Question

prove that div vectorB =0
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Answer 1

Answer:

Divergence of Curl of any Vector Field is Zero.

Or simply Curl of any Vector Field is Solenoidal.

By this assumption if u take Curl of any Vector Field that is not irrotational, you will end up with another Vector Field which have Zero Divergence at every point.

Now working backwards, given a Vector Field have Zero Divergence everywhere it may or may not be Curl of another Vector Field depending on the context of Physics. For instance, Magnetic Field is inherently Solenoidal so we have no obligation to find whether it is Curl of other Vector Field. On the other hand if for some other Physical condition if we were to find it , the Vector Field is not unique , as many Vector Field can have the identical derived Curl and it might depend on the initial condition.

Consider three distinct Vector Fields

Their Curl are the same

And Divergence of the Curl of the Vector Field is identically Zero.

As per your question, Yes if Divergence of Vector Field is Zero it can be Curl of another Vector Field. But it not unique.