Helmholtz's decomposition theorem

In vector calculus, Helmholtz’s decomposition theorem states that any vector field in three dimensions can be decomposed into the superposition of a curl-free field and a divergence-free field (called the Helmholtz decomposition or the Helmholtz representation). Formally:

$$\vec{F}={\vec{F}}_{flux}+{\vec{F}}_{circ}$$ $$\vec{F}=-\nabla \varphi +\nabla \times \vec{A}$$

The flux field ($-\nabla \varphi$) is called the irrotational or conservative field, where $\varphi$ is the scalar potential. The circulation field ($\nabla \times \vec{A}$) is called the source-free or solenoidal field.

I think it’s good to think about the flux field like an electric field, just like how the field lines radiate out from a positive point charge towards a negative point charge. Similarly for the divergence field like a magnetic field, where the field lines rotate. This is in fact the set-up for Maxwell’s equations.

Computations

If we take the curl of the field, then:

$$\nabla \times \vec{F}=-\nabla \times \nabla \varphi +\nabla \times \nabla \times \vec{A}$$ $$\nabla \times (\nabla \times \vec{A})={\mu }_0\vec{J}$$

where $\vec{J}$ is some flux density. If we take the divergence of the field:

$$\nabla \cdot \vec{F}=-\nabla \cdot \nabla \varphi =\rho$$

where $\rho$ is the scalar density.