18.013A

• Preface

• Chapter 0    The Spreadsheet

• Chapter 1    Philosophy, Numbers and Functions

• Chapter 2    The Exponential Function and Trigonometric Functions

• Chapter 3    Vectors, Dot Products, Matrix Multiplication and Distance

• Chapter 4    Area of a Parallelogram, Determinants, Volume and Hypervolume, the Vector Product

• Chapter 5    Vectors and Geometry in Two and Three Dimensions

• Chapter 6    Differentiable Functions, the Derivative and Differentials

• Chapter 7    Computation of Derivatives from their Definition

• Chapter 8    Calculation of Derivatives by Rule

• Chapter 9    Derivatives of Vector Fields and the Gradient in Polar Coordinates

• Chapter 10  Higher Derivatives, Taylor Series, Quadratic Approximations and Accuracy of Approximations

• Chapter 11  Quadratic Approximations in Several Dimensions

• Chapter 12  Applications of Differentiation: Direct Use of Linear Approximation

• Chapter 13  Solving Equations

• Chapter 14  Extrema

• Chapter 15  Curves

• Chapter 16  Some Important Examples and a Formulation in Physics

• Chapter 17  The Product Rule and Differentiating Vectors

• Chapter 18  Complex Numbers and Functions of Them

• Chapter 19  The Anti-derivative or Indefinite Integral

• Chapter 20  The Area under a Curve and its Many Generalizations

• Chapter 21  The Fundamental Theorem of Calculus in One Dimension

• Chapter 22  The Fundamental Theorem of Calculus in Higher Dimensions; Additive Measures, Stokes Theorem and the Divergence Theorem

• Chapter 23  Reducing a Line Integral to an Ordinary Integral and Related Reductions

• Chapter 24  Reducing a Surface Integral to a Multiple Integral and the Jacobian

• Chapter 25  Numerical Integration

• Chapter 26  Numerical Solution of Differential Equations

• Chapter 27  Doing Integrals

• Chapter 28  Introduction to Electric and Magnetic Fields

• Chapter 29  Magnetic Fields, Magnetic Induction and Electrodynamics

• Chapter 30  Series

• Chapter 31  Doing Area, Surface and Volume Integrals

• Chapter 32  Some Linear Algebra

• Chapter 33  Second Order Differential Equations

• Exercises

• Tools

• Chapter A  Series Summing Magic

• Chapter B  Fun with Determinants

• Glossary of Notations

• Index

Applets

  Precalculus

• Operations on Functions

• Trigonometric Functions

• Slope of a Line

  Single Variable Calculus

• Derivative and Tangent Line

• Constant, Linear, Quadratic and Cubic Approximations

• Newton's Method

• Numerical Integration

• Lagrange Multipliers with Two Variables

  Vectors and Algebra

• Rotating Coordinates

• Operations on Vectors

• Determinant and Vector Products

• Multiplication of a Vector by a Matrix

• Linear Transformations in Three Dimensions

  Application to 3D Linear Geometry

• Lines in Space

• Planes in Space

  Curves

• Polar Plotter

• Curves in Two Dimensions

• Curves in Three Dimensions

  Fields and Surfaces

• Directional Derivatives

• Contour Lines, Gradients and Directional Derivatives

• Curves and Surfaces

• Functions of Two Variables

• Newton's Method with Two Equations and Two Variables

  Complex Numbers and Functions

• Complex Numbers

• Complex Functions

  Integration on Curves

• Curves and Vector Fields

• Line Integrals

  Integration on Surfaces

• Flux Integrals

• Integration Bounds

  Differential Equations

• First Order ODE

• Second Order ODE

• System First Order ODE

  Applications

• Static Electric Fields in Two Dimensions

• Static Electric Fields in Three Dimensions

• Stationary Magnetic Fields in Two Dimensions

• Series RLC Circuit