Editing Conservative vector field (section)

==Definition==

A [[vector field]] <math>\mathbf{v}: U \to \R^n</math>, where <math>U</math> is an open subset of <math>\R^n</math>, is said to be conservative if  there exists a <math>C^1</math> ([[Smoothness#Multivariate differentiability classes|continuously differentiable]]) [[scalar field]] <math>\varphi</math><ref name=":1">For <math>\mathbf{v} = \nabla \varphi</math> to be [[Conservative vector field#Path independence|path-independent]], <math>\varphi</math> is not necessarily continuously differentiable, the condition of being differentiable is enough, since the [[Gradient theorem]], that proves the path independence of <math>\nabla \varphi</math>, does not require <math>\varphi</math> to be continuously differentiable.

There must be a reason for the definition of conservative vector fields to require <math>\varphi</math> to be [[Smoothness#Multivariate differentiability classes|continuously differentiable]].</ref> on <math>U</math> such that

<math display="block">\mathbf{v} = \nabla \varphi.</math>

Here, <math>\nabla \varphi</math> denotes the [[gradient]] of <math>\varphi</math>. Since <math>\varphi</math> is continuously differentiable, <math>\mathbf{v}</math> is continuous. When the equation above holds, <math>\varphi</math> is called a [[scalar potential]] for <math>\mathbf{v}</math>.

The [[Helmholtz decomposition|fundamental theorem of vector calculus]] states that, under some regularity conditions, any vector field can be expressed as the sum of a conservative vector field and a [[solenoidal field]].