91Ó°ÊÓ

The given integral is$\text{∮}\mathrm{B}×\mathrm{nd}\mathrm{σ}.$

Divergence theorem, often known as Gauss' theorem or Ostrogradsky's theorem, is a theorem that connects the flow of a vector field across a closed surface to the field's divergence in the volume enclosed. According to this theorem, the surface integral of a vector field over a closed surface, also known as the flux through the surface, equals the volume integral of the divergence over the region inside the surface.

Use the divergence theorem.

${\text{∮}}_a\mathrm{B}×\mathrm{n}d\mathrm{σ}={∫}_1 (\mathrm{∇}×\mathrm{B})d{\mathrm{τ}}_{\mathrm{I}}$

$B=\mathrm{∇}×A$

Then, the equation is ${\text{∮}}_{\mathrm{∂}v}\mathrm{B}×\mathrm{nd}\mathrm{σ}={∫}_V \mathrm{∇}×(\mathrm{∇}×\mathrm{A})d\mathrm{τ}$true.

${\text{∮}}_{\mathrm{∂}v}Bnd\mathrm{σ}={∫}_v \mathrm{∇}(\mathrm{∇}×A)d\mathrm{τ}$

But, $\mathrm{∇}×(\mathrm{∇}×A)=0$everywhere .

${\text{∮}}_{\mathrm{∂}v}\mathrm{B}×\mathrm{n}d\mathrm{σ}=0$

Hence, the solution of the integrals is ${\text{∮}}_{\mathrm{∂}v}\mathrm{B}×\mathrm{nd}\mathrm{σ}=0$.