Why is flux linked between a bar magnet and a coil same as the flux due to bar magnet from radius of coil to infinity? Why is flux linked between a bar magnet and a coil same as the flux due to bar magnet extending from radius of coil upto infinity?
It has been stated without any reasoning in  this video. at about 1 min.
 A: Imagine a volume consisting of a hemispherical surface and  the equatorial plane.Now  net magnetic flux passing through a closed surface (hemispherical+equatorial) is zero. Now let some flux $\Phi$ is crossing the equatorial plane and coming inside the volume then the same amount of flux must exit through the hemispherical surface too so that total flux coming out through the total surface is zero. Now the arguments hold if the hemispherical surface is changed to any arbitrary surface with any radius which makes a closed volume with the equatorial base.
A: Since there is no such thing as a magnetic monopole, the total flux across any infinite plane must be zero. If you cut out a circle (or any shape) and determine the flux through it, then the total flux through what was left must be equal and opposite to what was cut out.
A little more detail in response to Timaeus's comment (this starts to look like @Prof Shonku's answer... but it didn't start out that way): Absent a magnetic monopole, the divergence of the flux through any closed volume is zero - that's just how the mathematics works.  If you have a bar magnet, at sufficiently large distance (much greater than the size of the magnet) it gives you a dipole field. Now the field of a dipole magnet drops off as the inverse cube of distance; therefore, if you create a volume in the shape of a hemisphere, while the area of the hemisphere grows as the square of the radius, the flux drops with the cube; so for a sufficiently large hemisphere, the integral of the flux is zero. If flux through the hemisphere is zero, then the flux through the plane closing the hemisphere must also be zero (since the total flux out of the volume is zero). This was the part of the argument I failed to make explicit.
