Why is flux important?

Why is Magnetic flux a scalar and Magnetic flux density a vector? Why is the magnetic flux density represented by the field lines?

• I don't get how wb/m^2 is a vector and has direction. And what does the direction of the magnetic flux density say?

• Answer:

  The name "magnetic flux density" for BB\mathbf{B} or "magnetic field" is misleading. The magnetic field is more fundamental than the magnetic flux. The magnetic flux is the integral of the magnetic field dot the normal vector. Φ=∫Bâ‹…^ndSΦ=∫Bâ‹…n^dS\Phi = \int \mathbf{B} \cdot \mathbf{\hat{n}} \, dS or Φ=∫Bâ‹…dSΦ=∫Bâ‹…dS\Phi = \int \mathbf{B} \cdot d\mathbf{S} where ^nn^\mathbf{\hat{n}} is the normal vector and dSdSd\mathbf{S} is the infinitesimal area vector. Since the flux is an integral of a dot product, it is scalar. Intuitively, the flux is the net "amount" of  magnetic field "passing"  through a surface (or the net number of field lines crossing the surface), which should not have a direction, but should depend on the direction of the field and the orientation of the surface.