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Calculus

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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!

Precalculus

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1.1 Algebra 75% developed  as of 24 October 2020

1.2 Functions 75% developed  as of 24 October 2020

1.3 Trigonometric functions 75% developed  as of 16 November 2020

1.4 Graphing functions 75% developed  as of 20 November 2020

1.5 Rational functions

1.6 Conic sections 75% developed  as of 21 November 2020

1.7 Exercises

1.8 Hyperbolic logarithm and angles 75% developed

Limits

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2.1 An Introduction to Limits 75% developed

2.2 Finite Limits 50% developed

2.3 Infinite Limits 50% developed

2.4 Continuity 25% developed

2.5 Formal Definition of the Limit 25% developed

2.6 Proofs of Some Basic Limit Rules

2.7 Exercises

Differentiation

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Basics of Differentiation 75% developed

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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 50% developed

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3.11 L'Hôpital's Rule 75% developed

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

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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 Integration

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4.1 Definite integral 25% developed

4.2 Fundamental Theorem of Calculus 25% developed

4.3 Indefinite integral 25% developed

4.4 Improper Integrals

Integration Techniques

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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 Integration

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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

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Parametric Equations

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5.1 Introduction to Parametric Equations

5.2 Differentiation and Parametric Equations

5.3 Integration and Parametric Equations

Polar Equations

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5.5 Introduction to Polar Equations

5.6 Differentiation and Polar Equations

5.7 Integration and Polar Equations

Sequences and Series

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Sequences

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6.1 Definition of a Sequence

6.2 Sequences

Series and Tests

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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

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6.10 Absolute and Conditional Convergence

Series and Calculus

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6.11 Taylor series

6.12 Power series

6.13 Leibniz' formula for pi

Exercises

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6.14 Exercises


Multivariable Calculus

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

Introduction to Multivariable Calculus

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7.1 Vectors 75% developed

7.2 Curves and Surfaces in Space 75% developed  as of 9 Feb 2021

7.3 Vector Functions 75% developed  as of 11 Mar 2021

7.4 Introduction to multivariable calculus 50% developed

Differentiation

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7.5 Limits and Continuity

7.6 Partial Derivatives

7.7 The chain rule and Clairaut's theorem 50% developed

7.8 Chain Rule

7.9 Directional derivatives and the gradient vector

7.10 Derivatives of Multivariate Functions 50% developed

7.11 Inverse Function Theorem, Implicit Function Theorem (optional) 100% developed

Integration

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7.12 Multiple integration

7.13 Change of variables

Vector calculus

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7.14 Vector Calculus 100% developed

7.15 Vector Calculus Identities 100% developed

7.16 Inverting Vector Calculus Operators 100% developed

7.17 Points, Paths, Surfaces, and Volumes 75% developed

7.18 Helmholtz Decomposition Theorem 75% developed

7.19 Discrete Analog to Vector Calculus 100% developed

7.20 Exercises


Differential Equations

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8.1 Ordinary Differential Equations 25% developed

8.2 Partial Differential Equations 50% developed

Extensions

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Advanced Integration Techniques

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9.1 Complexifying

Further Analysis

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9.2 Systems of Ordinary Differential Equations 0% developed

Formal Theory of Calculus

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9.3 Real Numbers 25% developed

9.4 Complex Numbers 50% developed

9.5 Hyperbolic Angle 100% developed

References

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Acknowledgements and Further Reading

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

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