Why does $\vec \nabla \times \vec A = 0$ imply $\vec A = - \nabla B$ where $\vec A$ is a vector field and $B$ is a scalar field?
I see this in my Electricity and Magnetism textbooks all over the place and I just took it for granted. Is this a theorem or does it come from one?
I can verify in my head that this is valid, but I would love some context.
If $\vec{A}$ has a simply connected domain, any closed loop $C$ encloses a surface $S$ on which Stokes's theorem gives $\int_C \vec{A}\cdot d\vec{r}=\int_S\vec{\nabla}\times \vec{A}\cdot d\vec{S}=0$. But these vanishing loop integrals imply any infinitesimal $\vec{A}\cdot d\vec{r}$ is following a scalar field's value through space, i.e. $\vec{A}=-\vec{\nabla}B$ as required.