  We study the exterior derivative as a symmetric unbounded operator on square
integrable 1-forms on a 3D bounded domain $D$. We aim to identify boundary
conditions that render this operator self-adjoint. By the symplectic version of
the Glazman-Krein-Naimark theorem this amounts to identifying complete
Lagrangian subspaces of the trace space of H(curl) equipped with a symplectic
pairing arising from the $\wedge$-product of 1-forms on $\partial D$.
Substantially generalizing earlier results, we characterize Lagrangian
subspaces associated with closed and co-closed traces. In the case of
non-trivial topology of the domain, different contributions from co-homology
spaces also distinguish different self-adjoint extension. Finally, all
self-adjoint extensions discussed in the paper are shown to possess a discrete
point spectrum, and their relationship with curl curl-operators is discussed.
