Ampère's circuital law
Second, there are infinitely many possible surfaces S that have the curve C as their border. (Imagine a soap film on a wire loop, which can be deformed by moving the wire). Which of those surfaces is to be chosen? If the loop does not lie in a single plane, for example, there is no one obvious choice.  The answer is that it does not matter; it can be proven that any surface with boundary C can be chosen.