# Calculus

- Last updated

- Save as PDF

- Page ID
- 3227

- Supplemental Modules (Calculus)
- Differential Calculus
- Integral Calculus
- 1: Area and Volume
- 2: Techniques of Integration
- 3: L'Hopital's Rule and Improper Integrals
- 4: Transcendental Functions
- 5: Work and Force
- 6: Moments and Centroids

- Vector Calculus
- 1: Vector Basics
- 2: Vector-Valued Functions and Motion in Space
- 3: Multiple Integrals
- 3.1: Double and Iterated Integrals Over Rectangles
- 3.2: Area by Double Integration
- 3.3: Double Integrals Over General Regions
- 3.4: Double Integrals in Polar Form
- 3.5: Triple Integrals in Rectangular Coordinates
- 3.6: Triple Integrals in Cylindrical and Spherical Coordinates
- 3.7: Moments and Centers of Mass
- 3.8: Jacobians
- 3.9: Substitutions in Multiple Integrals

- 4: Integration in Vector Fields
- 4.1: Differentiation and Integration of Vector Valued Functions
- 4.2: Surfaces and Area
- 4.3: Line Integrals
- 4.4: Conservative Vector Fields and Independence of Path
- 4.5: Path Independence, Conservative Fields, and Potential Functions
- 4.6: Vector Fields and Line Integrals: Work, Circulation, and Flux
- 4.7: Surface Integrals
- 4.8: Green’s Theorem in the Plane
- 4.9: The Divergence Theorem and a Unified Theory
- 4.10: Stokes’ Theorem

- Multivariable Calculus

- Book: Calculus (OpenStax)
- 1: Functions and Graphs
- 2: Limits
- 3: Derivatives
- 3.0: Prelude to Derivatives
- 3.1: Defining the Derivative
- 3.2: The Derivative as a Function
- 3.3: Differentiation Rules
- 3.4: Derivatives as Rates of Change
- 3.5: Derivatives of Trigonometric Functions
- 3.6: The Chain Rule
- 3.7: Derivatives of Inverse Functions
- 3.8: Implicit Differentiation
- 3.9: Derivatives of Exponential and Logarithmic Functions
- 3.E: Derivatives (Exercises)
- 3.S: Derivatives (Summary)

- 4: Applications of Derivatives
- 4.0: Prelude to Applications of Derivatives
- 4.1: Related Rates
- 4.2: Linear Approximations and Differentials
- 4.3: Maxima and Minima
- 4.4: The Mean Value Theorem
- 4.5: Derivatives and the Shape of a Graph
- 4.6: Limits at Infinity and Asymptotes
- 4.7: Applied Optimization Problems
- 4.8: L’Hôpital’s Rule
- 4.9: Newton’s Method
- 4.10: Antiderivatives
- 4.E: Applications of Derivatives (Exercises)

- 5: Integration
- 5.1: Approximating Areas
- 5.2: The Definite Integral
- 5.3: The Fundamental Theorem of Calculus
- 5.4: Integration Formulas and the Net Change Theorem
- 5.5: Substitution
- 5.6: Integrals Involving Exponential and Logarithmic Functions
- 5.7: Integrals Resulting in Inverse Trigonometric Functions
- 5.E: Integration (Exercises)

- 6: Applications of Integration
- 6.0: Prelude to Applications of Integration
- 6.1: Areas between Curves
- 6.2: Determining Volumes by Slicing
- 6.3: Volumes of Revolution: Cylindrical Shells
- 6.4: Arc Length of a Curve and Surface Area
- 6.5: Physical Applications of Integration
- 6.6: Moments and Centers of Mass
- 6.7: Integrals, Exponential Functions, and Logarithms
- 6.8: Exponential Growth and Decay
- 6.9: Calculus of the Hyperbolic Functions
- 6.E: Applications of Integration (Exercises)

- 7: Techniques of Integration
- 8: Introduction to Differential Equations
- 9: Sequences and Series
- 10: Power Series
- 11: Parametric Equations & Polar Coordinates
- 12: Vectors in Space
- 13: Vector-Valued Functions
- 14: Differentiation of Functions of Several Variables
- 14.0: Prelude to Differentiation of Functions of Several Variables
- 14.1: Functions of Several Variables
- 14.2: Limits and Continuity
- 14.3: Partial Derivatives
- 14.4: Tangent Planes and Linear Approximations
- 14.5: The Chain Rule for Multivariable Functions
- 14.6: Directional Derivatives and the Gradient
- 14.7: Maxima/Minima Problems
- 14.8: Lagrange Multipliers
- 14.E: Differentiation of Functions of Several Variables (Exercises)

- 15: Multiple Integration
- 15.0: Prelude to Multiple Integration
- 15.1: Double Integrals over Rectangular Regions
- 15.2: Double Integrals over General Regions
- 15.3: Double Integrals in Polar Coordinates
- 15.4: Triple Integrals
- 15.5: Triple Integrals in Cylindrical and Spherical Coordinates
- 15.6: Calculating Centers of Mass and Moments of Inertia
- 15.7: Change of Variables in Multiple Integrals
- 15.E: Multiple Integration (Exercises)

- 16: Vector Calculus
- 17: Second-Order Differential Equations
- Appendices

- Book: Active Calculus (Boelkins et al.)
- 1: Understanding the Derivative
- 1.1: How do we Measure Velocity?
- 1.2: The Notion of Limit
- 1.3: The Derivative of a Function at a Point
- 1.4: The Derivative Function
- 1.5: Interpretating, Estimating, and Using the Derivative
- 1.6: The Second Derivative
- 1.7: Limits, Continuity, and Differentiability
- 1.8: The Tangent Line Approximation
- 1.E: Understanding the Derivative (Exercises)

- 2: Computing Derivatives
- 2.1: Elementary Derivative Rules
- 2.2: The Sine and Cosine Function
- 2.3: The Product and Quotient Rules
- 2.4: Derivatives of Other Trigonometric Functions
- 2.5: The Chain Rule
- 2.6: Derivatives of Inverse Functions
- 2.7: Derivatives of Functions Given Implicitely
- 2.8: Using Derivatives to Evaluate Limits
- 2.E: Computing Derivatives (Exercises)

- 3: Using Derivatives
- 4: The Definite Integral
- 5: Finding Antiderivatives and Evaluating Integrals
- 6: Using Definite Integrals
- 7: Differential Equations
- 8: Sequences and Series
- 9: Multivariable and Vector Functions
- 10: Derivatives of Multivariable Functions
- 11: Multiple Integrals

- 1: Understanding the Derivative
- Book: Calculus (Guichard)
- 1: Analytic Geometry
- 2: Instantaneous Rate of Change: The Derivative
- 3: Rules for Finding Derivatives
- 4: Transcendental Functions
- 4.1: Trigonometric Functions
- 4.2: The Derivative of 1/sin x
- 4.3: A Hard Limit
- 4.4: The Derivative of sin x - II
- 4.5: Derivatives of the Trigonometric Functions
- 4.6: Exponential and Logarithmic Functions
- 4.7: Derivatives of the Exponential and Logarithmic Functions
- 4.8: Implicit Differentiation
- 4.9: Inverse Trigonometric Functions
- 4.10: Limits Revisited
- 4.11: Hyperbolic Functions
- 4.E: Transcendental Functions (Exercises)

- 5: Curve Sketching
- 6: Applications of the Derivative
- 7: Integration
- 8: Techniques of Integration
- 9: Applications of Integration
- 10: Polar Coordinates & Parametric Equations
- 11: Sequences and Series
- 11.0: Prelude to Sequences and Series
- 11.1: Sequences
- 11.2: Series
- 11.3: The Integral Test
- 11.4: Alternating Series
- 11.5: Comparison Test
- 11.6: Absolute Convergence
- 11.7: The Ratio and Root Tests
- 11.8: Power Series
- 11.9: Calculus with Power Series
- 11.10: Taylor Series
- 11.11: Taylor's Theorem
- 11.12: Additional Exercises
- 11.E: Sequences and Series (Exercises)

- 12: Three Dimensions
- 13: Vector Functions
- 14: Partial Differentiation
- 15: Multiple Integration
- 16: Vector Calculus
- 17: Differential Equations

- Book: Calculus (Apex)
- 1: Limits
- 2: Derivatives
- 3: The Graphical Behavior of Functions
- 4: Applications of the Derivative
- 5: Integration
- 6: Techniques of Integration
- 7: Applications of Integration
- 8: Sequences and Series
- 9: Curves in the Plane
- 10: Vectors
- 11: Vector-Valued Functions
- 12: Functions of Several Variables
- 12.1: Introduction to Multivariable Functions
- 12.2: Limits and Continuity of Multivariable Functions
- 12.3: Partial Derivatives
- 12.4: Differentiability and the Total Differential
- 12.5: The Multivariable Chain Rule
- 12.6: Directional Derivatives
- 12.7: Tangent Lines, Normal Lines, and Tangent Planes
- 12.8: Extreme Values
- 12.E: Applications of Functions of Several Variables (Exercises)

- 13: Multiple Integration
- 14: Appendix

- Map: Calculus - Early Transcendentals (Stewart)
- 1: Functions and Models
- 2: Limits and Derivatives
- 3: Differentiation Rules
- 3.1: Derivatives of Polynomials and Exponential Functions
- 3.2: The Product and Quotient Rules
- 3.3: Derivatives of Trigonometric Functions
- 3.4: The Chain Rule
- 3.5: Implicit Differentiation
- 3.6: Derivatives of Logarithmic Functions
- 3.7: Rates of Change in the Natural and Social Sciences
- 3.8: Exponential Growth and Decay
- 3.9: Related Rates
- 3.10: Linear Approximations and Differentials
- 3.11: Hyperbolic Functions

- 4: Applications of Differentiation
- 5: Integrals
- 6: Applications of Integration
- 7: Techniques of Integration
- 8: Further Applications of Integration
- 9: Differential Equations
- 10: Parametric Equations And Polar Coordinates
- 11: Infinite Sequences And Series
- 11.1: Sequences
- 11.2: Series
- 11.3: The Integral Test and Estimates of Sums
- 11.4: The Comparison Tests
- 11.5: Alternating Series
- 11.6: Absolute Convergence and the Ratio and Root Test
- 11.7: Strategy for Testing Series
- 11.8: Power Series
- 11.9: Representations of Functions as Power Series
- 11.10: Taylor and Maclaurin Series
- 11.11: Applications of Taylor Polynomials

- 12: Vectors and The Geometry of Space
- 13: Vector Functions
- 14: Partial Derivatives
- 15: Multiple Integrals
- 15.1: Double Integrals over Rectangles
- 15.2: Double Integrals over General Regions
- 15.3: Double Integrals in Polar Coordinates
- 15.4: Applications of Double Integrals
- 15.5: Surface Area
- 15.6: Triple Integrals
- 15.7: Triple Integrals in Cylindrical Coordinates
- 15.8: Triple Integrals in Spherical Coordinates
- 15.9: Change of Variables in Multiple Integrals

- 16: Vector Calculus
- 17: Second-Order Differential Equations
- Chapter 18
- Chapter 19
- Chapter 20

- Map: University Calculus - Early Transcendentals (Hass et al.)
- 1: Functions
- 2: Limits and Continuity
- 3: Differentiation
- 3.1: Tangents and the Derivative at a Point
- 3.2: The Derivative as a Function
- 3.3: Differentiation Rules
- 3.4: The Derivative as a Rate of Change
- 3.5: Derivatives of Trigonometric Functions
- 3.6: The Chain Rule
- 3.7: Implicit Differentiation
- 3.8: Derivatives of Inverse Functions and Logarithms
- 3.9: Inverse Trigonometric Functions
- 3.10: Related Rates
- 3.11: Linearization and Differentials

- 4: Applications of Definite Integrals
- 5: Integration
- 6: Applications of Definite Integrals
- 7: Integrals and Transcendental Functions
- 8: Techniques of Integration
- 9: Infinite Sequence and Series
- 9.10: The Binomial Series and Applications of Taylor Series
- 9.1: Sequences
- 9.2: Infinite Series
- 9.3: The Integral Test
- 9.4: Comparison Tests
- 9.5: The Ratio and Root Tests
- 9.6: Alternating Series, Absolute and Conditional Convergence
- 9.7: Power Series
- 9.8: Taylor and Maclaurin Series
- 9.9: Convergence of Taylor Series

- 10: Parametric Equations and Polar Coordinates
- 11: Vectors and the Geometry of Space
- 12: Vector-Valued Functions and Motion in Space
- 13: Partial Derivatives
- 14: Multiple Integrals
- 14.1: Double and Iterated Integrals over Rectangles
- 14.2: Double Integrals over General Regions
- 14.3: Area by Double Integration
- 14.4: Double Integrals in Polar Form
- 14.5: Triple Integrals in Rectangular Coordinates
- 14.6: Moments and Centers of Mass
- 14.7: Triple Integrals in Cylindrical and Spherical Coordinates
- 14.8: Substitutions in Multiple Integrals

- 15: Integration in Vector Fields
- 15.1: Line Integrals
- 15.2: Vector Fields and Line Integrals: Work, Circulation, and Flux
- 15.3: Path Independence, Conservative Fields, and Potential Functions
- 15.4: Green's Theorem in the Plane
- 15.5: Surfaces and Area
- 15.6: Surface Integrals
- 15.7: Stokes' Theorem
- 15.8: The Divergence Theorem and a Unified Theory

- 16: First-Order Differential Equations
- 17: Second-Order Differential Equations

- Book: Vector Calculus (Corral)
- 1: Vectors in Euclidean Space
- 2: Functions of Several Variables
- 2.1: Functions of Two or Three Variables
- 2.2: Partial Derivatives
- 2.3: Tangent Plane to a Surface
- 2.4: Directional Derivatives and the Gradient
- 2.5: Maxima and Minima
- 2.6: Unconstrained Optimization: Numerical Methods
- 2.7: Constrained Optimization - Lagrange Multipliers
- 2.E: Functions of Several Variables (Exercises)

- 3: Multiple Integrals
- 3.1: Double Integrals
- 3.2: Double Integrals Over a General Region
- 3.3: Triple Integrals
- 3.4: Numerical Approximation of Multiple Integrals
- 3.5: Change of Variables in Multiple Integrals
- 3.6: Application: Center of Mass
- 3.7: Application: Probability and Expectation Values
- 3.E: Multiple Integrals (Exercises)

- 4: Line and Surface Integrals

Tue, 14 Feb 2017 16:52:59 GMT

Calculus

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.

*Thumbnail: The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus. Image used with permission (GNU Free Documentation License, Version 1.3 and CC- SA-BY 3.0; Wikipedia).*