  By making use of the $\phi $-mapping topological current theory, a novel
expression of $\nabla \times \vec{V}$ in BEC is obtained, which reveals the
inner topological structure of vortex lines characterized by Hopf indices and
Brouwer degrees. This expression is just that formula Landau and Feynman
expected to find out long time ago. In the case of superconductivity, the
decomposition theory of U(1) gauge potential in terms of the condensate wave
function gives a rigorous proof of London assumption, and shows that each
vortex line should carry a quantized flux. The $\phi $-mapping topological
current theory of $\nabla \times \vec{V}$ can also gives a precise bifurcation
theory of vortex lines in BEC.
