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 P: 3,883 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 $\vec B$ is definitely a vector function where $$\nabla \cdot \vec B\;=\;0\;\hbox { and }\; \nabla \times \vec B \;=\; \mu\vec J\;+\;\frac{\partial\vec D}{\partial t}$$ But $\Phi$ is scalar where: $$\Phi\;=\;\oint_s \vec B\cdot d\vec s$$ 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 $\Phi$ indicates.