Solenoid and non-conservative vector field

Question

Are solenoid vector fields non-coservative vector fields? If so, are all non-conservative vector fields solenoid fields?

Thanks

Answer 1

Certainly a solenoidal vector field is not always non-conservative; to take a simple example, any constant vector field is solenoidal. However, some solenoidal vector fields are non-conservative - in fact, lots of them. By the Fundamental Theorem of Vector Calculus, every vector field is the sum of a conservative vector field and a solenoidal field; so if you start with a non-conservative vector field, the solenoidal field in question will have to be non-conservative.

On the other hand, it is certainly not the case that all non-conservative vector fields are solenoidal. A solenoidal field is just a vector field with divergence zero; take any non-conservative vector field with nonzero divergence (say, $F(x,y) = \langle x,x\rangle$, for example) and it will not be solenoidal.