Abstract:  The uniqueness theorem of the electrostatic field plays an essential role in electrodynamics. It states that the electric field can be determined uniquely, given the charge distribution and dielectric distribution of the area of our concern and given the potential or its normal derivative on the boundary. The proof of uniqueness theorem is tricky. However, most textbooks do not discuss the proof method's physical significance and the generalization to other fields. This paper finds that the action closely relates to the proof of uniqueness theorem. Once one constructs the action obeying the corresponding equations of motion and boundary conditions, one can prove the uniqueness theorem for any field. Furthermore, we discuss more general equations of motion and boundary conditions and study the case violating the uniqueness theorem.

Key words: uniqueness theorem, action, electromagnetostatic field