# Vector tube

A closed set $ \Phi $
of points of a space $ \Omega $
in which there has been specified a vector field $ \mathbf a ( M) $
such that the normal vector $ \mathbf n $
is orthogonal to $ \mathbf a $
everywhere on its boundary surface $ S $.
The vector tube $ \Phi $
consists of vector lines $ \Gamma $
of the field $ \mathbf a $,
i.e. curves in $ \Omega $
at each point of which the tangent direction coincides with the direction of $ \mathbf a $.
A line $ \Gamma $
is completely contained in $ \Phi $
if one point of $ \Gamma $
is contained in $ \Phi $.
If $ \mathbf a $
is the field of velocities of a stationary liquid flow, then $ \Gamma $
is the trajectory of the liquid particles, while $ \Phi $
is the part of $ \Omega $"
swept along" by the motion of a given amount of liquid particles.

The intensity $ I $ of the tube $ \Phi $ in the cross-section $ S ^ \prime $ is the flux (cf. Vector analysis) of $ \mathbf a $ across $ S ^ \prime $:

$$ I( S ^ \prime ) = \int\limits \int\limits S ^ \prime ( \mathbf a , \mathbf n ) d \sigma , $$

where $ \mathbf n $ is the unit normal vector to $ S ^ \prime $. If the field $ \mathbf a $ is solenoidal ( $ \mathop{\rm div} \mathbf a = 0 $), the law of preservation of the intensity of the vector tube holds:

$$ I( S ^ \prime ) = I( S ^ {\prime\prime} ) . $$

Let $ a _ {1} ( x, y, z), a _ {2} ( x, y, z), a _ {3} ( x, y, z) $ be the orthogonal Cartesian coordinates of the vector $ \mathbf a = \mathbf a ( M), $ and let $ x, y, z $ be the coordinates of the point $ M $. Then the boundary of $ \Phi $ is locally defined by an equation $ F( x, y, z) = \textrm{ const } $, where $ F( x, y, z) $ satisfies the partial differential equation

$$ ( \mathbf a , \nabla F ) = $$

$$ = \ a _ {1} ( x, y, z) \frac{\partial F }{\partial x } + a _ {2} ( x, y, z) \frac{\partial F }{\partial y } + a _ {3} ( x, y, z) \frac{\partial F }{\partial z } = 0. $$

#### Comments

#### References

[a1] | A.P. Wills, "Vector analysis with an introduction to tensor analysis" , Dover, reprint (1958) pp. Sect. 45 |

**How to Cite This Entry:**

Vector tube.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Vector_tube&oldid=49142