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A closed set  $  \Phi $
of points of a space  $  \Omega $
in which there has been specified a [[Vector field|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|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.P. Wills,   "Vector analysis with an introduction to tensor analysis" , Dover, reprint  (1958)  pp. Sect. 45</TD></TR></table>