Editing Helmholtz decomposition (section)

=== Matrix approach ===

The generalization to <math>d</math> dimensions cannot be done with a vector potential, since the rotation operator and the [[cross product]] are defined (as vectors) only in three dimensions. 

Let <math>\mathbf{F}</math> be a vector field on a bounded domain <math>V\subseteq\mathbb{R}^d</math> which decays faster than <math>|\mathbf{r}|^{-\delta}</math> for <math>|\mathbf{r}| \to \infty</math> and <math>\delta > 2</math>.

The scalar potential is defined similar to the three dimensional case as:
<math display="block">\Phi(\mathbf{r}) = - \int_{\mathbb{R}^d} \operatorname{div}(\mathbf{F}(\mathbf{r}')) K(\mathbf{r}, \mathbf{r}') \mathrm{d}V' = - \int_{\mathbb{R}^d} \sum_i \frac{\partial F_i}{\partial r_i}(\mathbf{r}') K(\mathbf{r}, \mathbf{r}') \mathrm{d}V',</math>
where as the integration kernel <math>K(\mathbf{r}, \mathbf{r}')</math> is again the [[fundamental solution]] of [[Laplace's equation]], but in d-dimensional space:
<math display="block">K(\mathbf{r}, \mathbf{r}') = \begin{cases}  \frac{1}{2\pi} \log{ | \mathbf{r}-\mathbf{r}' | } &  d=2,  \\    \frac{1}{d(2-d)V_d} | \mathbf{r}-\mathbf{r}' | ^{2-d} &  \text{otherwise}, \end{cases}</math>
with <math>V_d = \pi^\frac{d}{2} / \Gamma\big(\tfrac{d}{2}+1\big)</math> the volume of the d-dimensional [[unit ball]]s and <math>\Gamma(\mathbf{r})</math> the [[gamma function]].

For <math>d = 3</math>, <math>V_d</math> is just equal to <math>\frac{4 \pi}{3}</math>, yielding the same prefactor as above.
The rotational potential is an [[antisymmetric matrix]] with the elements:
<math display="block">A_{ij}(\mathbf{r}) = \int_{\mathbb{R}^d} \left( \frac{\partial F_i}{\partial x_j}(\mathbf{r}') - \frac{\partial F_j}{\partial x_i}(\mathbf{r}') \right) K(\mathbf{r}, \mathbf{r}') \mathrm{d}V'. </math>
Above the diagonal are <math>\textstyle\binom{d}{2}</math> entries which occur again mirrored at the diagonal, but with a negative sign.
In the three-dimensional case, the matrix elements just correspond to the components of the vector potential <math>\mathbf{A} = [A_1, A_2, A_3] = [A_{23}, A_{31}, A_{12}]</math>.
However, such a matrix potential can be written as a vector only in the three-dimensional case, because <math>\textstyle\binom{d}{2} = d</math> is valid only for <math>d = 3</math>.

As in the three-dimensional case, the gradient field is defined as
<math display="block">
\mathbf{G}(\mathbf{r}) = - \nabla \Phi(\mathbf{r}).
</math>
The rotational field, on the other hand, is defined in the general case as the row divergence of the matrix:
<math display="block">\mathbf{R}(\mathbf{r}) = \left[ \sum\nolimits_k \partial_{r_k} A_{ik}(\mathbf{r}); {1 \leq i \leq d} \right].</math>
In three-dimensional space, this is equivalent to the rotation of the vector potential.<ref name="glotzl2023" /><ref name="glotzl2020" />