On the determination of the rotation from the stretch

In nonlinear kinematics the problem of determining the rotation from the stretch can be formulated through a system of partial differential equations in the group of rotations or in the space of skew-symmetric tensors. We examine the relationships between these two systems of equations with the aim of seeing under what circumstances they can be considered equivalent in the sense that they have the same integrability conditions and the solutions of one system correspond to those of the other. Then, we express the problem in the rotation group by means of a set of equations having a rather simple form, and show that its solutions give the sought rotation fields. We present two examples of integration of the proposed equations: the first one refers to a class of problems whose solution is known and has been obtained from the system in the space of skew tensors; the second one refers to a more general case.