stylistic:style_transfer, information:classification
Definition

A vector field v: U → R n {\displaystyle \mathbf {v}:U\to \mathbb {R} ^{n}}, where U {\displaystyle U} is an open subset of R n {\displaystyle \mathbb {R} ^{n}}, is said to be conservative if there exists a C 1 {\displaystyle C^{1}} (continuously differentiable) scalar field φ {\displaystyle \varphi } on U {\displaystyle U} such that

Here, ∇ φ {\displaystyle \nabla \varphi } denotes the gradient of φ {\displaystyle \varphi }. Since φ {\displaystyle \varphi } is continuously differentiable, v {\displaystyle \mathbf {v} } is continuous. When the equation above holds, φ {\displaystyle \varphi } is called a scalar potential for v {\displaystyle \mathbf {v} }.

The 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.