All too often in physics familiarity is a substitute for understanding, and
the beginner who lacks familiarity wonders which is at fault: physics or
himself. Physical mathematics provides well defined concepts and
techniques for the study of physical systems. It is more than mathematical
techniques used in the solution of problems which have already been
formulated; it helps in the very formulation of the laws of physical
systems and brings a better understanding of physics. Thus physical
mathematics includes mathematics which gives promise of being useful in
our analysis of physical phenomena. Attempts to use mathematics for this
purpose may fail because the mathematical tool is too crude; physics may
then indicate along which lines it should be sharpened. In fact, the
analysis of physical systems has spurred many a new mathematical
development.
Considerations of relevance to physics underlie the choice of material
included here. Any choice is necessarily arbitrary; we included first the
topics which we enjoy most but we soon recognized that instant
gratification is a short range criterion. We then included material which can be
appreciated only after a great deal of intellectual asceticism but which may
be farther reaching. Finally, this book gathers the starting points of some
great currents of contemporary mathematics. It is intended for an
advanced physical mathematics course.