Editing Conservative vector field (section)

==Irrotational vector fields==

[[File:Irrotational vector field.svg|300px|thumbnail|right|The above vector field <math>\mathbf{v} = \left( - \frac{y}{x^2 + y^2},\frac{x}{x^2 + y^2},0 \right)</math> defined on <math>U = \R^3 \setminus \{ (0,0,z) \mid z \in \R \}</math>, i.e., <math>\R^3</math> with removing all coordinates on the <math>z</math>-axis (so not a simply connected space), has zero curl in <math>U</math> and is thus irrotational. However, it is not conservative and does not have path independence.]]

Let <math>n = 3</math> (3-dimensional space), and let <math>\mathbf{v}: U \to \R^3</math> be a <math>C^1</math> ([[Smoothness#Multivariate differentiability classes|continuously differentiable]]) vector field, with an open subset <math>U</math> of <math>\R^n</math>. Then <math>\mathbf{v}</math> is called irrotational if  its [[Curl (mathematics)|curl]] is <math>\mathbf{0}</math> everywhere in <math>U</math>, i.e., if
<math display="block">\nabla \times \mathbf{v} \equiv \mathbf{0}.</math>

For this reason, such vector fields are sometimes referred to as curl-free vector fields or curl-less vector fields. They are also referred to as [[Helmholtz decomposition#Longitudinal and transverse fields |longitudinal vector fields]].

It is an [[Vector calculus identities#Curl of gradient is zero|identity of vector calculus]] that for any <math>C^2</math> ([[Smoothness#Multivariate differentiability classes|continuously differentiable up to the 2nd derivative]]) scalar field <math>\varphi</math> on <math>U</math>, we have
<math display="block">\nabla \times (\nabla \varphi) \equiv \mathbf{0}.</math>

Therefore, ''every <math>C^1</math> conservative vector field in <math>U</math> is also an irrotational vector field in <math>U</math>''. This result can be easily proved by expressing <math>\nabla \times (\nabla \varphi)</math> in a [[Cartesian coordinate system]] with [[Symmetry of second derivatives#Schwarz's theorem|Schwarz's theorem]] (also called Clairaut's theorem on equality of mixed partials).

Provided that <math>U</math> is a [[simply connected space|simply connected open space]] (roughly speaking, a single piece open space without a hole within it), the converse of this is also true: ''Every irrotational vector field in a simply connected open space <math>U</math> is a <math>C^1</math> conservative vector field in <math>U</math>''.

The above statement is ''not'' true in general if <math>U</math> is not simply connected. Let <math>U</math> be <math>\R^3</math> with removing all coordinates on the <math>z</math>-axis (so not a simply connected space), i.e., <math>U = \R^3 \setminus \{ (0,0,z) \mid z \in \R \}</math>. Now, define a vector field <math>\mathbf{v}</math> on <math>U</math> by
<math display="block">\mathbf{v}(x,y,z) ~ \stackrel{\text{def}}{=} ~ \left( - \frac{y}{x^2 + y^2},\frac{x}{x^2 + y^2},0 \right).</math>

Then <math>\mathbf{v}</math> has zero curl everywhere in <math>U</math> (<math>\nabla \times \mathbf{v} \equiv \mathbf{0}</math> at everywhere in <math>U</math>), i.e., <math>\mathbf{v}</math> is irrotational. However, the [[Circulation (physics)|circulation]] of <math>\mathbf{v}</math> around the [[unit circle]] in the <math>xy</math>-plane is <math>2 \pi</math>; in [[polar coordinates]], <math>\mathbf{v} = \mathbf{e}_{\phi} / r</math>, so the integral over the unit circle is
<math display="block">\oint_{C} \mathbf{v} \cdot \mathbf{e}_{\phi} ~ d{\phi} = 2 \pi.</math>

Therefore, <math>\mathbf{v}</math> does not have the path-independence property discussed above so is not conservative even if <math>\nabla \times \mathbf{v} \equiv \mathbf{0}</math> since <math>U</math> where <math>\mathbf{v}</math> is defined is not a simply connected open space. 

Say again, in a simply connected open region, an irrotational vector field <math>\mathbf{v}</math> has the path-independence property (so <math>\mathbf{v}</math> as conservative). This can be proved directly by using [[Stokes' theorem]],<math display="block">\oint _{P_c} \mathbf{v} \cdot d \mathbf {r} = \iint _{A}(\nabla \times \mathbf{v})\cdot d \mathbf {a} = 0</math>for any smooth oriented surface <math>A</math> which boundary is a simple closed path <math>P_c</math>. So, it is concluded that ''In a simply connected open region, any'' <math>C^1</math> ''vector field that has the path-independence property (so it is a conservative vector field.) must also be irrotational and vice versa.''

=== Abstraction ===
More abstractly, in the presence of a [[Riemannian metric]], vector fields correspond to [[differential form|differential {{nowrap|<math>1</math>-forms}}]]. The conservative vector fields correspond to the [[closed and exact differential forms|exact]] {{nowrap|<math>1</math>-forms}}, that is, to the forms which are the [[exterior derivative]] <math>d\phi</math> of a function (scalar field) <math>\phi</math> on <math>U</math>. The irrotational vector fields correspond to the [[closed and exact differential forms|closed]] {{nowrap|<math>1</math>-forms}}, that is, to the {{nowrap|<math>1</math>-forms}} <math>\omega</math> such that <math>d\omega = 0</math>. As {{nowrap|<math>d^2 = 0</math>,}} any exact form is closed, so any conservative vector field is irrotational. Conversely, all closed {{nowrap|<math>1</math>-forms}} are exact if <math>U</math> is [[simply connected]].

=== Vorticity ===
{{main article|Vorticity}}

The [[vorticity]] <math>\boldsymbol{\omega}</math> of a vector field can be defined by:
<math display="block">\boldsymbol{\omega} ~ \stackrel{\text{def}}{=} ~ \nabla \times \mathbf{v}.</math>

The vorticity of an irrotational field is zero everywhere.<ref>{{citation|title = Elements of Gas Dynamics|first1 = H.W.|last1 = Liepmann|author-link1 = Hans W. Liepmann|first2 = A.|last2 = Roshko|author-link2 = Anatol Roshko|publisher = Courier Dover Publications|year = 1993|orig-year = 1957|isbn = 0-486-41963-0}}, pp. 194–196.</ref> [[Kelvin's circulation theorem]] states that a fluid that is irrotational in an [[inviscid flow]] will remain irrotational. This result can be derived from the [[vorticity transport equation]], obtained by taking the curl of the [[Navier–Stokes equations]].

For a two-dimensional field, the vorticity acts as a measure of the ''local'' rotation of fluid elements. The vorticity does ''not'' imply anything about the global behavior of a fluid. It is possible for a fluid that travels in a straight line to have vorticity, and it is possible for a fluid that moves in a circle to be irrotational.