Mathematics for Physicists

Name: Mathematics for Physicists
Brand: Wiley
Price: 48.99 EUR
Availability: OnlineOnly

Brian R. Martin Graham Shaw(Author)

Wiley (Publisher)

1st Edition

Published on 9. April 2015

584 pages

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978-1-118-67663-9 (ISBN)

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Mathematics for Physicists is a relatively short volume covering all the essential mathematics needed for a typical first degree in physics, from a starting point that is compatible with modern school mathematics syllabuses. Early chapters deliberately overlap with senior school mathematics, to a degree that will depend on the background of the individual reader, who may quickly skip over those topics with which he or she is already familiar. The rest of the book covers the mathematics that is usually compulsory for all students in their first two years of a typical university physics degree, plus a little more. There are worked examples throughout the text, and chapter-end problem sets.

Mathematics for Physicists features:

Interfaces with modern school mathematics syllabuses
All topics usually taught in the first two years of a physics degree
Worked examples throughout
Problems in every chapter, with answers to selected questions at the end of the book and full solutions on a website

This text will be an excellent resource for undergraduate students in physics and a quick reference guide for more advanced students, as well as being appropriate for students in other physical sciences, such as astronomy, chemistry and earth sciences.

Brian Martin was a full-time member of staff of the Department of Physics & Astronomy at UCL from 1968 to 1995, including a decade from 1994 to 2004 as Head of the Department. I retired in 2005 and now hold the title of Emeritus Professor of Physics. I have extensive experience of teaching undergraduate mathematics classes at all levels and experience of other universities via external examining for first degrees at Imperial College and Royal Holloway College London. I was also the external member of the General Board of the Department of Physics at Cambridge University that reviewed the whole academic programme of that department, including teaching.

Graham Shaw is a full-time member of staff of the School of Physics & Astronomy at Manchester University and will retire in September 2009. I have extensive experience of teaching undergraduate physics and the associated mathematics, and have been a member of the department's Teaching Committee and the Course Director of the Honours School of Mathematics and Physics for many years.

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Content

Intro
Mathematics for Physicists
Contents
Editors' preface to the Manchester Physics Series
Authors' preface
Notes and website information
'Starred' material
Website
Examples, problems and solutions
1 Real numbers, variables and functions
1.1 Real numbers
1.1.1 Rules of arithmetic: rational and irrational numbers
1.1.2 Factors, powers and rationalisation
1.1.3 Number systems
1.2 Real variables
1.2.1 Rules of elementary algebra
1.2.2 Proof of the irrationality of V2
1.2.3 Formulas, identities and equations
1.2.4 The binomial theorem
1.2.5 Absolute values and inequalities
1.3 Functions, graphs and co-ordinates
1.3.1 Functions
1.3.2 Cartesian co-ordinates
Problems 1
2 Some basic functions and equations
2.1 Algebraic functions
2.1.1 Polynomials
2.1.2 Rational functions and partial fractions
2.1.3 Algebraic and transcendental functions
2.2 Trigonometric functions
2.2.1 Angles and polar co-ordinates
2.2.2 Sine and cosine
2.2.3 More trigonometric functions
2.2.4 Trigonometric identities and equations
2.2.5 Sine and cosine rules
2.3 Logarithms and exponentials
2.3.1 The laws of logarithms
2.3.2 Exponential function
2.3.3 Hyperbolic functions
2.4 Conic sections
Problems 2
3 Differential calculus
3.1 Limits and continuity
3.1.1 Limits
3.1.2 Continuity
3.2 Differentiation
3.2.1 Differentiability
3.2.2 Some standard derivatives
3.3 General methods
3.3.1 Product rule
3.3.2 Quotient rule
3.3.3 Reciprocal relation
3.3.4 Chain rule
3.3.5 More standard derivatives
3.3.6 Implicit functions
3.4 Higher derivatives and stationary points
3.4.1 Stationary points
3.5 Curve sketching
Problems 3
4 Integral calculus
4.1 Indefinite integrals
4.2 Definite integrals
4.2.1 Integrals and areas
4.2.2 Riemann integration
4.3 Change of variables and substitutions
4.3.1 Change of variables
4.3.2 Products of sines and cosines
4.3.3 Logarithmic integration
4.3.4 Partial fractions
4.3.5 More standard integrals
4.3.6 Tangent substitutions
4.3.7 Symmetric and antisymmetric integrals
4.4 Integration by parts
4.5 Numerical integration
4.6 Improper integrals
4.6.1 Infinite integrals
4.6.2 Singular integrals
4.7 Applications of integration
4.7.1 Work done by a varying force
4.7.2 The length of a curve
4.7.3 Surfaces and volumes of revolution
4.7.4 Moments of inertia
Problems 4
5 Series and expansions
5.1 Series
5.2 Convergence of infinite series
5.3 Taylors theorem and its applications
5.3.1 Taylors theorem
5.3.2 Small changes and l'Hôpitals rule
5.3.3 Newtons method
5.3.4 Approximation errors: Euler's number
5.4 Series expansions
5.4.1 Taylor and Maclaurin series
5.4.2 Operations with series
5.5 Proof of d'Alembert's ratio test
5.5.1 Positive series
5.5.2 General series
5.6 Alternating and other series
Problems 5
6 Complex numbers and variables
6.1 Complex numbers
6.2 Complex plane: Argand diagrams
6.3 Complex variables and series
6.3.1 Proof of the ratio test for complex series
6.4 Eulers formula
6.4.1 Powers and roots
6.4.2 Exponentials and logarithms
6.4.3 De Moivre's theorem
6.4.4 Summation of series and evaluation of integrals
Problems 6
7 Partial differentiation
7.1 Partial derivatives
7.2 Differentials
7.2.1 Two standard results
7.2.2 Exact differentials
7.2.3 The chain rule
7.2.4 Homogeneous functions and Euler's theorem
7.3 Change of variables
7.4 Taylor series
7.5 Stationary points
7.6 Lagrange multipliers
7.7 Differentiation of integrals
Problems 7
8 Vectors
8.1 Scalars and vectors
8.1.1 Vector algebra
8.1.2 Components of vectors: Cartesian co-ordinates
8.2 Products of vectors
8.2.1 Scalar product
8.2.2 Vector product
8.2.3 Triple products
8.2.4 Reciprocal vectors
8.3 Applications to geometry
8.3.1 Straight lines
8.3.2 Planes
8.4 Differentiation and integration
Problems 8
9 Determinants, Vectors and Matrices
9.1 Determinants
9.1.1 General properties of determinants
9.1.2 Homogeneous linear equations
9.2 Vectors in n Dimensions
9.2.1 Basis vectors
9.2.2 Scalar products
9.3 Matrices and linear transformations
9.3.1 Matrices
9.3.2 Linear transformations
9.3.3 Transpose, complex, and Hermitian conjugates
9.4 Square Matrices
9.4.1 Some special square matrices
9.4.2 The determinant of a matrix
9.4.3 Matrix inversion
9.4.4 Inhomogeneous simultaneous linear equations
Problems 9
10 Eigenvalues and eigenvectors
10.1 The eigenvalue equation
10.1.1 Properties of eigenvalues
10.1.2 Properties of eigenvectors
10.1.3 Hermitian matrices
10.2 Diagonalisation of matrices
10.2.1 Normal modes of oscillation
10.2.2 Quadratic forms
Problems 10
11 Line and multiple integrals
11.1 Line integrals
11.1.1 Line integrals in a plane
11.1.2 Integrals around closed contours and along arcs
11.1.3 Line integrals in three dimensions
11.2 Double integrals
11.2.1 Greens theorem in the plane and perfect differentials
11.2.2 Other co-ordinate systems and change of variables
11.3 Curvilinear co-ordinates in three dimensions
11.3.1 Cylindrical and spherical polar co-ordinates
11.4 Triple or volume integrals
11.4.1 Change of variables
Problems 11
12 Vector calculus
12.1 Scalar and vector fields
12.1.1 Gradient of a scalar field
12.1.2 Div, grad and curl
12.1.3 Orthogonal curvilinear co-ordinates
12.2 Line, surface, and volume integrals
12.2.1 Line integrals
12.2.2 Conservative fields and potentials
12.2.3 Surface integrals
12.2.4 Volume integrals: moments of inertia
12.3 The divergence theorem
12.3.1 Proof of the divergence theorem and Green's identities
12.3.2 Divergence in orthogonal curvilinear co-ordinates
12.3.3 Poissons equation and Gaus's theorem
12.3.4 The continuity equation
12.4 Stokes theorem
12.4.1 Proof of Stokes' theorem
12.4.2 Curl in curvilinear co-ordinates
12.4.3 Applications to electromagnetic fields
Problems 12
13 Fourier analysis
13.1 Fourier series
13.1.1 Fourier coefficients
13.1.2 Convergence
13.1.3 Change of period
13.1.4 Non-periodic functions
13.1.5 Integration and differentiation of Fourier series
13.1.6 Mean values and Parseval's theorem
13.2 Complex Fourier series
13.2.1 Fourier expansions and vector spaces
13.3 Fourier transforms
13.3.1 Properties of Fourier transforms
13.3.2 The Dirac delta function
13.3.3 The convolution theorem
Problems 13
14 Ordinary differential equations
14.1 First-order equations
14.1.1 Direct integration
14.1.2 Separation of variables
14.1.3 Homogeneous equations
14.1.4 Exact equations
14.1.5 First-order linear equations
14.2 Linear ODEs with constant coefficients
14.2.1 Complementary functions
14.2.2 Particular integrals: method of undetermined coefficients
14.2.3 Particular integrals: the D-operator method
14.2.4 Laplace Transforms
14.3 Eulers equation
Problems 14
15 Series solutions of ordinary differential equations
15.1 Series solutions
15.1.1 Series solutions about a regular point
15.1.2 Series solutions about a regular singularity: Frobenius method
15.1.3 Polynomial solutions
15.2 Eigenvalue equations
15.3 Legendres equation
15.3.1 Legendre functions and Legendre polynomials
15.3.2 The generating function
15.3.3 Associated Legendre equation
15.3.4 Rodrigues formula
15.4 Bessels equation
15.4.1 Bessel functions
15.4.2 Properties of non-singular Bessel functions J? (x)
Problems 15
16 Partial differential equations
16.1 Some important PDEs in physics
16.2 Separation of variables: Cartesian co-ordinates
16.2.1 The wave equation in one spatial dimension
16.2.2 The wave equation in three spatial dimensions
16.2.3 The diffusion equation in one spatial dimension
16.3 Separation of variables: polar co-ordinates
16.3.1 Plane-polar co-ordinates
16.3.2 Spherical polar co-ordinates
16.3.3 Cylindrical polar co-ordinates
16.4 The wave equation: d'Alembert's solution
16.5 Euler Equations
16.6 Boundary conditions and uniqueness
16.6.1 Laplace transforms
Problems 16
Answers to selected problems
Index
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Mathematics for Physicists

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