Guide to Essential Math

To the Student1 Mathematical Thinking 1.1 The NCAA Problem 1.2 Gauss and the Arithmetic Series 1.3 The Pythagorean Theorem 1.4 Torus Area and Volume 1.5 Einstein's Velocity Addition Law1.6 The Birthday Problem 1.7 p¼ in the Gaussian Integral 1.8 Function Equal to its Derivative 1.9 Log of N Factorial for Large N1.10 Potential and Kinetic Energies1.11 Lagrangian Mechanics1.12 Riemann Zeta Function and Prime Numbers 1.13 How to Solve It 1.14 A Note on Mathematical Rigor 2. Numbers2.1 Integers2.2 Primes 2.3 Divisibility 2.4 Fibonacci Numbers 2.5 Rational Numbers 2.6 Exponential Notation2.7 Powers of 10 2.8 Binary Number System 2.9 Infinity 3 Algebra 3.1 Symbolic Variables3.2 Legal and Illegal Algebraic Manipulations 3.3 Factor-Label Method 3.4 Powers and Roots3.5 Logarithms3.6 The Quadratic Formula3.7 Imagining i 3.8 Factorials, Permutations and Combinations3.9 The Binomial Theorem 3.10 e is for Euler 4 Trigonometry4.1 What Use is Trigonometry? 4.2 The Pythagorean Theorem 4.3 ¼ in the Sky 4.4 Sine and Cosine 4.5 Tangent and Secant 4.6 Trigonometry in the Complex Plane 4.7 De Moivre's Theorem4.8 Euler's Theorem4.9 Hyperbolic Functions5 Analytic Geometry 5.1 Functions and Graphs5.2 Linear Functions 5.3 Conic Sections 5.4 Conic Sections in Polar Coordinates6 Calculus 6.1 A Little Road Trip6.2 A Speedboat Ride6.3 Differential and Integral Calculus6.4 Basic Formulas of Differential Calculus6.5 More on Derivatives 6.6 Indefinite Integrals6.7 Techniques of Integration6.8 Curvature, Maxima and Minima6.9 The Gamma Function6.10 Gaussian and Error Functions7 Series and Integrals 7.1 Some Elementary Series 7.2 Power Series7.3 Convergence of Series 7.4 Taylor Series 7.5 L'H'opital's Rule7.6 Fourier Series7.7 Dirac Deltafunction7.8 Fourier Integrals 7.9 Generalized Fourier Expansions 7.10 Asymptotic Series 8 Differential Equations 8.1 First-Order Differential Equations8.2 AC Circuits8.3 Second-Order Differential Equations8.4 Some Examples from Physics8.5 Boundary Conditions8.6 Series Solutions8.7 Bessel Functions 8.8 Second Solution 9 Matrix Algebra9.1 Matrix Multiplication 9.2 Further Properties of Matrices 9.3 Determinants 9.4 Matrix Inverse9.5 Wronskian Determinant9.6 Special Matrices 9.7 Similarity Transformations9.8 Eigenvalue Problems9.9 Group Theory9.10 Minkowski Spacetime10 Multivariable Calculus 10.1 Partial Derivatives10.2 Multiple Integration10.3 Polar Coordinates10.4 Cylindrical Coordinates10.5 Spherical Polar Coordinates10.6 Differential Expressions10.7 Line Integrals 10.8 Green's Theorem11 Vector Analysis 11.1 Scalars and Vectors 11.2 Scalar or Dot Product11.3 Vector or Cross Product 11.4 Triple Products of Vectors11.5 Vector Velocity and Acceleration11.6 Circular Motion11.7 Angular Momentum 11.8 Gradient of a Scalar Field 11.9 Divergence of a Vector Field 11.10 Curl of a Vector Field11.11 Maxwell's Equations11.12 Covariant Electrodynamics11.13 Curvilinear Coordinates11.14 Vector Identities12 Special Functions 12.1 Partial Differential Equations 12.2 Separation of Variables12.3 Special Functions 12.4 Leibniz's Formula12.5 Vibration of a Circular Membrane12.6 Bessel Functions12.7 Laplace Equation in Spherical Coordinates12.8 Legendre Polynomials12.9 Spherical Harmonics12.10 Spherical Bessel Functions12.11 Hermite Polynomials12.12 Laguerre Polynomials13 Complex Variables 13.1 Analytic Functions13.2 Derivative of an Analytic Function 13.3 Contour Integrals 13.4 Cauchy's Theorem13.5 Cauchy's Integral Formula13.6 Taylor Series13.7 Laurent Expansions13.8 Calculus of Residues13.9 Multivalued Functions 13.10 Integral Representations for Special Functions