The book assumes next to no prior knowledge of the topic. The first part introduces the core mathematics, always in conjunction with the physical context. In the second part of the book, a series of examples showcases some of the more conceptually advanced areas of physics, the presentation of which draws on the developments in the first part. A large number of problems helps students to hone their skills in using the presented mathematical methods. Solutions to the problems are available to instructors on an associated password-protected website for lecturers.

The book assumes next to no prior knowledge of the topic. The first part introduces the core mathematics, always in conjunction with the physical context. In the second part of the book, a series of examples showcases some of the more conceptually advanced areas of physics, the presentation of which draws on the developments in the first part. A large number of problems helps students to hone their skills in using the presented mathematical methods. Solutions to the problems are available to instructors on an associated password-protected website for lecturers.

Preface page ixPart I: Mathematics 11 Functions of one variable 21.1 Limits 21.2 Elementary Calculus 61.2.1 Chain Rule 71.2.2 Differentiation products and quotients 81.2.3 Inverse Functions 91.3 Integration 101.4 The Binomial Expansion 131.5 Taylor's series 151.6 Extrema 171.7 Power Series 181.8 Basic Functions 191.8.1 Exponential 191.8.2 Logarithm 231.9 1st order ordinary differential equations 241.10 Trigonometric Functions 261.10.1 L'Hopital's rule 281.11 Problems 302 Complex numbers 332.1 Exponential function of a complex variable 342.2 Argrand Diagrams and the Complex Plane 362.3 Hyperbolic functions 382.4 The simple harmonic oscillator 402.4.1 Mechanics in one dimension 422.5 Problems 453 Functions of Several Variables 483.1 Partial derivatives 483.1.1 Definition of the partial derivative 483.1.2 Total derivatives 513.1.3 Some relations 533.1.4 Change of variables 553.1.5 Mechanics again 563.1.6 Exact differentials and thermodynamics 583.2 Extrema under constraint 603.3 Multiple Integrals 623.3.1 Triple Integrals 663.3.2 Change of variables 673.4 Problems 694 Vectors in R3 724.1 Basic operation 724.1.1 scalar triple product 794.1.2 Vector equations of lines and planes 804.2 Kinematics in three dimensions 814.2.1 Differentiation 814.2.2 Motion in a uniform magnetic field 814.3 Coordinate systems 834.3.1 Polar coordinates 834.4 Central Forces 844.5 Rotating Frames 884.5.1 Larmor Effect 914.6 Problems 935 Vector fields and operators 965.1 The gradient operator 965.1.1 Coordinate Systems 975.2 Work and energy in vectorial mechanics 1015.2.1 Line integrals 1045.3 A little fluid dynamics 1075.3.1 Rotational motion 1115.3.2 Fields 1145.3.3 Surface integrals 1155.4 The divergence theorem 1185.5 Stokes' Theorem 1215.5.1 Conservative Forces 1235.6 Problems 1266 Generalized Functions. 1306.1 The Dirac delta function 1306.2 Problems 1397 Vector Spaces 1407.1 Formal Definition of a vector space 1407.2 Fourier Series 1457.3 Linear Operators 1487.4 Change of basis 1607.5 Problems 163Part II: Physics 1688 Maxwell's Equations: A very short Introduction 1698.1 Electrostatic: Gauss's Law 1698.2 Gauss's Law for a magnetic field 1738.3 Ampere's Law 1738.3.1 Gauge conditions 1748.4 Problems 1779 Special Relativity:4-vector formalism 1799.1 Lorentz transformation 1799.1.1 Inertial frames 1799.1.2 Properties and consequences of the Lorentz transformation1829.2 Minkowski space 1839.2.1 Four vectors 1839.2.2 Time Dilation 1899.3 Four velocity 1909.3.1 Four momentum 1919.4 Electrodynamics 1979.4.1 Maxwell's equations in 4-vector form 1979.4.2 Field of a moving point charge 2009.5 Problems 20210 Quantum Theory 20510.1 Formalism 20510.1.1 Dirac notation 20610.2 Probabilistic interpretation 20710.2.1 Commutator relations 20810.2.2 Functions of observables 21010.3 The Stern-Gerlach experiment 21010.3.1 Spin space 210viii Contents10.3.2 Explicit matrix representation 21010.3.3 EPR paradox 21010.3.4 Bell's Theorem 21010.4 Quantization 21010.4.1 Time evolution 21010.4.2 The harmonic oscillator, coherent states 21010.5 Problems 21211 Atoms, molecules, solids, wave mechanics in one dimension21611.1 Atom 21711.1.1 The square well 21811.1.2 The delta function potential 21911.2 Molecules 22111.3 Solids 22311.3.1 Block's Theorem 22411.3.2 Band structure 22612 An informal treatment of variational principles and theirhistory 229Appendix 1 Conic Sections 230Appendix 2 Vector Relations 232Index 237