I am trying to proof Helmholtz’s theorem and I am currently using David J. Griffith’s book on electrodynamics to do this. Within the book’s Appendix B, (I have not finished the book, and am on the last sub chapter of the first chapter, the theory of vector fields) I have read along and understood until, “However, it so happens that there is no function with zero divergence and zero curl everywhere and goes to zero at infinity. (See sect 3.1.5)”.

Where sect 3.1.5 is “Boundary conditions and Uniqueness theorem”. Which gives the first uniqueness theorem as “The solution to Laplace’s equation in some volume V is uniquely determined if V is specified on boundary surface S”. While I understand individually what they mean, I don’t see how it proves the original quote. Any help is appreciated.