vector analysis---WRONG!!!

[kolahalb]

What is the wrong in the proof that there is no magnetic field?
We know:
div B=0 and B=curl A where B is the magnetic field and A is its vector potential
Then,∫div B dV=0=∫B.n dA(divergence theorem)
Thus, 0=∫(curl A).n dA
=∫A.dr(Stokes' theorem)
The last will be a cyclic integral.
Since the cyclic line integral is zero,A is conservative;
or,A=grad f
Then,B=curl A=curl grad f=0
or,B=0

I think the problem lies between the equality in Stokes' theorem.In original Stokes' theorem LHS is a non-cyclic integral whereas here it comes directly from divergence theorem and is a cyclic integral...
Please let me know if i am correct;if not,then where did I go wrong.
Thank you in advance

 

[pmonon]

You are right that you have to be careful about closed and non closed line/path integral. Further more it will be an error to assume that
∫div B dV=0, rather this integral should be constant. Any way, since there are more than one flaws in the proof the argument is not proved and hence false.

 

[kolahalb]

I did not understand why ∫div B dV is not zero.For a closed volume there is no magnetic monopole that will make the divergence non-zero.

 

[pmonon]

I did not understand why ∫div B dV is not zero.For a closed volume there is no magnetic monopole that will make the divergence non-zero.

Yes, I was trying to say it is zero or constant in general and in this case it is zero when you work with

∫© B.da which will be zero.