Editing Vector field (section)

==Operations on vector fields==

===Line integral===
{{Main|Line integral}}
A common technique in physics is to integrate a vector field along a [[differential geometry of curves|curve]], also called determining its [[line integral]]. Intuitively this is summing up all vector components in line with the tangents to the curve, expressed as their scalar products. For example, given a particle in a force field (e.g. gravitation), where each vector at some point in space represents the force acting there on the particle, the line integral along a certain path is the work done on the particle, when it travels along this path. Intuitively, it is the sum of the scalar products of the force vector and the small tangent vector in each point along the curve.

The line integral is constructed analogously to the [[Riemann integral]] and it exists if the curve is rectifiable (has finite length) and the vector field is continuous.

Given a vector field {{mvar|V}} and a curve {{mvar|γ}}, [[parametric equation|parametrized]] by {{mvar|t}} in {{closed-closed|''a'', ''b''}} (where {{mvar|a}} and {{mvar|b}} are [[real number]]s), the line integral is defined as
<math display="block">\int_\gamma V(\mathbf {x}) \cdot \mathrm{d}\mathbf {x}  = \int_a^b  V(\gamma(t)) \cdot \dot \gamma(t)\, \mathrm{d}t.</math>

To show vector field topology one can use [[line integral convolution]].

===Divergence===
{{Main|Divergence}}
The [[divergence]] of a vector field on Euclidean space is a function (or scalar field).  In three-dimensions, the divergence is defined by
<math display="block">\operatorname{div} \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z},</math>

with the obvious generalization to arbitrary dimensions.  The divergence at a point represents the degree to which a small volume around the point is a [[sources and sinks|source or a sink]] for the vector flow, a result which is made precise by the [[divergence theorem]].

The divergence can also be defined on a [[Riemannian manifold]], that is, a manifold with a [[Riemannian metric]] that measures the length of vectors.

===Curl in three dimensions===
{{Main|Curl  (mathematics)}}
The [[Curl (mathematics)|curl]] is an operation which takes a vector field and produces another vector field. The curl is defined only in three dimensions, but some properties of the curl can be captured in higher dimensions with the [[exterior derivative]].  In three dimensions, it is defined by
<math display="block">\operatorname{curl}\mathbf{F} = \nabla \times \mathbf{F} = \left(\frac{\partial F_3}{\partial y} - \frac{\partial F_2}{\partial z}\right)\mathbf{e}_1 - \left(\frac{\partial F_3}{\partial x} - \frac{\partial F_1}{\partial z}\right)\mathbf{e}_2 + \left(\frac{\partial F_2}{\partial x}- \frac{\partial F_1}{\partial y}\right)\mathbf{e}_3.</math>

The curl measures the density of the [[angular momentum]] of the vector flow at a point, that is, the amount to which the flow circulates around a fixed axis.  This intuitive description is made precise by [[Stokes' theorem]].

===Index of a vector field===
The index of a vector field is an integer that helps describe its behaviour around an isolated zero (i.e., an isolated singularity of the field). In the plane, the index takes the value −1 at a saddle singularity but +1 at a source or sink singularity.

Let  ''n be'' the dimension of the manifold on which the vector field is defined. Take a closed surface (homeomorphic to the (n-1)-sphere) S around the zero, so that no other zeros lie in the interior of S. A map from this sphere to a unit sphere of dimension ''n''&nbsp;−&nbsp;1 can be constructed by dividing each vector on this sphere by its length to form a unit length vector, which is a point on the unit sphere S<sup>''n''−1</sup>. This defines a continuous map from S to S<sup>''n''−1</sup>. The index of the vector field at the point is the [[Degree of a continuous mapping#Differential topology|degree]] of this map. It can be shown that this integer does not depend on the choice of S, and therefore depends only on the vector field itself.

The index is not defined at any non-singular point (i.e., a point where the vector is non-zero). It is equal to +1 around a source, and more generally equal to (−1)<sup>''k''</sup> around a saddle that has ''k'' contracting dimensions and ''n''−''k'' expanding dimensions.

'''The index of the vector field''' as a whole is defined when it has just finitely many zeroes. In this case, all zeroes are isolated, and the index of the vector field is defined to be the sum of the indices at all zeroes.

For an ordinary (2-dimensional) sphere in three-dimensional space, it can be shown that the index of any vector field on the sphere must be 2. This shows that every such vector field must have a zero. This implies the [[hairy ball theorem]].

For a vector field on a compact manifold with finitely many zeroes, the [[Poincaré-Hopf theorem]] states that the vector field’s index is the manifold’s [[Euler characteristic]].