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
Syllabus
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