Introduction

Multiplicative integration was initiated by Volterra [33] in 1887 as a method of solving systems of linear differential equations. The early history of the subject bears the impress of this initiation. Schlesinger, who took up the further development of the theory and became its chief proponent, stressed the link with differential equations in his 1908 lectures on the subject [26]. But the historical growth of mathematics only roughly displays its authentic design as a logical edifice. From a logical or architectural standpoint multiplicative integration belongs to the area of Lie groups and Lie algebras. This was perceived by Birkhoff [2] in the late 1930s, and has been borne out by subsequent work. Accordingly, our first task here will be to lay bare the intrinsic Lie group-theoretic aspect of the subject. For this it is very convenient, as again Birkhoff indicated [2, Sec. 4], to consider the kinematics of fluid flows. It is with this that we shall begin.

Our exposition will not be rigorous in every detail. We shall not, for instance, define what “smooth” means in each case. Nor shall we tarry over questions concerning the existence or interchange of limits. We shall assume that the reader has the empathy to see that our assertions can be made correct by the imposition of reasonable restraints and that rigorization is feasible.