $A(x_1,\ldots,x_n)$. The partial derivative of $A$ with respect to any one of the variables is denoted by
$$\frac{\partial A}{\partial x_i}$$
where $1 \le i \le n$. By definition, the exterior derivative of the differential $0$-form, i.e. of the function, $A$ is
$$\operatorname{d}\!A = \frac{\partial A}{\partial x_1}\, \operatorname{d}\!x_1 + \cdots + \frac{\partial A}{\partial x_n}\, \operatorname{d}\!x_n$$
Each of the $\operatorname{d}\!x_i$ are differential 1-forms. They take a tangent vector and give the $i$-th component of that vector. If you want to get very technical then each of the $\operatorname{d}\!x_i$ are linearly independent covectors. Since each of the $\operatorname{d}\!x_i$ are linearly independent, $\operatorname{d}\!A = 0$ if and only if each of the partial derivatives are zero.

