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 of 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