---
date: '2024-11-27'
description: decomposition of sufficiently smooth, rapidly decaying vector fields
id: Helmholtz decomposition
modified: 2026-06-05 15:08:30 GMT-04:00
tags:
  - math
title: Helmholtz decomposition
created: '2024-11-27'
published: '2024-11-27'
pageLayout: default
slug: thoughts/Helmholtz-decomposition
permalink: https://aarnphm.xyz/thoughts/Helmholtz-decomposition.md
generator:
  quartz: v4.6.0
  hostedProvider: Cloudflare
  baseUrl: aarnphm.xyz
full: https://aarnphm.xyz/llms-full.txt
---
> certain differentiable vector fields can be resolved into sum of an _irrotational_ vector field and _solenoidal_ vector vield

> \[!definition\] Definition 1.
>
> 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 [[thoughts/Vector calculus#divergence]] of the vector field $R$