On page 289 of the text "Fundamentals of Fluid Mechanics" by Munson et al., the authors give the following definition of the normal stress acting on the surface of a fluid element:
At any arbitrary location within a fluid mass, the force acting on a small area, $\delta A$, that lies in an arbitrary surface, can be represented by $\delta \mathbf{F}_{s}$. . . $\delta \mathbf{F}_{s} $ can be resolved into three components, $\delta F_n, \delta F_1$, and $\delta F_2$, where $\delta F_n$ is normal to the area . . . the normal stress, $\sigma_n$, is defined as
$$ \sigma_n = \lim_{\delta A \to 0} \frac{\delta F_n}{\delta A}.$$
Now, it will be recognized immediately that $\lim_{\delta A \to 0} \frac{\delta F_n}{\delta A}$ is nothing more than the derivative of $F_n$ with respect to $A$, i.e. $F_n'(A)$.
However, does it make sense to think of stress in this way? Let us imagine a surface containing two concentric circles, both with center at $\mathbf{x}_0$, with areas $A_1$ and $A_2$, respectively, such that $A_2 - A_1 = \delta A$. Further, suppose that the total force $F_n$ on both areas is the same - that is to say, $F_n(A_1) = F_n(A_2)$. If this is the case, then
$$ \frac{\delta F_n}{\delta A} = \frac{F_n(A_2) - F_n(A_1)}{A_2 - A_1} = 0. $$
therefore, $F'(A) = 0$ (since $F$ is the same on both area elements), which implies that $\sigma_n = 0$.
However, this does not agree with our common-use definition of the term "normal stress", in which we would have that $\sigma_n(A_1) = F/A_1$ and $\sigma_n(A_2) = F/A_2$. How can I reconcile this apparent inconsistency?
