Proving Vector Integral Identity

[TSC]

$$\oint {d{\bf{R}} \times {\bf{A}}} = \mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc} <br /> {\left( {d{\bf{S}} \times \nabla } \right) \times {\bf{A}}}$$

 

[ShayanJ]

I would try Stokes' theorem. Just try to find apropriate fields and field derivatives instead of the original fields in the theorem. I'm almost sure it will work.

 

[TSC]

I am trying to prove this using the common vector identities and Stoke's theorem, without going into tensor notation at every step. It seems to be quite impossible.