Vector tube

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

The intensity of the tube in the cross-section is the flux (cf. Vector analysis) of across :

where is the unit normal vector to . If the field is solenoidal (), the law of preservation of the intensity of the vector tube holds:

Let be the orthogonal Cartesian coordinates of the vector and let be the coordinates of the point . Then the boundary of is locally defined by an equation , where satisfies the partial differential equation



[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. Yu.P. Pyt'ev (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098