Editing Vector field (section)

===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]].