Math 2.0

Real functions Definition

Conservative vector field

Builds on Vector fieldGradientC^k function characterised by continuous derivatives

Definition

Let U⊆ℝnU \subseteq \mathbb{R}^n be open, and let F ⁣:U→ℝnF \colon U \to \mathbb{R}^n be a continuous vector field. The field FF is conservative on UU if there is a function ϕ ⁣:U→ℝ\phi \colon U \to \mathbb{R} of class C^1 with F=∇ϕF = \nabla \phi on UU, that is, Fi=∂ϕ/∂xiF_i = \partial \phi / \partial x_i for i=1,…,ni = 1, \dots, n.

Remarks

A function ϕ\phi with ∇ϕ=F\nabla \phi = F is a scalar potential of FF. Physics texts often use −ϕ-\phi instead. The field F(x,y)=(2xy,x2)F(x, y) = (2 x y, x^2) is conservative on ℝ2\mathbb{R}^2, with ϕ(x,y)=x2y\phi(x, y) = x^2 y. The field F(x,y)=(−y,x)F(x, y) = (-y, x) is not conservative on ℝ2\mathbb{R}^2. Its line integral around the unit circle is 2π≠02\pi \ne 0, which contradicts path independence. If FF is conservative and of class C1C^1, then its curl is zero. The converse holds on convex sets but fails in general.

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  15. C^k function characterised by continuous derivatives statement
  16. Vector field definition
  17. Conservative vector field definition

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18 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Conservative_vector_field, Definition
About this page
Tag05R9NF2J49
URIm2://05R9NF2J49
Kinddefinition
WritingW1 stated
MSC26B12
DefinesConservative vector field
UsesVector field, Gradient, C^k function characterised by continuous derivatives

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