Editing Helmholtz decomposition (section)

== Generalization to higher dimensions ==

=== 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" />

=== Tensor approach ===

In a <math>d</math>-dimensional vector space with <math>d\neq 3</math>, <math display="inline">-\frac{1}{4\pi\left|\mathbf{r}-\mathbf{r}'\right|}</math> can be replaced by the appropriate [[Green's function#Green's functions for the Laplacian|Green's function for the Laplacian]], defined by
<math display="block">
\nabla^2 G(\mathbf{r},\mathbf{r}') = \frac{\partial}{\partial r_\mu}\frac{\partial}{\partial r_\mu}G(\mathbf{r},\mathbf{r}') = \delta^d(\mathbf{r}-\mathbf{r}')
</math>
where [[Einstein notation|Einstein summation convention]] is used for the index <math>\mu</math>. For example, <math display="inline">G(\mathbf{r},\mathbf{r}')=\frac{1}{2\pi}\ln\left|\mathbf{r}-\mathbf{r}'\right|</math> in 2D.

Following the same steps as above, we can write
<math display="block">
F_\mu(\mathbf{r}) = \int_V F_\mu(\mathbf{r}') \frac{\partial}{\partial r_\mu}\frac{\partial}{\partial r_\mu}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'
 = \delta_{\mu\nu}\delta_{\rho\sigma}\int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r_\rho}\frac{\partial}{\partial r_\sigma}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'
</math>
where <math>\delta_{\mu\nu}</math> is the [[Kronecker delta]] (and the summation convention is again used). In place of the definition of the vector Laplacian used above, we now make use of an identity for the [[Levi-Civita symbol]] <math>\varepsilon</math>,
<math display="block">
\varepsilon_{\alpha\mu\rho}\varepsilon_{\alpha\nu\sigma} = (d-2)!(\delta_{\mu\nu}\delta_{\rho\sigma} - \delta_{\mu\sigma}\delta_{\nu\rho})
</math>
which is valid in <math>d\ge 2</math> dimensions, where <math>\alpha</math> is a <math>(d-2)</math>-component [[Multi-index notation|multi-index]]. This gives
<math display="block">
F_\mu(\mathbf{r}) = \delta_{\mu\sigma}\delta_{\nu\rho}\int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r_\rho}\frac{\partial}{\partial r_\sigma}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'
+ \frac{1}{(d-2)!}\varepsilon_{\alpha\mu\rho}\varepsilon_{\alpha\nu\sigma} \int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r_\rho}\frac{\partial}{\partial r_\sigma}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'
</math>

We can therefore write
<math display="block">
F_\mu(\mathbf{r}) = -\frac{\partial}{\partial r_\mu} \Phi(\mathbf{r}) + \varepsilon_{\mu\rho\alpha}\frac{\partial}{\partial r_\rho} A_{\alpha}(\mathbf{r})
</math>
where
<math display="block">
\begin{aligned}
\Phi(\mathbf{r}) &= -\int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r_\nu}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'\\
A_{\alpha} &= \frac{1}{(d-2)!}\varepsilon_{\alpha\nu\sigma} \int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r_\sigma}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'
\end{aligned}
</math>
Note that the vector potential is replaced by a rank-<math>(d-2)</math> tensor in <math>d</math> dimensions.

Because <math>G(\mathbf{r},\mathbf{r}')</math> is a function of only <math>\mathbf{r}-\mathbf{r}'</math>, one can replace <math>\frac{\partial}{\partial r_\mu}\rightarrow - \frac{\partial}{\partial r'_\mu}</math>, giving
<math display="block">
\begin{aligned}
\Phi(\mathbf{r}) &= \int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r'_\nu}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'\\
A_{\alpha} &= -\frac{1}{(d-2)!}\varepsilon_{\alpha\nu\sigma} \int_V F_\nu(\mathbf{r}') \frac{\partial}{\partial r_\sigma'}G(\mathbf{r},\mathbf{r}') \,\mathrm{d}^d \mathbf{r}'
\end{aligned}
</math>
[[Integration_by_parts#Higher_dimensions|Integration by parts]] can then be used to give
<math display="block">
\begin{aligned}
\Phi(\mathbf{r}) &= -\int_V G(\mathbf{r},\mathbf{r}')\frac{\partial}{\partial r'_\nu}F_\nu(\mathbf{r}') \,\mathrm{d}^d \mathbf{r}' + \oint_{S} G(\mathbf{r},\mathbf{r}') F_\nu(\mathbf{r}') \hat{n}'_\nu \,\mathrm{d}^{d-1} \mathbf{r}'\\
A_{\alpha} &= \frac{1}{(d-2)!}\varepsilon_{\alpha\nu\sigma} \int_V G(\mathbf{r},\mathbf{r}') \frac{\partial}{\partial r_\sigma'}F_\nu(\mathbf{r}')  \,\mathrm{d}^d \mathbf{r}'- \frac{1}{(d-2)!}\varepsilon_{\alpha\nu\sigma} \oint_{S} G(\mathbf{r},\mathbf{r}') F_\nu(\mathbf{r}')  \hat{n}'_\sigma \,\mathrm{d}^{d-1} \mathbf{r}'
\end{aligned}
</math>
where <math>S=\partial V</math> is the boundary of <math>V</math>. These expressions are analogous to those given above for [[#Three-dimensional_space|three-dimensional space]].

For a further generalization to manifolds, see the discussion of [[Hodge decomposition]] [[#Differential forms|below]].