Introduction Preface Chapter 1 Variational Method 1.1. Functional and Its Extremal Problems 1.1.1. The conception of functional 1.1.2. The extremes of functionals 1.2. The Variational of Functionals and the Simplest Euler Equation 1.2.1. The variational of functionals 1.2.2. The simplest Euler equation 1.3. The Cases of Multifunctions and Multivariates 1.3.1. Multifunctions 1.3.2. Multivariates 1.4. Functional Extremes under Certain Conditions 1.4.1. Isoperimetric problem 1.4.2. Geodesic problem 1.5. Natural Boundary Conditions 1.6. Variational Principle 1.6.1. Variational principle of classical mechanics 1.6.2. Variational principle of quantummechanics 1.7. The Applications of the Variational Method in Physics 1.7.1. The applications in classical physics 1.7.2. The applications in quantum mechanics Exercises
Chapter 2 Hilbert Space 2.1. Linear Space, Inner Product Spaceand Hilbert Space 2.1.1. Linear space 2.1.2. Inner product space 2.1.3. Hilbert space 2.2. Operators in Inner Product Spaces 2.2.1. Operators and adjoint operators 2.2.2. Self-adjoint operators 2.2.3. The alternative theorem for the solutions of linear algebraic equations 2.3. Complete Set of Orthonormal Functions 2.3.1. Three kinds of convergences 2.3.2. The completeness of a set of functions 2.3.3. N-dimensional space and Hilbert function space 2.3.4. Orthogonal polynomials 2.4. Polynomial Approximation 2.4.1. Weierstrass theorem 2.4.2. Polynomial approximation Exercises
Chapter 3 Linear Ordinary Differential Equations of Second Order 3.1. General Theory 3.1.1. The existence and uniqueness of solutions 3.1.2. The structure of solutions of homogeneous equations 3.1.3. The solutions of inhomogeneous equations 3.2. Sturm-Liouville Eigenvalue Problem 3.2.1. The form of Sturm-LiouviUe equations …… Chapter 4 Bessel Functions Chapter 5 The Dirac Delta Function Chapter 6 Green's Function Chapter 7 Norm Chapter 8 Integral Equations Chapter 9 Application of Number Theory in Inverse Problems in Physics Chapter 10 Fundamental Equations in Spaces with Arbitrary Dimensions References Answers of Selected Exercises Author Index Subject Index
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