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curl B = 0

and

div B = 0

The problem was really about the maths, but it was phrased as if the field were a magnetic field. Now, I'm just wondering what the physical implications of this would be. Obviously if B is a magnetic field then div B = 0 is just one of Maxwells equations, but what if curl B = 0 as well? If B=0 both conditions are satisfied, but is there a non-zero vector/magnetic field that satisfies these conditions?

I'm really at a loss here...