Calculus

Welcome to the Wikibook of
Calculus

This wikibook aims to be a high quality calculus textbook through which users can master the discipline. Standard topics such as limits, differentiation and integration are covered, as well as several others. Please contribute wherever you feel the need. You can simply help by rating individual sections of the book that you feel were inappropriately rated!

PrecalculusEdit

1.1 Algebra  

1.2 Functions  

1.3 Trigonometric functions  

1.4 Graphing functions  

1.5 Rational functions

1.6 Conic sections  

1.7 Exercises

1.8 Hyperbolic logarithm and angles  

LimitsEdit

2.1 An Introduction to Limits  

2.2 Finite Limits  

2.3 Infinite Limits  

2.4 Continuity  

2.5 Formal Definition of the Limit  

2.6 Proofs of Some Basic Limit Rules

2.7 Exercises

DifferentiationEdit

Basics of Differentiation  Edit

3.1 Differentiation Defined

3.2 Product and Quotient Rules

3.3 Derivatives of Trigonometric Functions

3.4 Chain Rule

3.5 Higher Order Derivatives: an introduction to second order derivatives

3.6 Implicit Differentiation

3.7 Derivatives of Exponential and Logarithm Functions

3.8 Derivatives of Hyperbolic Functions

3.9 Some Important Theorems

3.10 Exercises

Applications of Derivatives  Edit

3.11 L'Hôpital's Rule  

3.12 Extrema and Points of Inflection

3.13 Newton's Method

3.14 Related Rates

3.15 Optimization

3.16 Euler's Method

3.17 Approximating Values of Functions

3.18 Exercises


IntegrationEdit

 
The definite integral of a function f(x) from x=0 to x=a is equal to the area under the curve from 0 to a.

Basics of IntegrationEdit

4.1 Definite integral  

4.2 Fundamental Theorem of Calculus  

4.3 Indefinite integral  

4.4 Improper Integrals

Integration TechniquesEdit

 
From bottom to top:
  • an acceleration function a(t);
  • the integral of the acceleration is the velocity function v(t);
  • and the integral of the velocity is the distance function s(t).

4.5 Infinite Sums

4.6 Derivative Rules and the Substitution Rule

4.7 Integration by Parts

4.8 Trigonometric Substitutions

4.9 Trigonometric Integrals

4.10 Rational Functions by Partial Fraction Decomposition

4.11 Tangent Half Angle Substitution

4.12 Reduction Formula

4.13 Irrational Functions

4.14 Numerical Approximations

4.15 Exercises

Applications of IntegrationEdit

4.16 Area

4.17 Volume

4.18 Volume of Solids of Revolution

4.19 Arc Length

4.20 Surface Area

4.21 Work

4.22 Center of Mass

4.23 Pressure and Force

4.24 Probability

Parametric and Polar EquationsEdit

Parametric EquationsEdit

5.1 Introduction to Parametric Equations

5.2 Differentiation and Parametric Equations

5.3 Integration and Parametric Equations

Polar EquationsEdit

5.5 Introduction to Polar Equations

5.6 Differentiation and Polar Equations

5.7 Integration and Polar Equations

Sequences and SeriesEdit

SequencesEdit

6.1 Definition of a Sequence

6.2 Sequences

SeriesEdit

6.3 Definition of a Series

6.4 Series

6.5 Limit Test for Convergence

6.6 Comparison Test for Convergence

6.7 Integral Test for Convergence

Series and calculusEdit

6.8 Taylor series

6.9 Power series

6.10 Leibniz' formula for pi

ExercisesEdit

6.11 Exercises


Multivariable CalculusEdit

 
This is an example of using spherical coordinates in 3 dimensions to calculate the volume of a given shape

Introduction to Multivariable CalculusEdit

7.1 Vectors  

7.2 Curves and Surfaces in Space  

7.3 Vector Functions  

7.4 Introduction to multivariable calculus  

DifferentiationEdit

7.5 Limits and Continuity

7.6 Partial Derivatives

7.7 The chain rule and Clairaut's theorem  

7.8 Chain Rule

7.9 Directional derivatives and the gradient vector

7.10 Derivatives of Multivariate Functions  

7.11 Inverse Function Theorem, Implicit Function Theorem (optional)  

IntegrationEdit

7.12 Multiple integration

7.13 Change of variables

Vector calculusEdit

7.14 Vector Calculus  

7.15 Vector Calculus Identities  

7.16 Inverting Vector Calculus Operators  

7.17 Points, Paths, Surfaces, and Volumes  

7.18 Helmholtz Decomposition Theorem  

7.19 Discrete Analog to Vector Calculus  

7.20 Exercises


Differential EquationsEdit

8.1 Ordinary Differential Equations  

8.2 Partial Differential Equations  

ExtensionsEdit

Advanced Integration TechniquesEdit

9.1 Complexifying

Further AnalysisEdit

9.2 Systems of Ordinary Differential Equations  

Formal Theory of CalculusEdit

9.3 Real Numbers  

9.4 Complex Numbers  

9.5 Hyperbolic Angle  

ReferencesEdit

Acknowledgements and Further ReadingEdit