This textbook is a comprehensive introduction to the key disciplines of mathematics - linear algebra, calculus, and geometry - needed in the undergraduate physics curriculum. Its leitmotiv is that success in learning these subjects depends on a good balance between theory and practice. Reflecting this belief, mathematical foundations are explained in pedagogical depth, and computational methods are introduced from a physicist's perspective and in a timely manner. This original approach presents concepts and methods as inseparable entities, facilitating in-depth understanding and making even advanced mathematics tangible. The book guides the reader from high-school level to advanced subjects such as tensor algebra, complex functions, and differential geometry. It contains numerous worked examples, info sections providing context, biographical boxes, several detailed case studies, over 300 problems, and fully worked solutions for all odd-numbered problems. An online solutions manual for all even-numbered problems will be made available to instructors.
Author(s): Alexander Altland, Jan von Delft
Edition: 1
Publisher: Cambridge University Press
Year: 2019
Language: English
Commentary: True PDF
Pages: 720
City: Cambridge, UK
Tags: Linear Algebra; Mathematics; Calculus; Vector Calculus
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
