Meaning of Weak and Strong in Mathematics
Description
In mathematics, “weak” means “(logically) loose, less strict, less rigorous”. Being “less” something is to be understood in a relative sense. Conversely, “strong” means that the condition is (relatively) strict.
In simple terms, “weak” can be translated as “in effect, frankly”. For example, if we talk about college entrance examination scores, the top cumulative 4% of scores are awarded the 1st grade. Strictly, accurately speaking, it’s correct that only up to 4% are in the 1st grade, and the score report reflects this. However, if my percentile is 4.05%, although my score report will show 2nd grade, couldn’t we say, in effect, that I’m in the 1st grade? Although not officially, frankly speaking, it’s not completely untrue to say I am in the 1st grade. From this perspective, a student in the top 4.05% can be referred to as being in the weak 1st grade.
Weak Derivative
Following the explanation above, a weak derivative can be considered as an “in effect derivative”. Consider the two functions displayed here. Regarding $x \in \mathbb{R}$,
$$ u(x) = |x| \quad \text{and} \quad v(x) = \begin{cases} 1, & 0 \lt x \\ 0, & x = 0 \\ -1, & x \lt 0 \end{cases} $$
As is well-known, $u$ is not differentiable at $x = 0$, therefore, the derivative of $u$ as defined over $\mathbb{R}$ does not exist. However, if you observe $v$, it satisfies $u^{\prime}(x) = v(x)$ at all points where $x \ne 0$ exists. If you were $v$, wouldn’t it be too unfair to be denied the status of a derivative just because of one point $x = 0$ in the real space? That’s why we console $v$ by calling it the “in effect derivative” of $u$.