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Preface; Acknowledgements; Contents; Notation; Part I Analysis; Chapter 1 Background in Analysis; 1.1 Asymptotic Expansions; 1.2 Gaussian Integrals; 1.3 Laplace Integrals; 1.4 Fourier Transform; 1.5 Laplace Transform; 1.6 Mellin Transform; 1.7 Derivative of Determinants; 1.8 Hamiltonian Systems; 1.9 Hilbert Spaces; 1.10 Functional Spaces; 1.11 Self-Adjoint and Unitary Operators; 1.12 Integral Operators; 1.13 Resolvent and Spectrum; 1.14 Spectral Resolution; 1.15 Functions of Operators; 1.16 Spectral Functions; 1.17 Heat Semigroups; 1.17.1 Definition and Basic Properties
1.17.2 Duhamel's Formula and Volterra's Series1.17.3 Chronological Exponent; 1.17.4 Campbell-Hausdorff Formula; 1.17.5 Heat Semigroup for Time Dependent Operators; 1.18 Notes; Chapter 2 Introduction to Partial Differential Equations; 2.1 First Order Partial Differential Equations; 2.2 Second Order Partial Differential Equations; 2.2.1 Elliptic Partial Differential Operators; 2.2.2 Classification of Second-Order Partial Differential Equations; 2.2.3 Elliptic Equations; 2.2.4 Parabolic Equations; 2.2.5 Hyperbolic Equations; 2.2.6 Boundary Conditions; 2.2.7 Gauss Theorem
2.2.8 Existence and Uniqueness of Solutions2.3 Partial Differential Operators; 2.3.1 Adjoint Operator; 2.3.2 Adjoint Boundary Conditions; 2.3.3 Spectral Theorem; 2.4 Heat Kernel; 2.4.1 Heat Kernel; 2.4.2 Heat Kernel of Time-dependent Operators; 2.4.3 Cauchy Problem; 2.4.4 Boundary Value Problem; 2.5 Differential Operators with Constant Coefficients; 2.5.1 Ordinary Differential Equations; 2.5.2 Indegro-Differential Equations; 2.5.3 Elliptic Partial Differential Operators; 2.5.4 Parabolic Partial Differential Equations; 2.5.5 Ordinary Differential Equations on Half-line
2.6 Differential Operators with Linear Coefficients2.7 Homogeneous Differential Operators; 2.8 Notes; Part II Geometry; Chapter 3 Introduction to Differential Geometry; 3.1 Differentiable Manifolds; 3.1.1 Basic Definitions; 3.1.2 Vector Fields; 3.1.3 Covector Fields; 3.1.4 Riemannian Metric; 3.1.5 Arc Length; 3.1.6 Riemannian Volume Element; 3.1.7 Tensor Fields; 3.1.8 Permutations of Tensors; 3.1.9 Einstein Summation Convention; 3.1.10 Levi-Civita Symbol; 3.1.11 Lie Derivative; 3.2 Connection; 3.2.1 Covariant Derivative; 3.2.2 Parallel Transport; 3.2.3 Geodesics; 3.3 Curvature
3.3.1 Riemann Tensor3.3.2 Properties of Riemann Tensor; 3.4 Geometry of Two-dimensional Manifolds; 3.4.1 Gauss Curvature; 3.4.2 Two-dimensional Constant Curvature Manifolds; 3.5 Killing Vectors; 3.6 Synge Function; 3.6.1 Definition and Basic Properties; 3.6.2 Derivatives of Synge Function; 3.6.3 Van Vleck-Morette Determinant; 3.7 Operator of Parallel Transport; 3.7.1 Definition and Basic Properties; 3.7.2 Derivatives of the Operator of Parallel Transport; 3.7.3 Generalized Operator of Parallel Transport; 3.8 Covariant Expansions of Two-Point Functions
1.17.2 Duhamel's Formula and Volterra's Series1.17.3 Chronological Exponent; 1.17.4 Campbell-Hausdorff Formula; 1.17.5 Heat Semigroup for Time Dependent Operators; 1.18 Notes; Chapter 2 Introduction to Partial Differential Equations; 2.1 First Order Partial Differential Equations; 2.2 Second Order Partial Differential Equations; 2.2.1 Elliptic Partial Differential Operators; 2.2.2 Classification of Second-Order Partial Differential Equations; 2.2.3 Elliptic Equations; 2.2.4 Parabolic Equations; 2.2.5 Hyperbolic Equations; 2.2.6 Boundary Conditions; 2.2.7 Gauss Theorem
2.2.8 Existence and Uniqueness of Solutions2.3 Partial Differential Operators; 2.3.1 Adjoint Operator; 2.3.2 Adjoint Boundary Conditions; 2.3.3 Spectral Theorem; 2.4 Heat Kernel; 2.4.1 Heat Kernel; 2.4.2 Heat Kernel of Time-dependent Operators; 2.4.3 Cauchy Problem; 2.4.4 Boundary Value Problem; 2.5 Differential Operators with Constant Coefficients; 2.5.1 Ordinary Differential Equations; 2.5.2 Indegro-Differential Equations; 2.5.3 Elliptic Partial Differential Operators; 2.5.4 Parabolic Partial Differential Equations; 2.5.5 Ordinary Differential Equations on Half-line
2.6 Differential Operators with Linear Coefficients2.7 Homogeneous Differential Operators; 2.8 Notes; Part II Geometry; Chapter 3 Introduction to Differential Geometry; 3.1 Differentiable Manifolds; 3.1.1 Basic Definitions; 3.1.2 Vector Fields; 3.1.3 Covector Fields; 3.1.4 Riemannian Metric; 3.1.5 Arc Length; 3.1.6 Riemannian Volume Element; 3.1.7 Tensor Fields; 3.1.8 Permutations of Tensors; 3.1.9 Einstein Summation Convention; 3.1.10 Levi-Civita Symbol; 3.1.11 Lie Derivative; 3.2 Connection; 3.2.1 Covariant Derivative; 3.2.2 Parallel Transport; 3.2.3 Geodesics; 3.3 Curvature
3.3.1 Riemann Tensor3.3.2 Properties of Riemann Tensor; 3.4 Geometry of Two-dimensional Manifolds; 3.4.1 Gauss Curvature; 3.4.2 Two-dimensional Constant Curvature Manifolds; 3.5 Killing Vectors; 3.6 Synge Function; 3.6.1 Definition and Basic Properties; 3.6.2 Derivatives of Synge Function; 3.6.3 Van Vleck-Morette Determinant; 3.7 Operator of Parallel Transport; 3.7.1 Definition and Basic Properties; 3.7.2 Derivatives of the Operator of Parallel Transport; 3.7.3 Generalized Operator of Parallel Transport; 3.8 Covariant Expansions of Two-Point Functions