Swapping partial derivative and curl operator

Question

The third Maxwell equation states that $$\nabla \times \mathbf E = -\frac{\partial\mathbf B}{\partial t}.$$

Then I have in my notes: $$\nabla \times (\nabla \times \mathbf E) = -\nabla \times \frac{\partial\mathbf B}{\partial t} \color{red}{=} -\frac{\partial}{\partial t}\nabla \times \mathbf B.$$

I don't understand why it's possible to swap the two operators. What are the conditions that have to be satisfied in order to do so?

Answer 1

Since both $\frac{\partial}{\partial t}$ and $\nabla$ are first-order differential operators (that is, linear operators), the sought condition boils down to the symmetry of second partial derivatives.

Schwarz's theorem is relevant here.

If $f\ \colon \mathbb R^n \to \mathbb R$ has continuous second partial derivatives at $\mathbf x \in \mathbb R^n$, then $$\frac{\partial^2 f}{\partial x_i\partial x_j}(\mathbf x) = \frac{\partial^2 f}{\partial x_j\partial x_i}(\mathbf x)$$ for every $i, j = \{1, 2, \ldots, n\}$.

Now, since both $\mathbf E$ and $\mathbf B$ have continuous second partial derivatives, the two linear differential operators can be swapped.