On permanent vector-lines in 𝑁 dimensions

Abstract

Prim and Truesdell have recently given a simple vector proof of Zorawski's condition for the permanence of vector-lines in a moving fluid.' Their argument is necessarily by its form confined to threedimensional space, and it seems of interest to investigate the problem in a Euclidean space of N dimensions, using a slightly different approach. We consider a Euclidean space EN with rectangular cartesians xi. Two vector fields are given, vi(x, t), ci(x, t), where t is a parameter (the time); we shall call v* the primary and ci the secondary vector fields; vi plays the part of velocity. Any vector field defines for given t a congruence of curves in an obvious way, the direction of the curve at each point coinciding with that of the vector field. We are not particularly interested here in the congruence defined by the primary vector field (stream-lines), and we shall use the expression vector-lines to refer exclusively to the congruence defined by the secondary field, that is, those curves which satisfy