Mathematics for physicists - Instructors' manual (solution to even numbered problems)

✍ Scribed by Alexander Altland, Jan von Delft

• Publisher: Cambridge University Press
• Year: 2019
• Tongue: English
• Leaves: 145
• Category: Library

✦ Table of Contents

EvenNumberedSolutions
EvenNumberedSolutionsContents
EvenNumberedSolutions
Even-numbered solutions
Problems: Linear Algebra
Mathematics before numbers
Sets and Maps
Groups
Fields
Vector spaces
Vector spaces: examples
Basis and dimension
Euclidean geometry
Normalization and orthogonality
Inner product spaces
Vector product
Algebraic formulation
Further properties of the vector product
Linear Maps
Linear maps
Matrix multiplication
The inverse of a matrix
General linear maps and matrices
Matrices describing coordinate changes
Determinants
Computing determinants
Matrix diagonalization
Matrix diagonalization
Functions of matrices
Unitarity and Hermiticity
Unitarity and orthogonality
Hermiticity and symmetry
Relation between Hermitian and unitary matrices
Multilinear algebra
Direct sum and direct product of vector spaces
Dual space
Tensors
Alternating forms
Wedge product
Inner derivative
Pullback
Metric structures
Solutions: Calculus
Differentiation of one-dimensional functions
Definition of differentiability
Differentiation rules
Derivatives of selected functions
Integration of one-dimensional functions
One-dimensional integration
Integration rules
Practical remarks on one-dimensional integration
Partial differentiation
Partial derivative
Multiple partial derivatives
Chain rule for functions of several variables
Multi-dimensional integration
Cartesian area and volume integrals
Curvilinear area integrals
Curvilinear volume integrals
Curvilinear integration in arbitrary dimensions
Changes of variables in higher-dimensional integration
Taylor series
Complex Taylor series
Finite-order expansion
Solving equations by Taylor expansion
Higher-dimensional Taylor series
Fourier calculus
The -Function
Fourier series
Fourier transform
Case study: Frequency comb for high-precision measurements
Differential equations
Separable differential equations
Linear first-order differential equations
Systems of linear first-order differential equations
Linear higher-order differential equations
General higher-order differential equations
Linearizing differential equations
Functional calculus
Euler-Lagrange equations
Calculus of complex functions
Holomorphic functions
Complex integration
Singularities
Residue theorem
Solutions: Vector Calculus
Curves
Curve velocity
Curve length
Line integral
Curvilinear Coordinates
Cylindrical and spherical coordinates
Local coordinate bases and linear algebra
Fields
Definition of fields
Scalar fields
Extrema of functions with constraints
Gradient fields
Sources of vector fields
Circulation of vector fields
Practical aspects of three-dimensional vector calculus
Introductory concepts of differential geometry
Differentiable manifolds
Tangent space
Alternating differential forms
Cotangent space and differential one-forms
Pushforward and Pullback
Forms of higher degree
Integration of forms
Riemannian differential geometry
Definition of the metric on a manifold
Volume form and Hodge star
Differential forms and electrodynamics
Laws of electrodynamics II: Maxwell equations
Invariant formulation
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