Helmholtz decomposition

certain differentiable vector fields can be resolved into sum of an irrotational vector field and solenoidal vector vield

definition

for a vector field $\mathbf{F} \in C^1 (V, \mathbb{R}^n)$ defined on a domain $V \subseteq \mathbb{R}^n$, a Helmholtz decomposition is a pair of vector fields $\mathbf{G} \in C^1 (V, \mathbb{R}^n)$ and $\mathbf{R} \in C^1 (V, \mathbb{R}^n)$ such that:

$$\begin{aligned} \mathbf{F}(\mathbf{r}) &= \mathbf{G}(\mathbf{r}) + \mathbf{R}(\mathbf{r}) \\ \mathbf{G}(\mathbf{r}) &= - \nabla \Phi (\mathbf{r}) \\ \nabla \cdot \mathbf{R}(\mathbf{r}) &= 0 \end{aligned}$$

Here $\Phi \in C^2(V, \mathbb{R})$ is a scalar potential, $\nabla \Phi$ is its gradient, and $\nabla \cdot \mathbf{R}$ is the divergence of the vector field $R$