This text summarizes the core undergraduate physics curriculum together with the mathematics frequently encountered in engineering and physics calculations. The text focuses exclusively on core material relevant to practical problem solving, such as algebra, geometry, exponential, logarithmic functions and trigonometry. The book also examines particle mechanics, fluid mechanics, special relativity and electromagnetism. This book presents fundamental formulas and derivations and provides simple, coherent explanations of underlying concepts.

Table of Contents

• 1 Introduction
• 2 Problem Solving
• 2.1 Analysis
• 2.2 Test-Taking Techniques
• 2.2.1 Dimensional Analysis
• 3 Scientific Programming
• 3.1 Language Fundamentals
• 3.1.1 Octave Programming
• 4 Elementary Mathematics
• 4.1 Algebra
• 4.1.1 Equation Manipulation
• 4.1.2 Linear Equation Systems
• 4.1.3 Factoring
• 4.1.4 Inequalities
• 4.1.5 Sum Formulas
• 4.1.6 Binomial Theorem
• 4.2 Geometry
• 4.2.1 Angles
• 4.2.2 Triangles
• 4.2.3 Right Triangles
• 4.2.4 Polygons
• 4.2.5 Circles
• 4.3 Exponential, Logarithmic Functions, and Trigonometry
• 4.3.1 Exponential Functions
• 4.3.2 Inverse Functions and Logarithms
• 4.3.3 Hyperbolic Functions
• 4.3.4 Complex Numbers and Harmonic Functions
• 4.3.5 Inverse Harmonic and Hyperbolic Functions
• 4.3.6 Trigonometric Identities
• 4.4 Analytic Geometry
• 4.4.1 Lines and Planes
• 4.4.2 Conic Sections
• 4.4.3 Areas, Volumes, and Solid Angles
• 5 Vectors and Matrices
• 5.1 Matrices and Matrix Products
• 5.2 Equation Systems
• 5.3 Traces and Determinants
• 5.4 Vectors and Inner Products
• 5.5 Cross and Outer Products
• 5.6 Vector Identities
• 5.7 Rotations and Orthogonal Matrices
• 5.8 Groups and Matrix Generators
• 5.9 Eigenvalues and Eigenvectors
• 5.10 Similarity Transformations
• 6 Calculus of a Single Variable
• 6.1 Derivatives
• 6.2 Integrals
• 6.3 Series
• 7 Calculus of Several Variables
• 7.1 Partial Derivatives
• 7.2 Multidimensional Taylor Series and Extrema
• 7.3 Multiple Integration
• 7.4 Volumes and Surfaces of Revolution
• 7.5 Change of Variables and Jacobians
• 8 Calculus of Vector Functions
• 8.1 Generalized Coordinates
• 8.2 Vector Differential Operators
• 8.3 Vector Differential Identities
• 8.4 Gauss's and Stokes’ Laws and Green's Identities
• 8.5 Lagrange Multipliers
• 9 Probability Theory and Statistics
• 9.1 Random Variables, Probability Density, and Distributions
• 9.2 Mean, Variance, and Standard Deviation
• 9.3 Variable Transformations
• 9.4 Moments and Moment-Generating Function
• 9.5 Multivariate Probability Distributions, Covariance, and Correlation
• 9.6 Gaussian, Binomial, and Poisson Distributions
• 9.7 Least Squares Regression
• 9.8 Error Propagation
• 9.9 Numerical Models
• 10 Complex Analysis
• 10.1 Functions of a Complex Variable
• 10.2 Derivatives, Analyticity, and the Cauchy–Riemann elations
• 10.3 Conformal Mapping
• 10.4 Cauchy's Theorem and Taylor and Laurent Series
• 10.5 Residue Theorem
• 10.6 Dispersion Relations
• 10.7 Method of Steepest Decent
• 11 Differential Equations
• 11.1 Linearity, Superposition, and Initial and Boundary Values
• 11.2 Numerical Solutions
• 11.3 First-Order Differential Equations
• 11.4 Wronskian
• 11.5 Factorization
• 11.6 Method of Undetermined Coefficients
• 11.7 Variation of Parameters
• 11.8 Reduction of Order
• 11.9 Series Solution and Method of Frobenius
• 11.10 Systems of Equations, Eigenvalues, and Eigenvectors
• 12 Transform Theory
• 12.1 Eigenfunctions and Eigenvectors
• 12.2 Sturm–Liouville Theory
• 12.3 Fourier Series
• 12.4 Fourier Transforms
• 12.5 Delta Functions
• 12.6 Green's Functions
• 12.7 Laplace Transforms
• 12.8 z-Transforms
• 13 Partial Differential Equations and Special Functions
• 13.1 Separation of Variables and Rectangular Coordinates
• 13.2 Legendre Polynomials
• 13.3 Spherical Harmonics
• 13.4 Bessel Functions
• 13.5 Spherical Bessel Functions
• 14 Integral Equations and the Calculus of Variations
• 14.1 Volterra and Fredholm Equations
• 14.2 Calculus of Variations the Euler-Lagrange Equation
• 15 Particle Mechanics
• 15.1 Newton's Laws
• 15.2 Forces
• 15.3 Numerical Methods
• 15.4 Work and Energy
• 15.5 Lagrange Equations
• 15.6 Three-Dimensional Particle Motion
• 15.7 Impulse
• 15.8 Oscillatory Motion
• 15.9 Rotational Motion About a Fixed Axis
• 15.10 Torque and Angular Momentum
• 15.11 Motion in Accelerating Reference Systems
• 15.12 Gravitational Forces and Fields
• 15.13.Celestial Mechanics
• 15.14 Dynamics of Systems of Particles
• 15.15 Two-Particle Collisions and Scattering
• 15.16 Mechanics of Rigid Bodies
• 15.17 Hamilton's Equation and Kinematics
• 16 Fluid Mechanics
• 16.1. Continuity Equation
• 16.2. Euler's Equation
• 16.3. Bernoulli's Equation
• 17 Special Relativity
• 17.1 Four-Vectors and Lorentz Transformation
• 17.2 Length Contraction, Time Dilation, and Simultaneity
• 17.3 Covariant Notation
• 17.4 Casuality and Minkowski Diagrams
• 17.5 Velocity Addition and Doppler Shift
• 17.6 Energy and Momentum
• 18 Electromagnetism
• 18.1 Maxwell's Equations
• 18.2 Gauss's Law
• 18.3 Electric Potential
• 18.4 Current and Resistivity
• 18.5 Dipoles and Polarization
• 18.6 Boundary Conditions and Green's Functions
• 18.7 Multipole Expansion
• 18.8 Relativistic Formulation of Electromagnetism, Gauge Transformations, and Magnetic Fields
• 18.9 Magnetostatics
• 18.10 Magnetic Dipoles
• 18.11 Magnetization
• 18.12 Induction and Faraday's Law
• 18.13 Circuit Theory and Kirchoff's Laws
• 18.14 Conservation Laws and the Stress Tensor
• 18.15 Lienard–Wiechert Potentials
• 18.16 Radiation from Moving Charges
• 19 Wave Motion
• 19.1 Wave Equation
• 19.2 Propagation of Waves
• 19.3 Planar Electromagnetic Waves
• 19.4 Polarization
• 19.5 Superposition and Interference
• 19.6 Multipole Expansion for Radiating Fields
• 19.7 Phase and Group Velocity
• 19.8 Minimum Time Principle and Ray Optics
• 19.9 Refraction and Snell's Law
• 19.10 Lenses
• 19.11 Mechanical Reflection
• 19.12 Doppler Effect and Shock Waves
• 19.13 Waves in Periodic Media
• 19.14 Conducting Media
• 19.15 Dielectric Media
• 19.16 Reflection and Transmission
• 19.17 Diffraction
• 19.18 Waveguides and Cavities
• 20 Quantum Mechanics
• 20.1 Fundamental Principles
• 20.2 Particle–Wave Duality
• 20.3 Interference of Quantum Waves
• 20.4 Schrödinger Equation
• 20.5 Particle Flux and Reflection
• 20.6 Wave Packet Propagation
• 20.7 Numerical Solutions
• 20.8 Quantum Mechanical Operators
• 20.9 Heisenberg Uncertainty Relation
• 20.10 Hilbert Space Representation
• 20.11 Square Well and Delta Function Potentials
• 20.12 WKB Method
• 20.13 Harmonic Oscillators
• 20.14 Heisenberg Representation
• 20.15 Translation Operators
• 20.16 Perturbation Theory
• 20.17 Adiabatic Theorem
• 21 Atomic Physics
• 21.1 Properties of Fermions
• 21.2 Bohr Model
• 21.3 Atomic Spectra and X-Rays
• 21.4 Atomic Units
• 21.5 Angular Momentum
• 21.6 Spin
• 21.7 Interaction of Spins
• 21.8 Hydrogenic Atoms
• 21.9 Atomic Structure
• 21.10 Spin–Orbit Coupling
• 21.11 Atoms in Static Electric and Magnetic Fields
• 21.12 Helium Atom and the Molecule
• 21.13 Interaction of Atoms with Radiation
• 21.14 Selection Rules
• 21.15 Scattering Theory
• 22 Nuclear and Particle Physics
• 22.1 Nuclear Properties
• 22.2 Radioactive Decay
• 22.3 Nuclear Reactions
• 22.4 Fission and Fusion
• 22.5 Fundamental Properties of Elementary Particles
• 23 Thermodynamics and Statistical Mechanics
• 23.1 Entropy
• 23.2 Ensembles
• 23.3 Statistics
• 23.4 Partition Functions
• 23.5 Density of States
• 23.6 Temperature and Energy
• 23.7 Phonons and Photons
• 23.8 The Laws of Thermodynamics
• 23.9 The Legendre Transformation and Thermodynamic Quantities
• 23.10 Expansion of Gases
• 23.11 Heat Engines and the Carnot Cycle
• 23.12 Thermodynamic Fluctuations
• 23.13 Phase Transformations
• 23.14 The Chemical Potential and Chemical Reactions
• 23.15 The Fermi Gas
• 23.16 Bose–Einstein Condensation
• 23.17 Physical Kinetics and Transport Theory
• 24 Condensed Matter Physics
• 24.1 Crystal Structure
• 24.2 X-Ray Diffraction
• 24.3 Thermal Properties
• 24.4 Electron Theory of Metals
• 24.5 Superconductors
• 24.6 Semiconductors
• 25 Laboratory Methods
• 25.1 Interaction of Particles with Matter
• 25.2 Radiation Detection and Counting Statistics
• 25.3 Lasers