Conclusion

The first Helmholtz theorem explains why the $div$ and $curl$ are so ubiquitous in vector fields. It also explains the structure of Maxwell equations, usually written in such a way that the first member of each is either a $div$ or a $curl$. The second explains the meanings, and so also the names, of those operators. Considering that Hermann von Helmholtz graduated in Medicine, not in Physics, that was not a bad job at all! I wish to thank Walter F. Wreszinski for enlightening comments.