Preface
The calculus of variations is a branch of mathematics developed at the end of the
seventeenth century. In a sense, it can be said that the calculus of variation is a
science refined from the other sciences. This book mainly introduces the classical
calculus of variations, its theory is complete, it has a wide range of applications in
many aspects, such as mechanics, physics, chemistry, optics, acoustics, heat
transfer, materials science, tribology, economics, aerospace theory, information
theory, automatic control theory, image processing, medicine and biology, etc. The
finite element method developed in the middle of the twentieth century, one of its
mathematical foundations is the calculus of variations. Now the calculus of variations has become an essential mathematical foundation of graduate students,
college students, engineers, technicians and scientific workers.
The purpose of writing this book is to provide a textbook or teaching reference
book learning the calculus of variations for the graduate students and senior grade
college students of institutions of higher learning, such that them can be familiar
with the basic concepts and calculation methods of the calculus of variations, in
which it includes the preliminary knowledge, variational problems with fixed
boundaries, sufficient conditions of extrema of functionals, problems with variable
boundaries, variational problems of conditional extrema, variational problems in
parametric forms, variational principles, direct methods of variational problems,
variational principles in mechanics and their applications and variational problems
of functionals with vector, tensor and Hamiltonian operators. Quite a part of contents in this book are the authors’ research achievements. Especially the authors
proposed the extremal function theorem of the complete functional, unified various
kinds of Euler equations in the calculus of variations, originated the variational
theory of functionals with vector, modulus of vector, arbitrary order tensor and
Hamiltonian operators, gave the corresponding Euler equations and natural
boundary conditions, expanded the application range of the calculus of variations.
This book can also be used as relevant professional teachers, scientific research
personnel and engineering technical personnel’s reference.
vii
The calculus of variations is a branch of mathematics developed at the end of the
seventeenth century. In a sense, it can be said that the calculus of variation is a
science refined from the other sciences. This book mainly introduces the classical
calculus of variations, its theory is complete, it has a wide range of applications in
many aspects, such as mechanics, physics, chemistry, optics, acoustics, heat
transfer, materials science, tribology, economics, aerospace theory, information
theory, automatic control theory, image processing, medicine and biology, etc. The
finite element method developed in the middle of the twentieth century, one of its
mathematical foundations is the calculus of variations. Now the calculus of variations has become an essential mathematical foundation of graduate students,
college students, engineers, technicians and scientific workers.
The purpose of writing this book is to provide a textbook or teaching reference
book learning the calculus of variations for the graduate students and senior grade
college students of institutions of higher learning, such that them can be familiar
with the basic concepts and calculation methods of the calculus of variations, in
which it includes the preliminary knowledge, variational problems with fixed
boundaries, sufficient conditions of extrema of functionals, problems with variable
boundaries, variational problems of conditional extrema, variational problems in
parametric forms, variational principles, direct methods of variational problems,
variational principles in mechanics and their applications and variational problems
of functionals with vector, tensor and Hamiltonian operators. Quite a part of contents in this book are the authors’ research achievements. Especially the authors
proposed the extremal function theorem of the complete functional, unified various
kinds of Euler equations in the calculus of variations, originated the variational
theory of functionals with vector, modulus of vector, arbitrary order tensor and
Hamiltonian operators, gave the corresponding Euler equations and natural
boundary conditions, expanded the application range of the calculus of variations.
This book can also be used as relevant professional teachers, scientific research
personnel and engineering technical personnel’s reference.
vii