# 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!
* Introduction
* Contributing
* Resources

## Precalculus

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

## Limits

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

## Differentiation
### Basics of Differentiation

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

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

## Integration
### Basics of Integration

4.1 Definite integral
4.2 Fundamental Theorem of Calculus
4.3 Indefinite integral
4.4 Improper Integrals

### Integration Techniques

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 Integration

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 Equations
### Parametric Equations

5.1 Introduction to Parametric Equations
5.2 Differentiation and Parametric Equations
5.3 Integration and Parametric Equations

### Polar Equations

5.5 Introduction to Polar Equations
5.6 Differentiation and Polar Equations
5.7 Integration and Polar Equations

## Sequences and Series
### Sequences

6.1 Definition of a Sequence
6.2 Sequences

### Series and Tests

6.3 Definition of a Series
6.4 Series
6.5 Divergence Test
6.6 Ratio Test
6.7 Limit Comparison Test
6.8 Direct Comparison Test
6.9 Integral Test

### Convergence

6.10 Absolute and Conditional Convergence

### Series and Calculus

6.11 Taylor series
6.12 Power series
6.13 Leibniz' formula for pi

### Exercises

6.14 Exercises

## Multivariable Calculus
### Introduction to Multivariable Calculus

7.1 Vectors
7.2 Curves and Surfaces in Space
7.3 Vector Functions
7.4 Introduction to multivariable calculus

### Differentiation

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
* Multivariate optimisation: The Lagrangian
* Calculus on matrices
7.10 Derivatives of Multivariate Functions
7.11 Inverse Function Theorem, Implicit Function Theorem (optional)

### Integration

7.12 Multiple integration
* Old: Double Integrals
7.13 Change of variables

### Vector calculus

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 Equations

8.1 Ordinary Differential Equations
8.2 Partial Differential Equations

## Extensions
### Advanced Integration Techniques

9.1 Complexifying

### Further Analysis

9.2 Systems of Ordinary Differential Equations

### Formal Theory of Calculus

9.3 Real Numbers
9.4 Complex Numbers
9.5 Hyperbolic Angle

## References

* Lester R. Ford, Sr. & Jr. (1963) Calculus, McGraw-Hill via HathiTrust
* w:Mellen W. Haskell (1895) On the introduction of the notion of hyperbolic functions Bulletin of the American Mathematical Society 1(6):155–9.
* Table of Trigonometry
* Summation notation
* Tables of Derivatives
* Tables of Integrals

## Acknowledgements and Further Reading

Navigation:
Main Page · Precalculus · Limits · Differentiation · Integration · Parametric and Polar Equations · Sequences and Series · Multivariable Calculus · Extensions · References