Calculus/Helmholtz Decomposition Theorem

The Helmholtz Decomposition Theorem, regarded as the fundamental theorem of vector calculus, dictates that any vector field ${\displaystyle \mathbf {F} }$ can be expressed as the sum of a conservative vector field ${\displaystyle \mathbf {G} }$ and a divergence free vector field ${\displaystyle \mathbf {H} }$: ${\displaystyle \mathbf {F} =\mathbf {G} +\mathbf {H} }$.

Given a vector field ${\displaystyle \mathbf {F} }$, the vector field ${\displaystyle \mathbf {G} (\mathbf {q} )={\frac {1}{4\pi }}\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {(\nabla \cdot \mathbf {F} )|_{\mathbf {q} '}(\mathbf {q} -\mathbf {q} ')}{|\mathbf {q} -\mathbf {q} '|^{3}}}dV'}$ has the same divergence as ${\displaystyle \mathbf {F} }$, and is also conservative: ${\displaystyle \nabla \cdot \mathbf {G} =\nabla \cdot \mathbf {F} }$ and ${\displaystyle \nabla \times \mathbf {G} =\mathbf {0} }$. The vector field ${\displaystyle \mathbf {H} =\mathbf {F} -\mathbf {G} }$ is divergence free.

Therefore ${\displaystyle \mathbf {F} =\mathbf {G} +\mathbf {H} }$ where ${\displaystyle \mathbf {G} (\mathbf {q} )={\frac {1}{4\pi }}\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {(\nabla \cdot \mathbf {F} )|_{\mathbf {q} '}(\mathbf {q} -\mathbf {q} ')}{|\mathbf {q} -\mathbf {q} '|^{3}}}dV'}$ and ${\displaystyle \mathbf {H} =\mathbf {F} -\mathbf {G} }$. Vector field ${\displaystyle \mathbf {G} }$ is conservative and ${\displaystyle \mathbf {H} }$ is divergence free.

Given a vector field ${\displaystyle \mathbf {F} }$, the vector field ${\displaystyle \mathbf {H} (\mathbf {q} )={\frac {1}{4\pi }}\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {(\nabla \times \mathbf {F} )|_{\mathbf {q} '}\times (\mathbf {q} -\mathbf {q} ')}{|\mathbf {q} -\mathbf {q} '|^{3}}}dV'}$ has the same curl as ${\displaystyle \mathbf {F} }$, and is also divergence free: ${\displaystyle \nabla \times \mathbf {H} =\nabla \times \mathbf {F} }$ and ${\displaystyle \nabla \cdot \mathbf {H} =0}$. The vector field ${\displaystyle \mathbf {G} =\mathbf {F} -\mathbf {H} }$ is conservative.

Therefore ${\displaystyle \mathbf {F} =\mathbf {G} +\mathbf {H} }$ where ${\displaystyle \mathbf {H} (\mathbf {q} )={\frac {1}{4\pi }}\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {(\nabla \times \mathbf {F} )|_{\mathbf {q} '}\times (\mathbf {q} -\mathbf {q} ')}{|\mathbf {q} -\mathbf {q} '|^{3}}}dV'}$ and ${\displaystyle \mathbf {G} =\mathbf {F} -\mathbf {H} }$. Vector field ${\displaystyle \mathbf {G} }$ is conservative and ${\displaystyle \mathbf {H} }$ is divergence free.

The Helmholtz decomposition can be derived as follows:

Given an arbitrary point ${\displaystyle \mathbf {q} '}$, the divergence of the vector field ${\displaystyle {\frac {\mathbf {q} -\mathbf {q} '}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}}$ is ${\displaystyle \nabla _{\mathbf {q} }\cdot {\frac {\mathbf {q} -\mathbf {q} '}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}=\delta (\mathbf {q} ;\mathbf {q} ')}$ where ${\displaystyle \delta (\mathbf {q} ;\mathbf {q} ')}$ is the Dirac delta function centered on ${\displaystyle \mathbf {q} '}$ (The subscript ${\displaystyle _{\mathbf {q} }}$ clarifies that ${\displaystyle \mathbf {q} }$ as opposed to ${\displaystyle \mathbf {q} '}$ is the parameter that the differential operator is being applied to). Since ${\displaystyle \nabla _{\mathbf {q} }({\frac {-1}{|\mathbf {q} -\mathbf {q} '|}})={\frac {\mathbf {q} -\mathbf {q} '}{|\mathbf {q} -\mathbf {q} '|^{3}}}}$, it is the case that ${\displaystyle \nabla _{\mathbf {q} }^{2}{\frac {-1}{4\pi |\mathbf {q} -\mathbf {q} '|}}=\nabla _{\mathbf {q} }\cdot {\frac {\mathbf {q} -\mathbf {q} '}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}=\delta (\mathbf {q} ;\mathbf {q} ')}$

Alongside the identities ${\displaystyle \nabla \cdot (f\mathbf {G} )=(\nabla f)\cdot \mathbf {G} +f(\nabla \cdot \mathbf {G} )}$, and ${\displaystyle \nabla \times (f\mathbf {G} )=(\nabla f)\times \mathbf {G} +f(\nabla \times \mathbf {G} )}$, and most importantly ${\displaystyle \nabla \times (\nabla \times \mathbf {F} )=\nabla (\nabla \cdot \mathbf {F} )-\nabla ^{2}\mathbf {F} }$, the following can be derived:

${\displaystyle \mathbf {F} (\mathbf {q} )=\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}\delta (\mathbf {q} ;\mathbf {q} ')\mathbf {F} (\mathbf {q} ')dV'}$ ${\displaystyle =\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}(\nabla _{\mathbf {q} }^{2}{\frac {-1}{4\pi |\mathbf {q} -\mathbf {q} '|}})\mathbf {F} (\mathbf {q} ')dV'}$ ${\displaystyle =\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}(\nabla _{\mathbf {q} }^{2}{\frac {-\mathbf {F} (\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|}})dV'}$

${\displaystyle =\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}(\nabla _{\mathbf {q} }(\nabla _{\mathbf {q} }\cdot {\frac {-\mathbf {F} (\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|}})-\nabla _{\mathbf {q} }\times (\nabla _{\mathbf {q} }\times {\frac {-\mathbf {F} (\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|}}))dV'}$

${\displaystyle =\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}(\nabla _{\mathbf {q} }({\frac {(\mathbf {q} -\mathbf {q} ')\cdot \mathbf {F} (\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}})-\nabla _{\mathbf {q} }\times ({\frac {(\mathbf {q} -\mathbf {q} ')\times \mathbf {F} (\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}))dV'}$

${\displaystyle =\nabla _{\mathbf {q} }\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\cdot (\mathbf {q} -\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}dV'+\nabla _{\mathbf {q} }\times \iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\times (\mathbf {q} -\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}dV'}$

${\displaystyle \mathbf {G} (\mathbf {q} )=\nabla _{\mathbf {q} }\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\cdot (\mathbf {q} -\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}dV'}$ is the gradient of a scalar field, and so is conservative.

${\displaystyle \mathbf {H} (\mathbf {q} )=\nabla _{\mathbf {q} }\times \iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\times (\mathbf {q} -\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}dV'}$ is the curl of a vector field, and so is divergence free.

In summary, ${\displaystyle \mathbf {F} =\mathbf {G} +\mathbf {H} }$ where ${\displaystyle \mathbf {G} (\mathbf {q} )=\nabla _{\mathbf {q} }\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\cdot (\mathbf {q} -\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}dV'}$ is conservative and ${\displaystyle \mathbf {H} (\mathbf {q} )=\nabla _{\mathbf {q} }\times \iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\times (\mathbf {q} -\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}dV'}$ is divergence free.