---
tag: 05R9NF2J49
kind: definition
name: Conservative vector field
msc: [26B12]
defines: 0PKG97PX9Y
uses: [0N67NBKQ5T, 0NS2QYVC4Y, 0TCBZMEYGN]
sources:
  - url: https://en.wikipedia.org/wiki/Conservative_vector_field
    locator: Definition
---

## Definition

Let $U \subseteq \mathbb{R}^n$ be open, and let $F \colon U \to \mathbb{R}^n$ be a continuous [[vector field|0N67NBKQ5T]].
The field $F$ is conservative on $U$ if there is a function $\phi \colon U \to \mathbb{R}$ of [[class C^1|0TCBZMEYGN]] with $F = \nabla \phi$ on $U$, that is, $F_i = \partial \phi / \partial x_i$ for $i = 1, \dots, n$.

## Remarks

A function $\phi$ with $\nabla \phi = F$ is a [[scalar potential|0FP4BVDAAQ]] of $F$. Physics texts often use $-\phi$ instead.
The field $F(x, y) = (2 x y, x^2)$ is conservative on $\mathbb{R}^2$, with $\phi(x, y) = x^2 y$.
The field $F(x, y) = (-y, x)$ is not conservative on $\mathbb{R}^2$. Its line integral around the unit circle is $2\pi \ne 0$, which contradicts [[path independence|0BFN21TF5F]].
If $F$ is conservative and of class $C^1$, then its [[curl is zero|0M2JRW1J0N]]. The converse holds on convex sets but [[fails in general|0GRM3BS0DD]].

## Explanations

### L1 human

A conservative field is the slope field of some landscape $\phi$.
Going around a loop in a landscape brings you back to the same height. So the total work of the field around the loop is zero.