Chapter 6.11, Problem 16P

What is wrong with the following "proof" that there are no magnetic fields? By electromagnetic theory, $\nabla \cdot B=0,$ and $B=\nabla \times A.$ (The error is not in these equations.) Using them, we find ${\displaystyle \iiint \nabla }\cdot Bd\tau =0={\displaystyle \iint B}\cdot nd\sigma \text{ (by the divergence theorem) }$

$={\displaystyle \iint (\nabla \times A)}\cdot nd\sigma ={\displaystyle \int A}\cdot dr(\text{byStoke <msup> <mo> ′ </mo> </msup> theorem})$ since ${\displaystyle \int A}\cdot dr=0,A$ is conservative, or $A=\nabla \psi .$ Then $B=\nabla \times A=\nabla \times \nabla \psi =0$ so $B=0$.