  The vector potential is a fundamental concept widely applied across various
fields. This paper presents an existence theorem of a vector potential for
divergence-free functions in $W^{m,p}(\mathbb{R}^N,\mathbb{T})$ with general
$m,p,N$. Based on this theorem, one can establish the space decomposition
theorem for functions in $W^{m,p}_0(\operatorname{curl};\Omega,\mathbb{R}^N)$
and the trace theorem for functions in $W^{m,p}(\Omega)$ within the Lipschitz
domain $\Omega \subset \mathbb{R}^N$. The methods of proof employed in this
paper are straightforward, natural, and consistent.
