Mathematics for Physicists: Introductory Concepts and Methods

Contents
Preface
L Linear Algebra
L1 Mathematics before numbers
	L1.1 Sets and maps
	L1.2 Groups
	L1.3 Fields
	L1.4 Summary and outlook
L2 Vector spaces
	L2.1 The standard vector space
	L2.2 General definition of vector spaces
	L2.3 Vector spaces: examples
	L2.4 Basis and dimension
	L2.5 Vector space isomorphism
	L2.6 Summary and outlook
L3 Euclidean geometry
	L3.1 Scalar product of
	L3.2 Normalization and orthogonality
	L3.3 Inner product spaces
	L3.4 Complex inner product
	L3.5 Summary and outlook
L4 Vector product
	L4.1 Geometric formulation
	L4.2 Algebraic formulation
	L4.3 Further properties of the vector product
	L4.4 Summary and outlook
L5 Linear maps
	L5.1 Linear maps
	L5.2 Matrices
	L5.3 Matrix multiplication
	L5.4 The inverse of a matrix
	L5.5 General linear maps and matrices
	L5.6 Matrices describing coordinate changes
	L5.7 Summary and outlook
L6 Determinants
	L6.1 Definition and geometric interpretation
	L6.2 Computing determinants
	L6.3 Properties of determinants
	L6.4 Some applications
	L6.5 Summary and outlook
L7 Matrix diagonalization
	L7.1 Eigenvectors and eigenvalues
	L7.2 Characteristic polynomial
	L7.3 Matrix diagonalization
	L7.4 Functions of matrices
	L7.5 Summary and outlook
L8 Unitarity and Hermiticity
	L8.1 Unitarity and orthogonality
	L8.2 Hermiticity and symmetry
	L8.3 Relation between Hermitian and unitary matrices
	L8.4 Case study: linear algebra in quantum mechanics
	L8.5 Summary and outlook
L9 Linear algebra in function spaces
	L9.1 Bases of function space
	L9.2 Linear operators and eigenfunctions
	L9.3 Self-adjoint linear operators
	L9.4 Function spaces with unbounded support
	L9.5 Summary and outlook
L10 Multilinear algebra
	L10.1 Direct sum and direct product of vector spaces
	L10.2 Dual space
	L10.3 Tensors
	L10.4 Alternating forms
	L10.5 Visualization of alternating forms
	L10.6 Wedge product
	L10.7 Inner derivative
	L10.8 Pullback
	L10.9 Metric structures
	L10.10 Summary and outlook
PL Problems: Linear Algebra
	P.L1 Mathematics before numbers
	P.L2 Vector spaces
	P.L3 Euclidean geometry
	P.L4 Vector product
	P.L5 Linear maps
	P.L6 Determinants
	P.L7 Matrix diagonalization
	P.L8 Unitarity and hermiticity
	P.L10 Multilinear algebra
C Calculus
Introductory remarks
C1 Differentiation of one-dimensional functions
	C1.1 Definition of differentiability
	C1.2 Differentiation rules
	C1.3 Derivatives of selected functions
	C1.4 Summary and outlook
C2 Integration of one-dimensional functions
	C2.1 The concept of integration
	C2.2 One-dimensional integration
	C2.3 Integration rules
	C2.4 Practical remarks on one-dimensional integration
	C2.5 Summary and outlook
C3 Partial differentiation
	C3.1 Partial derivative
	C3.2 Multiple partial derivatives
	C3.3 Chain rule for functions of several variables
	C3.4 Extrema of functions
	C3.5 Summary and outlook
C4 Multidimensional integration
	C4.1 Cartesian area and volume integrals
	C4.2 Curvilinear area integrals
	C4.3 Curvilinear volume integrals
	C4.4 Curvilinear integration in arbitrary dimensions
	C4.5 Changes of variables in higher-dimensional integration
	C4.6 Summary and outlook
C5 Taylor series
	C5.1 Taylor expansion
	C5.2 Complex Taylor series
	C5.3 Finite-order expansions
	C5.4 Solving equations by Taylor expansion
	C5.5 Higher-dimensional Taylor series
	C5.6 Summary and outlook
C6 Fourier calculus
	C6.1 The δ-function
	C6.2 Fourier series
	C6.3 Fourier transform
	C6.4 Case study: frequency comb for high-precision measurements
	C6.5 Summary and outlook
C7 Differential equations
	C7.1 Typology of differential equations
	C7.2 Separable differential equations
	C7.3 Linear first-order differential equations
	C7.4 Systems of linear first-order differential equations
	C7.5 Linear higher-order differential equations
	C7.6 General higher-order differential equations
	C7.7 Linearizing differential equations
	C7.8 Partial differential equations
	C7.9 Summary and outlook
C8 Functional calculus
	C8.1 Definitions
	C8.2 Functional derivative
	C8.3 Euler–Lagrange equations
	C8.4 Summary and outlook
C9 Calculus of complex functions
	C9.1 Holomorphic functions
	C9.2 Complex integration
	C9.3 Singularities
	C9.4 Residue theorem
	C9.5 Essential singularities
	C9.6 Riemann surfaces
	C9.7 Summary and outlook
PC Problems: Calculus
	P.C1 Differentiation of one-dimensional functions
	P.C2 Integration of one-dimensional functions
	P.C3 Partial differentiation
	P.C4 Multidimensional integration
	P.C5 Taylor series
	P.C6 Fourier calculus
	P.C7 Differential equations
	P.C8 Functional calculus
	P.C9 Calculus of complex functions
V Vector Calculus
Introductory remarks
V1 Curves
	V1.1 Definition
	V1.2 Curve velocity
	V1.3 Curve length
	V1.4 Line integral
	V1.5 Summary and outlook
V2 Curvilinear coordinates
	V2.1 Polar coordinates
	V2.2 Coordinate basis and local basis
	V2.3 Cylindrical and spherical coordinates
	V2.4 A general perspective of coordinates
	V2.5 Local coordinate bases and linear algebra
	V2.6 Summary and outlook
V3 Fields
	V3.1 Definition of fields
	V3.2 Scalar fields
	V3.3 Extrema of functions with constraints
	V3.4 Gradient fields
	V3.5 Sources of vector fields
	V3.6 Circulation of vector fields
	V3.7 Practical aspects of three-dimensional vector calculus
	V3.8 Summary and outlook
V4 Introductory concepts of differential geometry
	V4.1 Differentiable manifolds
	V4.2 Tangent space
	V4.3 Summary and outlook
V5 Alternating differential forms
	V5.1 Cotangent space and differential one-forms
	V5.2 Pushforward and pullback
	V5.3 Forms of higher degree
	V5.4 Integration of forms
	V5.5 Summary and outlook
V6 Riemannian differential geometry
	V6.1 Definition of the metric on a manifold
	V6.2 Volume form and Hodge star
	V6.3 Vectors vs. one-forms vs. two-forms in
	V6.4 Case study: metric structures in general relativity
	V6.5 Summary and outlook
V7 Case study: differential forms and electrodynamics
	V7.1 The ingredients of electrodynamics
	V7.2 Laws of electrodynamics I: Lorentz force
	V7.3 Laws of electrodynamics II: Maxwell equations
	V7.4 Invariant formulation
	V7.5 Summary and outlook
PV Problems: Vector Calculus
	P.V1 Curves
	P.V2 Curvilinear coordinates
	P.V3 Fields
	P.V4 Introductory concepts of differential geometry
	P.V5 Alternating differential forms
	P.V6 Riemannian differential geometry
	P.V7 Differential forms and electrodynamics
S Solutions
SL Solutions: Linear Algebra
	S.L1 Mathematics before numbers
	S.L2 Vector spaces
	S.L3 Euclidean geometry
	S.L4 Vector product
	S.L5 Linear Maps
	S.L6 Determinants
	S.L7 Matrix diagonalization
	S.L8 Unitarity and Hermiticity
	S.L10 Multilinear algebra
SC Solutions: Calculus
	S.C1 Differentiation of one-dimensional functions
	S.C2 Integration of one-dimensional functions
	S.C3 Partial differentiation
	S.C4 Multidimensional integration
	S.C5 Taylor series
	S.C6 Fourier calculus
	S.C7 Differential equations
	S.C8 Functional calculus
	S.C9 Calculus of complex functions
SV Solutions: Vector Calculus
	S.V1 Curves
	S.V2 Curvilinear coordinates
	S.V3 Fields
	S.V4 Introductory concepts of differential geometry
	S.V5 Alternating differential forms
	S.V6 Riemannian differential geometry
	S.V7 Differential forms and electrodynamics
Index