My book has introduced the idea of gauge invariance in terms of classical electrodynamics (attached file). However, I am not sure I completely understand how it works. On the one hand they use a lot of time on specifying how you can add to the vector potential the gradient of any scalar, whilst at the same time subtracting the time derivative of this scalar from the generalized electrical potential. On the other hand when it comes to the use of this property, as in the section "The coloumb Gauge" they just PICK the divergence of A to be zero. I know that is always possible but shouldn't V be adjusted at the same time? I think I'm getting something wrong. I don't think I understand what the original potentials always are... Because gauge transformations speaks of adjusting these, but can you really say that there are original potentials?