A common explanation for the reason why displacement current is needed is in the following diagram (Giancoli):
I can appreciate the reason why we need displacement current, however I really don't get the reasoning they used. The closed loop integral can have a "membrane" to catch the current and can be stretched over the capacitor and then the integral would give 0 if it wasn't for the displacement current.
I am quite confused as to why it is valid to stretch the membrane and still expect the same result. Unfortunately, I have only done single variable calculus, but I still would really like to know why it is valid that this membrane can be stretched.
