Why are integral and differential operators commutative? For instance, let's assume a constant 3D surface over time $S$.
$$
\frac{d}{dt}\iint_S \mathbf B \cdot \mathbf{ds} 
\quad=\quad 
\iint_S\frac{\partial \mathbf B}{\partial t}\cdot \mathbf{ds} 
$$

Why is that? Why can I commute $\frac{d}{dt}$ and $\iint_S$?

I would appreciate formal answers. If you can provide references too, I would like.
You can do it when the function $\boldsymbol B$, and the function $\dfrac{\partial \boldsymbol B}{\partial t}$ are continuous.

Why? Here is the answer.
Here is a short answer. Integral are sums (Riemann sums) and derivatives are differences (to the limit). Sums and differences are commutative.