I am only an engineer, just happen to come to Classical Physics to ask a question of my own and see this post. This is my understanding: Magnetic flux density [itex]\vec B[/itex] is definitely a vector function where
[tex]\nabla \cdot \vec B\;=\;0\;\hbox { and }\; \nabla \times \vec B \;=\; \mu\vec J\;+\;\frac{\partial\vec D}{\partial t}[/tex]
But [itex] \Phi[/itex] is scalar where:
[tex]\Phi\;=\;\oint_s \vec B\cdot d\vec s[/tex]
In words, B is line density and has direction. The total B flux line that cross an area gives you the TOTAL ( scalar) flux through the area as the equation of [itex] \Phi[/itex] indicates.
