Beberapa materi yang terdapat dalam ebook Mathematics for Physics I ini adalah :
1 Calculus of Variations
1.1 What is it good for?
1.2 Functionals
1.2.1 The functional derivative.
1.2.2 The Euler-Lagrange equation.
1.2.3 Some applications.
1.2.4 First integral
1.3 Lagrangian Mechanics.
1.3.1 One degree of freedom.
1.3.2 Noether’s theorem
1.3.3 Many degrees of freedom.
1.3.4 Continuous systems
1.4 Variable End Points
1.5 Lagrange Multipliers
1.6 Maximum or Minimum?.
1.7 Further Exercises and Problems.
2 Function Spaces
2.1 Motivation.
2.1.1 Functions as vectors
2.2 Norms and Inner Products
2.2.1 Norms and convergence.
2.2.2 Norms from integrals
2.2.3 Hilbert space
2.2.4 Orthogonal polynomials.
2.3 Linear Operators and Distributions
2.3.1 Linear operators .
2.3.2 Distributions and test-functions
2.4 Further Exercises and Problems
3 Linear Ordinary Differential Equations
3.1 Existence and Uniqueness of Solutions
3.1.1 Flows for first-order equations .
3.1.2 Linear independence . .
3.1.3 The Wronskian . .
3.2 Normal Form .
3.3 Inhomogeneous Equations .
3.3.1 Particular integral and complementary function
3.3.2 Variation of parameters
3.4 Singular Points
3.4.1 Regular singular points .
3.5 Further Exercises and Problems
4 Linear Differential Operators
4.1 Formal vs. Concrete Operators
4.1.1 The algebra of formal operators .
4.1.2 Concrete operators
4.2 The Adjoint Operator
4.2.1 The formal adjoint
4.2.2 A simple eigenvalue problem
4.2.3 Adjoint boundary conditions
4.2.4 Self-adjoint boundary conditions .
4.3 Completeness of Eigenfunctions
4.3.1 Discrete spectrum .
4.3.2 Continuous spectrum
4.4 Further Exercises and Problems
5 Green Functions
5.1 Inhomogeneous Linear equations
5.1.1 Fredholm alternative
5.2 Constructing Green Functions .
5.2.1 Sturm-Liouville equation
5.2.2 Initial-value problems
5.2.3 Modified Green function
5.3 Applications of Lagrange’s Identity
5.3.1 Hermiticity of Green function .
5.3.2 Inhomogeneous boundary conditions .
5.4 Eigenfunction Expansions .
5.5 Analytic Properties of Green Functions . .
5.5.1 Causality implies analyticity
5.5.2 Plemelj formulæ . .
5.5.3 Resolvent operator
5.6 Locality and the Gelfand-Dikii equation
5.7 Further Exercises and problems .
6 Partial Differential Equations
6.1 Classification of PDE’s . .
6.2 Cauchy Data .
6.2.1 Characteristics and first-order equations
6.2.2 Second-order hyperbolic equations
6.3 Wave Equation .
6.3.1 d’Alembert’s Solution . .
6.3.2 Fourier’s Solution .
6.3.3 Causal Green Function .
6.3.4 Odd vs. Even Dimensions . .
6.4 Heat Equation . .
6.4.1 Heat Kernel .
6.4.2 Causal Green Function .
6.4.3 Duhamel’s Principle . .
6.5 Potential Theory . .
6.5.1 Uniqueness and existence of solutions . .
6.5.2 Separation of Variables
6.5.3 Eigenfunction Expansions . .
6.5.4 Green Functions .
6.5.5 Boundary-value problems . .
6.5.6 Kirchhoff vs. Huygens .
6.6 Further Exercises and problems .
7 The Mathematics of Real Waves
7.1 Dispersive waves .
7.1.1 Ocean Waves .
7.1.2 Group Velocity .
7.1.3 Wakes
7.1.4 Hamilton’s Theory of Rays
7.2 Making Waves
7.2.1 Rayleigh’s Equation
7.3 Non-linear Waves
7.3.1 Sound in Air .
7.3.2 Shocks
7.3.3 Weak Solutions
7.4 Solitons
7.5 Further Exercises and Problems .
8 Special Functions
8.1 Curvilinear Co-ordinates
8.1.1 Div, Grad and Curl in Curvilinear Co-ordinates
8.1.2 The Laplacian in Curvilinear Co-ordinates
8.2 Spherical Harmonics .
8.2.1 Legendre Polynomials
8.2.2 Axisymmetric potential problems
8.2.3 General spherical harmonics
8.3 Bessel Functions
8.3.1 Cylindrical Bessel Functions
8.3.2 Orthogonality and Completeness
8.3.3 Modified Bessel Functions
8.3.4 Spherical Bessel Functions
8.4 Singular Endpoints
8.4.1 Weyl’s Theorem
8.5 Further Exercises and Problems .
9 Integral Equations
9.1 Illustrations .
9.2 Classification of Integral Equations
9.3 Integral Transforms
9.3.1 Fourier Methods
9.3.2 Laplace Transform Methods
9.4 Separable Kernels .
9.4.1 Eigenvalue problem
9.4.2 Inhomogeneous problem
9.5 Singular Integral Equations .
9.5.1 Solution via Tchebychef Polynomials .
9.6 Wiener-Hopf equations I .
9.7 Some Functional Analysis . .
9.7.1 Bounded and Compact Operators .
9.7.2 Closed Operators .
9.8 Series Solutions . .
9.8.1 Liouville-Neumann-Born Series .
9.8.2 Fredholm Series .
9.9 Further Exercises and Problems . .
A Linear Algebra Review
A.1 Vector Space
A.1.1 Axioms
A.1.2 Bases and components
A.2 Linear Maps .
A.2.1 Matrices
A.2.2 Range-nullspace theorem
A.2.3 The dual space .
A.3 Inner-Product Spaces
A.3.1 Inner products .
A.3.2 Euclidean vectors
A.3.3 Bra and ket vectors
A.3.4 Adjoint operator
A.4 Sums and Differences of Vector Spaces .
A.4.1 Direct sums
A.4.2 Quotient spaces .
A.4.3 Projection-operator decompositions
A.5 Inhomogeneous Linear Equations .
A.5.1 Rank and index .
A.5.2 Fredholm alternative .
A.6 Determinants
A.6.1 Skew-symmetric n-linear Forms .
A.6.2 The adjugate matrix .
A.6.3 Differentiating determinants
A.7 Diagonalization and Canonical Forms
A.7.1 Diagonalizing linear maps .
A.7.2 Diagonalizing quadratic forms
A.7.3 Block-diagonalizing symplectic forms .
B Fourier Series and Integrals.
B.1 Fourier Series
B.1.1 Finite Fourier series
B.1.2 Continuum limit
B.2 Fourier Integral Transforms .
B.2.1 Inversion formula
B.2.2 The Riemann-Lebesgue lemma .
B.3 Convolution .
B.3.1 The convolution theorem
B.3.2 Apodization and Gibbs’ phenomenon .
B.4 The Poisson Summation Formula .
Ebook Fisika matematik di atas dapat diunduh secara gratis menggunakan link berikut:
http://webusers.physics.illinois.edu/~m-stone5/mma/notes/amaster.pdf
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