Calculus/Helmholtz Decomposition Theorem

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

Approach #1

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

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

Approach #2

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

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

Approach #3

The Helmholtz decomposition can be derived as follows:

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

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

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

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

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

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

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

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

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