1. Getting Started 1.1 Introduction 1.1.1 Computers as a Tool 1.1.2 A Note About Notation and Style 1.1.3 Notation and Symbolize (For experts only) 1.2 Arithmetic and Algebra 1.2.1 Arithmetic and Notation 1.2.2 Algebraic Manipulations 1.2.3 PowerExpand 1.2.4 Simple Rules 1.2.5 A Home-made Complex Conjugate with SuperStar 1.2.6 Immediate and Delayed Substitutions 1.2.7 Selecting Parts of Expressions 1.2.8 Algebraic Equations 1.3 Functions and Procedures 1.3.1 Built-In Functions 1.3.2 User-Defined Functions 1.3.3 Pure Functions 1.3.4 Assigning Rules and Restrictions to Functions 1.3.5 Module 1.3.6 rootPlot 1.4 Packages 1.4.1 Loading Packages 1.4.2 Contexts 1.4.3 Shadowing 1.5 Calculus 1.5.1 Derivatives and Integrals 1.5.2 Differential Equations 1.5.3 Changing Variables and Pure Functions 1.5.4 Numerical Solutions of Differential Equations 1.6 Graphics 1.6.1 Using the Plot Command 1.6.2 Animated Plots 1.6.3 Vector Field Plots 1.6.4 Three-dimensional Graphics using Plot3D and ParametricPlot3D 1.7 Exercises Exercise 1: Real Expressions Exercise 2: Series Exercise 3: Integration Exercise 4: Superposition of Waves Exercise 5: Tables of Zeros for Several Functions Exercise 6: Plot of Roots Exercise 7: Coupled Equations Exercise 8: Statistical Data Exercise 9: Differential Equation Exercise 10: Coupled Differential Equations Exercise 11: Trajectories and Vector Field Plots for Coupled Differential Equations Exercise 12: Heat Equation Exercise 13: Wave Equation Exercise 14: Fit to Data Exercise 15: Rules used to solve problems Exercise 16: Procedures to select numbers Exercise 17: Procedure to Select certain Pairs of Points Exercise 18: Change Exercise 19: Toss of a Die Exercise 20: Three Dice Exercise 21: Playing Darts and the Calculation of \[Pi] Exercise 22: Dropping Needles Exercise 23: Pacal's Wager Exercise 24: The birthday problem 2. GENERAL PHYSICS Introduction 2.1 Newtonian Mechanics in Inertial Frames Overview of Newtonian Mechanics in Inertial Frames Problem 1: Escape Velocity Problem 2: Projectile in a Uniform Gravity Field Problem 3: Reflecting Trajectories Problem 4: Falling Projectile with Linear Drag Problem 5: Projectile with Quadratic Drag Problem 6: Rocket with varying Mass Problem 7: Keplerian Orbits 2.2 Newtonian Mechanics in Rotating Frames Overview of Newtonian Mechanics in Rotating Frames Problem 1: Projectile Motion as Measured by an Observer on Earth Problem 2: Foucault Pendulum 2.3 Electricity and Magnetism Overview of Electricity and Magnetism Problem 1: Charged Disk Problem 2: Uniformly Charged Sphere Problem 3: Electric Dipole Problem 4: Magnetic Vector Potential for a Linear Current Problem 5: Motion of a Charged Particle in a Uniform B Field Problem 6: Motion of a Charged Particle in a Uniform B Field and Time Varying Electric Field 2.4 Modern Physics Problem 1: Carbon Dating Problem 2: Stable Isotopes Problem 3: The Bohr Atom Problem 4: Relativistic Collision 2.5 Exercises Exercise 1: Exploding Projectiles Exercise 2: Intersecting Trajectories Exercise 3: Principle of Galilean Invariance Exercise 4: Elastic Collision Exercise 5: Projectile and Earth's Rotation Exercise 6: Projectile Thrown up an Incline Plane Exercise 7: Falling Object with Air Resistance as Observed from the Rotating Earth Exercise 8: Projectile Thrown up with Air Resistance Exercise 9: Quadrupole Exercise 10: Motion of a Charged Particle Exercise 11: B Field of a Spinning Disk 3. Oscillating Systems Introduction 3.1 Linear Oscillations Overview of Linear Oscillations 1. Phase plot for one-dimensional system 2. Time behavior of phase plot for a one-dimensional system 3. Combination of the phase plot and the time behavior of the phase plot Problem 1: Linear Oscillator Problem 2: Series expansion Solution Problem 3: Potential and Phase Diagrams for the Linear Oscillator Problem 4: Damped Linear Oscillator Problem 5: Damped Harmonic Oscillator and Driving Forces 3.2 Small Oscillations Overview of Small Oscillations and Normal Modes Eigenvalues and eigenvectors for small oscillating systems Problem 1: Two Coupled Oscillators along a Line Problem 2: Three Coupled Oscillators along a Line Problem 3: Three Coupled Oscillators along a Circle Problem 4: Double Pendulum Problem 5: Understanding the User-defined procedure smallOsc[] 3.3 Oscillating Circuits Overview of Circuits Problem 1: Series RC Circuit Problem 2: Series RL Loop Problem 3 : RLC Loop 3.4 Exercises Exercise 1: Blocks connect with a Spring on a Table Exercise 2: Falling Blocks connect with a Spring Exercise 3: Parabolic coordinates and Harmonic Oscillator Exercise 4: Forced Oscillator Exercise 5: Pendulums and Spring Exercise 6: Five Particles connected with Springs Exercise 7: Vibrating Atoms on an Equilateral Triangle Exercise 8: Vibrating Charged Spheres Exercise 9: Quadratic Damping Exercise 10: Motion of a Charged Particle in a Harmonic Potential 4. NonLinear Oscillating Systems Introduction 4.1 Nonlinear Pendulum Overview of the Nonlinear Pendulum Problem 1: Analytic Solution for the Planar Pendulum Problem 2: Damped Pendulum Problem 3: Periodic Solutions for the Driven Pendulum Problem 4: Looping Solutions for the Driven Pendulum Problem 5: Chaotic motion for the Driven Pendulum 4.2 Duffing Equation Overview of the Duffing Equation 1. User-defined procedure to plot the Duffing displacement motion 2. User-defined procedure to plot the Duffing phase 3. User-defined procedure to plot the Duffing Poincaré map Problem 1: Potential and Phase Diagrams for the Duffing Oscillator Problem 2: Phase Diagram and Orbits for the Damped, Duffing Equation Problem 3: Driven Duffing Orbits with No Damping Problem 4: Two Well Driven Duffing Oscillator with Damping 4.3 Exercises Exercise 1: van der Pol Oscillator and Limiting Cycles Exercise 2: Springs Exercise 3: Buckling column system Exercise 4: Nonlinear Equation Exercise 5: Inverted Pendulum Exercise 6: Driven Nonlinear Equation 5. Discrete Dynamical Systems Introduction 5.1 Logistic Map Overview of the Logistic Map Problem 1: Logistic Map Problem 2: Logistic Fixed Points Problem 3: Logistic Cobwebs. Problem 4: Logistic Bifurcations Problem 5: Logistic Lyapunov Exponent and Entropy 5.2 Other Maps Overview of Other Maps Problem 1: Salmon Map Problem 2: Sine-Circle Map Problem 3: Taylor-Greene-Chirikov map Problem 4: Henon Map 5.3 Fractals Overview of Fractals Problem 1: Mandelbrot Set Problem 2: Julia Set 5.4 Exercises Exercise 1: Gaussian Map Exercise 2: Return Maps for the Henon Map Exercise 3: Cubic Maps Exercise 4: Two Dimensional Map Exercise 5: Tent map 6. Lagrangians and Hamiltonians Introduction 6.1 Lagrangian Problems without Lagrange Multipliers Overview of Lagrangian Problems without Lagrange Multipliers Problem 1: Particle Sliding on a Movable Incline Problem 2: Bead Sliding on a Rotating Wire Problem 3: Bead on a Rotating Hoop Problem 4: Springs Mounted on Top of a Carriage Problem 5: Mass Falling Through a Hole in a Table Problem 6: Spring Pendulum 6.2 Lagrangian Problems with Lagrange Multipliers Overview of Nonholonomic Constraints and Lagrange Multipliers Problem 1: Atwood Machine Problem 2: Hoop Rolling on an Incline Problem 3: Sphere Rolling on a Fixed Sphere 6.3 Hamiltonian Problems Overview of Hamilton's Equations Problem 1: Harmonic Oscillator Problem 2: Nonlinear Oscillator Problem 3: Cylindrical Coordinates Problem 4: Swinging Atwood Machine Problem 5: Spherical Pendulum 6.4 Hamilton-Jacobi Problems Overview Problem 1: Harmonic Oscillator Problem 2: Particle in a Constant Gravity Field Problem 3: Kepler's Problem and Hamilton-Jacobi Equations 6.5 Exercises Exercise 1: Atwood Machine Exercise 2: Double Atwood machine Exercise 3: Spherical Pendulum Exercise 4: Double Pendulum Exercise 5: Spring Pendulum Exercise 6: Hamilton-Jacobi in parabolic coordinates Exercise 7: How Hamilton Works Exercise 8: How HamiltonJacobi Works Exercise 9: How firstDiffSeries Works Exercise 10: How firstOrderPert Works 7. Orbiting Bodies Introduction 7.1 Two Body Problem Overview of the Two-Body Problem Problem 1: Equivalent One-body Problem Problem 2: Kepler Orbits Problem 3: Precessing Ellipse Problem 4: Numerical Solution 7.2 Restricted Three Body Problem Overview of the Three Body Problem Problems on the Equal Mass Primaries (\[Mu]=1/2 ) Problem 1: Lagrangian Points for Equal Mass Binaries (\[Mu]=1/2 ) Problem 2: Looping Motion in an Equal Mass Binary System (\[Mu]=1/2) Problem 3: Symmetric orbits about the y-axis for \[Mu]=1/2 Problem 4: Mass exchange Between Equal Mass Binaries Problems on the Sun-Jupiter System (\[Mu]=.000954) Problem 5: Lagrangian Points for the Sun-Jupiter System Problem 6: Numerical Solution for the Trojan Asteroids Problem 7: Perturbative Solution for the Trojan Asteroids Problems on the Earth-Moon System (\[Mu]= .01215) Problem 8: Lagrangian Points for the Earth-Moon System Problem 9: Motion about L[4] in the Earth-Moon System Problem 10: Orbit around the Earth and Moon 7.3 Exercises Exercise 1: Central force problems Exercise 2: Central force procedure Exercise 3: Central forces and elliptical solutions Exercise 4: Attractive Inverse fifth power force Exercise 5: Eccentric Anomaly Exercise 6: Eccentric Anomaly Exercise 7: Kepler problem with drag Exercise 8: Lagrange points Exercise 9: Orbit around the Sun and Jupiter Exercise 10: Exact Solution for the Three-Body Problem 8. Electrostatics Introduction Mathematica Commands for All Sections Example: Equipotential surface and electric field of two-point charges 8.1 Point Charges, Multipoles, and Image Charges Overview of Point Charges, Multipoles, and Image Charges Mathematica Commands for Section 8.1 Monopole TrigToY TrigToP Problem 1: Superposition of point charges Problem 2: Point charges and grounded plane Problem 3: Point charges and grounded sphere Problem 4: Line charge and grounded plane Problem 5: Multipole expansion of a charge distribution 8.2 Laplace's Equation in Cartesian and Cylindrical Coordinates Overview of Cartesian and Cylindrical Coordinates Problem 1: Separation of variables in Cartesian and cylindrical coordinates Problem 2: Potential in a rectangular groove Problem 3: Rectangular conduit Problem 4: Potential inside a rectangular box with five sides at zero potential Problem 5: Conducting cylinder with a potential on the surface 8.3 Laplace's Equation in Spherical Coordinates Overview of Spherical Coordinates Problem 1: A charged ring Problem 2: Grounded sphere in an electric field Problem 3: Sphere with an axially symmetric charge distribution Problem 4: Sphere with a given axially symmetric potential Problem 5: Sphere with upper hemisphere V0 and lower hemisphere -V0 8.4 Exercises Exercise 1: Parallel plates and strips Exercise 2: Parallel plates and a point charge Exercise 3: Cylinder divided into segments Exercise 4: Elliptical cylinder in a uniform electric field Exercise 5: Cylindrical box and n charges Exercise 6: Distorted sphere with a potential Exercise 7: Concentric spheres with different potentials on the two hemispheres Exercise 8: Ring charge Exercise 9: Laplace's equations in other coordinates 9. Quantum Mechanics Introduction 9.1 One-Dimensional Schroedinger's Equation Problem 1: Particle bound in an Infinite Potential Well Problem 2: Particle bound in a Finite Potential Well Problem 3: Particle Hitting a Finite Step Potential Problem 4: Particle Propagating Towards a Rectangular Potential Problem 5: The One-Dimensional Harmonic Oscillator 9.2 Three-Dimensional Schroedinger's Equation Problem 1: Three-Dimensional Harmonic Oscillator in Cartesian Coordinates Problem 2: Schroedinger's Equation for Spherically Symmetric Potentials Problem 3: Particle in an Infinite, Spherical Well Problem 4: Particle in a Finite, Spherical Well Problem 5: The Hydrogen Atom in Spherical Coordinates 9.3 Exercises Exercise 1: Infinite Potential Well with Rectangular Perturbation Exercise 2: Tilted Square Well Exercise 3: The Wentzel-Kramers-Brillouin Approximation Exercise 4: Plot of the Electron Probability Density Exercise 5: Perturbed Harmonic Oscillator Exercise 6: Perturbation Theory Exercise 7: Separation of variables in Cylindrical Coordinates 10. Relativity and Cosmology Introduction 10.1 Special Relativity Overview of Special Relativity Problem 1: Decay of a particle Problem 2: Two-particle collision Problem 3: Compton scattering Problem 4: Moving mirror and generalized Snell's law Problem 5: One-dimensional motion of a relativistic particle with constant acceleration Problem 6: Two-dimensional motion of a relativistic particle in a uniform electric field 10.2 General Relativity Overview of General Relativity Problem 1: Killing vectors and Spherical Symmetry Problem 2: Schwarzschild solution Problem 3: Geodesics for the Schwarzschild metric Problem 4: Time it takes to fall into a black hole Problem 5: Circular geodesics for the Schwarzschild metric Problem 6: Precession of the Perihelion 10.3 Cosmology Overview of Friedmann, Robertson, and Walker Cosmology Problem 1: Field equations for Friedmann-Robertson-Walker cosmology Problem 2: Zero-pressure (Dust) Cosmological Models Problem 3: The Expansion and Age for the Friedmann-Robertson-Walker models 10.4 Exercises Exercise 1: Lorentz boosts Exercise 2: Radioactive decay of moving Nucleus Exercise 3: Schwarzschild solution Exercise 4: Potential analysis for timelike geodesics Exercise 5: Schwarzschild solution in null coordinates Exercise 6: Null geodesics for the Schwarzschild metric Exercise 7: Mercury's Anomalous Precession and Modification of Newton's Gravity Exercise 8: Cosmological Force and Mercury