  For $\partial \Omega$ the boundary of a bounded and connected strongly
Lipschitz domain in $\mathbb{R}^{d}$ with $d\geq3$, we prove that any field
$f\in L^{2} (\partial \Omega ; \mathbb{R}^{d})$ decomposes, in an unique way,
as the sum of three silent vector fields---fields whose magnetic potential
vanishes in one or both components of $\mathbb{R}^d\setminus\partial \Omega$.
Moreover, this decomposition is orthogonal if and only if $\partial \Omega$ is
a sphere. We also show that any $f$ in $L^{2} (\partial \Omega ;
\mathbb{R}^{d})$ is uniquely the sum of two silent fields and a Hardy function,
in which case the sum is orthogonal regardless of $\partial \Omega$; we express
the corresponding orthogonal projections in terms of layer potentials. When
$\partial \Omega$ is a sphere, both decompositions coincide and match what has
been called the Hardy-Hodge decomposition in the literature.
