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So, I have a basic understanding of the curl of a vector field. Defined as the closed loop line integral divided by the infinitesimal area it encloses, curl is a vector field itself, offering some "rotation related" information of the vector field from which it is derived.

For a velocity field, the curl of the field at a point would be proportional to the field angular velocity at this point. For a force field, it would be proportional to the angular acceleration of the field at this point.

So far so good. Now, what means taking the curl of the curl of a force field ? Why is this justified when deriving EM waves ? Let's say you never looked at the EM wave equation derivation, and you just know what the curl of a field is. You stare at Maxwell's equation in their differential form, in vacuum.

Why would you take the curl of the curl of the fields ? What's the reason behind that move ? Can't just be a random idea.

The only logical interpretation I can see is that the curl of the curl of say the E field, is the rate at which the angular acceleration of the field changes at the given point. But that doesn't tell me why one would be interested with that.