Choquet-Bruhat Y., DeWitt-Morette C., Dillard-Bleick M. - Analysis, Manifolds and Physics Vol.1
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 techni-
ques 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 gratifica-
tion 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.
Chapters I and II are two preliminary chapters included here to spare
the reader the task of looking up in several specialized books the
definitions and the basic theorems used in the subsequent chapters.
Chapter I is merely a review of fundamental notions of algebra, topology,
integration, and analysis. Chapter II treats the essentials of differential
calculus and calculus of variations on Banach spaces. Each of the
following chapters introduces a mathematical structure and exploits it
until it is sufficiently familiar to become an "instrument de pens6e":
Chapter III, differentiable manifolds, tangent bundles and their use in Lie
groups; Chapter IV, exterior derivation and the solutions of exterior
differential systems; Chapter V, Riemannian structures which, together
with the previous structures provide the basic geometric notions needed
in physics; Chapter VI, distributions and the Sobolev spaces with recent
applications to the theory of partial differential equations. The last chapter covers some selected topics in the theory of infinite dimensional
manifolds.
At the end of each chapter, several problems are worked out. Most of
them show how the concepts and the theorems introduced in the text can
be used in physics. They should be of interest both to physicists and
mathematicians. A sentence like "The Lagrangian is a function defined on
the tangent bundle of the configuration space" helps explain to the
physicist what a tangent bundle is and tells a mathematician what a
Lagrangian is. A sentence like "The strain tensor is the Lie derivative of
the metric with respect to the deformation" helps a physicist to under-
stand the concept of Lie derivatives and defines the strain tensor to a
mathematician. To both, they bring an added pleasure.
The pleasure of physical mathematics is well described by Hilbert:
learning that some genetic laws of the fruit fly had been derived by the
application of a certain set of axioms he exclaimed "So simple and precise
and at the same time so miraculous that no daring fantasy could have
imagined it "
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 techni-
ques 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 gratifica-
tion 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.
Chapters I and II are two preliminary chapters included here to spare
the reader the task of looking up in several specialized books the
definitions and the basic theorems used in the subsequent chapters.
Chapter I is merely a review of fundamental notions of algebra, topology,
integration, and analysis. Chapter II treats the essentials of differential
calculus and calculus of variations on Banach spaces. Each of the
following chapters introduces a mathematical structure and exploits it
until it is sufficiently familiar to become an "instrument de pens6e":
Chapter III, differentiable manifolds, tangent bundles and their use in Lie
groups; Chapter IV, exterior derivation and the solutions of exterior
differential systems; Chapter V, Riemannian structures which, together
with the previous structures provide the basic geometric notions needed
in physics; Chapter VI, distributions and the Sobolev spaces with recent
applications to the theory of partial differential equations. The last chapter covers some selected topics in the theory of infinite dimensional
manifolds.
At the end of each chapter, several problems are worked out. Most of
them show how the concepts and the theorems introduced in the text can
be used in physics. They should be of interest both to physicists and
mathematicians. A sentence like "The Lagrangian is a function defined on
the tangent bundle of the configuration space" helps explain to the
physicist what a tangent bundle is and tells a mathematician what a
Lagrangian is. A sentence like "The strain tensor is the Lie derivative of
the metric with respect to the deformation" helps a physicist to under-
stand the concept of Lie derivatives and defines the strain tensor to a
mathematician. To both, they bring an added pleasure.
The pleasure of physical mathematics is well described by Hilbert:
learning that some genetic laws of the fruit fly had been derived by the
application of a certain set of axioms he exclaimed "So simple and precise
and at the same time so miraculous that no daring fantasy could have
imagined it "
