Editing Vector field (section)

===Gradient field in Euclidean spaces===
[[File:Irrotationalfield.svg|thumb|300px|A vector field that has circulation about a point cannot be written as the gradient of a function.]]
{{further|Gradient}}

Vector fields can be constructed out of [[scalar field]]s using the [[gradient]] operator (denoted by the [[del]]: ∇).<ref>{{cite book|author=Dawber, P.G. | title=Vectors and Vector Operators| publisher=CRC Press| isbn=978-0-85274-585-4| year=1987| page=29 |url=https://books.google.com/books?id=luBlL7oGgUIC&pg=PA29}}</ref>

A vector field ''V'' defined on an open set ''S'' is called a '''gradient field''' or a '''[[conservative field]]''' if there exists a real-valued function (a scalar field) ''f'' on ''S'' such that
<math display="block">V = \nabla f = \left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \frac{\partial f}{\partial x_3}, \dots ,\frac{\partial f}{\partial x_n}\right).</math>

The associated [[Flow (mathematics)|flow]] is called the '''{{visible anchor|gradient flow}}''', and is used in the method of [[gradient descent]].

The [[line integral|path integral]] along any [[closed curve]] ''γ'' (''γ''(0) = ''γ''(1)) in a conservative field is zero:
<math display="block"> \oint_\gamma V(\mathbf {x})\cdot \mathrm{d}\mathbf {x} = \oint_\gamma \nabla f(\mathbf {x}) \cdot \mathrm{d}\mathbf {x}  = f(\gamma(1)) - f(\gamma(0)).</math>