Calculus
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- 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
- 4.1: Logs and Derivatives
- 4.2: Logs and Integrals
- 4.3: Exponentials With Other Bases
- 4.4: Inverse Functions
- 4.5: The Derivative and Integral of the Exponential Function
- 4.6: Exponential Growth and Decay
- 4.7: Inverse Trigonometric Derivatives
- 4.8: Integrals Involving Arctrig Functions
- 4.9: Hyperbolic 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
- Map: University Calculus (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
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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).