Reference15 articles.
1. For a discussion of the special case in which the only surface singularities are in vs, see N. D. Mermin, “Games to Play with 3He-A,” Physica, to be published (Proceedings of the Sussex Symposium on Superfluid 3He).
2. The analysis here is based on the discussion of vs given by N. D. Mermin and Tin-Lun Ho, Phys. Rev. Lett. 36, 594 (1976).
3. The importance of these considerations for the stability of persistent currents was brought home to me by several very stimulating remarks of P. W. Anderson, and a ferocious lunch-time discussion with M. E. Fisher.
4. For a more detailed exposition of these points, see Ref. 2. It should be emphasized that
$${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {v} _s}$$
and the components of the gradient of
$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\ell }$$
, together with the constraint (3) constitute the standard set of order parameter gradients for the hydro-dynamic description of a system whose order is characterized by an orthonormal triad of vectors. The constraint on the curl of
$${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {v} _s}$$
is simply the local integrability condition insuring that the
$${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {\phi } ^{\left( i \right)}}$$
can be reconstructed from a knowledge of (4) (cont.)
$${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {v} _s}$$
and the gradients of (in analogy to
$$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {\nabla }$$
x
$${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {v} _s}$$
= 0 in 4He-II, which is the constraint necessary to permit the reconstruction of the phase). The field
$${\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}\to {v} _s}$$
appears in the classical differential geometry of surfaces (see, for example, the discussion of the Gauss-Bonnet theorem in Ref. 7). It has been used to discuss singularities in nematics by M. Kleman, Phil. Mag. 27, 1057 (1973), and
5. M. Kleman, J. de Physique 34, 931 (1973).
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