Why are integral and differential operators commutative?

Question

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.

Answer 1

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.

Answer 2

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