Mathematical Physics

Mathematical Physics is the discipline dedicated to the development and application of rigorous mathematical methods to problems in physics, serving as a crucial bridge between abstract mathematics and theoretical physics. It focuses on creating the formal structures and frameworks necessary to describe physical phenomena, employing advanced concepts from areas such as differential geometry, functional analysis, and group theory to provide the precise language for fundamental theories like general relativity, quantum mechanics, and string theory. By doing so, it not only helps in solving complex physical problems but also ensures the logical consistency and deepens the conceptual understanding of the fundamental laws of the universe.

1. Foundations in Mathematical Methods
   1. Vector Spaces and Linear Algebra
      1. Abstract Vector Spaces
         1. Definition and Axioms
         2. Examples of Vector Spaces
         3. Vector Addition Properties
         4. Scalar Multiplication Properties
         5. Zero Vector and Additive Inverses
      2. Subspaces
         1. Definition and Criteria
         2. Examples of Subspaces
         3. Intersection and Sum of Subspaces
      3. Linear Independence and Spanning
         1. Linear Independence
         2. Spanning Sets
         3. Basis of a Vector Space
         4. Dimension Theorem
         5. Finite and Infinite Dimensional Spaces
      4. Inner Product Spaces
         1. Definition of Inner Product
         2. Properties of Inner Products
         3. Norm Induced by Inner Product
         4. Cauchy-Schwarz Inequality
         5. Triangle Inequality
         6. Parallelogram Law
      5. Orthogonality
         1. Orthogonal Vectors
         2. Orthonormal Sets
         3. Orthogonal Complements
         4. Projection onto Subspaces
      6. Gram-Schmidt Process
         1. Orthogonalization Algorithm
         2. Applications to Function Spaces
      7. Hilbert Spaces
         1. Completeness and Cauchy Sequences
         2. Examples of Hilbert Spaces
         3. Riesz Representation Theorem
      8. Linear Operators and Transformations
         1. Definition and Properties
         2. Kernel and Image
         3. Rank-Nullity Theorem
         4. Matrix Representation
         5. Change of Basis
         6. Similarity Transformations
      9. Eigenvalues and Eigenvectors
         1. Characteristic Polynomial
         2. Algebraic and Geometric Multiplicity
         3. Diagonalizability Conditions
         4. Jordan Canonical Form
      10. Spectral Theory for Finite Dimensions
          1. Spectral Theorem for Hermitian Matrices
          2. Spectral Theorem for Normal Matrices
          3. Applications in Physics
      11. Tensor Algebra
          1. Dual Spaces and Linear Functionals
          2. Tensor Products
          3. Multilinear Maps
          4. Symmetric and Antisymmetric Tensors
          5. Tensor Contraction
          6. Einstein Summation Convention
          7. Covariant and Contravariant Components
   2. Complex Analysis
      1. Complex Numbers and the Complex Plane
         1. Cartesian and Polar Forms
         2. Modulus and Argument
         3. Complex Exponential Function
         4. Roots of Unity
      2. Complex Functions
         1. Limits and Continuity
         2. Complex Differentiability
         3. Analytic Functions
         4. Power Series Representation
      3. Cauchy-Riemann Equations
         1. Derivation and Conditions
         2. Harmonic Functions
         3. Conjugate Harmonic Functions
      4. Complex Integration
         1. Contour Integrals
         2. Parametrization of Curves
         3. Path Independence
      5. Cauchy's Theorems
         1. Cauchy's Integral Theorem
         2. Cauchy's Integral Formula
         3. Derivatives of Analytic Functions
         4. Maximum Modulus Principle
      6. Series Expansions
         1. Taylor Series
         2. Laurent Series
         3. Radius of Convergence
         4. Classification of Singularities
      7. Residue Theory
         1. Residues at Poles
         2. Residue Theorem
         3. Evaluation of Real Integrals
         4. Principal Value Integrals
      8. Conformal Mappings
         1. Definition and Properties
         2. Möbius Transformations
         3. Applications to Boundary Value Problems
   3. Differential Equations
      1. Ordinary Differential Equations
         1. First-Order Equations
            1. Separable Equations
            2. Linear First-Order Equations
            3. Integrating Factor Method
            4. Exact Equations
            5. Bernoulli Equations
         2. Second-Order Linear Equations
            1. Homogeneous Equations with Constant Coefficients
            2. Characteristic Equation Method
            3. Nonhomogeneous Equations
            4. Method of Undetermined Coefficients
            5. Variation of Parameters
         3. Series Solutions
            1. Power Series Method
            2. Frobenius Method
            3. Regular and Irregular Singular Points
            4. Recurrence Relations
         4. Sturm-Liouville Theory
            1. Self-Adjoint Differential Operators
            2. Eigenvalue Problems
            3. Orthogonality of Eigenfunctions
            4. Eigenfunction Expansions
         5. Systems of Linear ODEs
            1. Matrix Methods
            2. Fundamental Matrix Solutions
            3. Phase Portraits
      2. Partial Differential Equations
         1. Classification of Second-Order PDEs
            1. Elliptic Equations
            2. Parabolic Equations
            3. Hyperbolic Equations
         2. Method of Separation of Variables
            1. Product Solutions
            2. Boundary and Initial Conditions
            3. Eigenfunction Expansions
         3. Transform Methods
            1. Fourier Transform Solutions
            2. Laplace Transform Solutions
         4. Green's Functions
            1. Construction Methods
            2. Applications to Inhomogeneous Problems
         5. Boundary Value Problems
            1. Dirichlet Problems
            2. Neumann Problems
            3. Mixed Boundary Conditions
            4. Uniqueness and Existence Theorems
         6. Important PDEs in Physics
            1. Laplace's Equation
            2. Poisson's Equation
            3. Heat Equation
            4. Wave Equation
            5. Schrödinger Equation
   4. Fourier Analysis and Integral Transforms
      1. Fourier Series
         1. Periodic Functions
         2. Fourier Coefficients
         3. Convergence Properties
         4. Gibbs Phenomenon
         5. Even and Odd Functions
         6. Parseval's Identity
      2. Fourier Transform
         1. Definition and Basic Properties
         2. Inverse Fourier Transform
         3. Convolution Theorem
         4. Plancherel Theorem
         5. Applications to Differential Equations
      3. Laplace Transform
         1. Definition and Properties
         2. Inverse Laplace Transform
         3. Convolution and Transfer Functions
         4. Applications to Initial Value Problems
      4. Other Integral Transforms
         1. Mellin Transform
         2. Hankel Transform
         3. Hilbert Transform
   5. Calculus of Variations
      1. Functionals and Variations
         1. Definition of Functionals
         2. First Variation
         3. Fundamental Lemma of Calculus of Variations
      2. Euler-Lagrange Equation
         1. Derivation from First Principles
         2. Boundary Conditions
         3. Applications to Physics Problems
      3. Constrained Optimization
         1. Holonomic Constraints
         2. Method of Lagrange Multipliers
         3. Isoperimetric Problems
      4. Second Variation and Stability
         1. Sufficient Conditions for Extrema
         2. Jacobi Equation
         3. Conjugate Points
      5. Noether's Theorem
         1. Symmetries and Conservation Laws
         2. Applications in Mechanics and Field Theory
      6. Hamilton's Principle
         1. Principle of Stationary Action
         2. Connection to Mechanics