Mathematics for Physicists Introductory Concepts and Methods 1st edition by Alexander Altland, Jan von Delft ISBN 1108471226 978-1108471220

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 R[sup(n)]

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 R[sup(n)]

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 deter

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 R[sup(3)]

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