(4.71) |

A *vortex line* is a line whose tangent is everywhere parallel to the local vorticity vector.
The vortex lines drawn through each point of a closed curve
constitute the surface of a *vortex tube*. Finally, a
*vortex filament* is a vortex tube whose cross-section is of infinitesimal dimensions.

Consider a section
of a vortex filament. The filament is bounded by the curved surface that forms the filament
wall, as well as two plane surfaces, whose vector areas are
and
(say), which form the ends of the section at
points
and
, respectively. (See Figure 4.9.) Let the plane surfaces have outward pointing normals
that are parallel (or anti-parallel) to the vorticity vectors,
**
**
and
**
**
, at points
and
,
respectively.
The divergence theorem (see Section A.20), applied to the section, yields

(4.72) |

where is an outward directed surface element, and a volume element. However,

(4.73) |

[see Equation (A.173)], implying that

(4.74) |

Now,

(4.75) |

This result is essentially an equation of continuity for vortex filaments. It implies that the product of the magnitude of the vorticity and the cross-sectional area, which is termed the

Because a vortex tube can be regarded as a bundle of vortex filaments whose net intensity is the sum of the intensities of the constituent filaments, we conclude that the intensity of a vortex tube remains constant along the tube.