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Interactive Calculus

From Visualization to Simulation

Master calculus through interactive visualizations and simulations. Cover differential and integral calculus, multivariable calculus, differential equations, and PDEs with real-world applications in physics, engineering, economics, and biology.

53Chapters
402Sections
150hReading
10Parts
Start chapter 01Browse curriculum
Part IFoundations32Part IIDifferential Calculus39Part IIIIntegral Calculus40Part IVSeries24Part VMultivariable38Part VIODEs42Part VIIPDEs72Part VIIINumerical Methods25Part IXAdvanced Topics19Part XApplications24
Curriculum

53 chapters— in publication order.

Part I

Foundations

Functions, limits, and continuity

  1. 01Mathematical Functions - The Building Blocks15 sections · 275m

    Essential functions with interactive visualizations: exponential, logarithmic, trigonometric, and more

    1.1–1.15
    1.1What is a Function? Visualizing Input-Output Relationships1.2Linear Functions and Rate of Change Preview1.3Polynomial Functions: Shapes and Behaviors1.4Power Functions and Scaling Laws1.5Exponential Functions: The Mathematics of Growth1.6Logarithmic Functions: Inverting Exponentials1.7Trigonometric Functions: Circular Motion1.8Inverse Trigonometric Functions1.9Hyperbolic Functions: The Other Trigonometry1.10Parametric Representations1.11Piecewise and Absolute Value Functions1.12Function Transformations: Shift, Scale, Reflect1.13Composition and Inverse Functions1.14How Derivatives and Integrals Connect: The Two Directions of Change1.15The Second Derivative: How Change Itself Changes
  2. 02Limits - Approaching the Infinite9 sections · 140m

    The conceptual foundation for all of calculus

    2.1–2.9
    2.1The Intuition of Limits: Getting Arbitrarily Close2.2One-Sided Limits and Jump Discontinuities2.3Limits at Infinity: Horizontal Asymptotes2.4Infinite Limits: Vertical Asymptotes2.5The Formal Definition: Epsilon-Delta2.6Limit Laws and Computation Strategies2.7The Squeeze Theorem2.8Special Limits: sin(x)/x and (1+1/n)^n2.9L'Hôpital's Rule Preview
  3. 03Continuity - No Breaks, No Jumps8 sections · 125m

    Connecting limits to function behavior

    3.1–3.8
    3.1Intuitive Continuity: Drawing Without Lifting3.2The Three Conditions for Continuity3.3Types of Discontinuities3.4Properties of Continuous Functions3.5The Intermediate Value Theorem3.6The Extreme Value Theorem3.7Uniform Continuity3.8Applications: Root Finding Algorithms

Part II

Differential Calculus

Derivatives and applications

  1. 04The Derivative - Instantaneous Rate of Change10 sections · 169m

    The central concept of differential calculus

    4.1–4.10
    4.1From Average to Instantaneous: The Derivative Concept4.2The Derivative as a Function4.3The Formal Definition: Limit of Difference Quotient4.4Differentiability and Corners4.5Basic Derivative Rules: Power, Sum, Constant4.6The Product Rule4.7The Quotient Rule4.8The Chain Rule: Derivatives of Compositions4.9Implicit Differentiation4.10Higher-Order Derivatives
  2. 05Derivatives of Transcendental Functions10 sections · 147m

    Exponential, logarithmic, and trigonometric derivatives

    5.1–5.10
    5.1Derivative of e^x: The Special Exponential5.2Derivatives of General Exponentials: a^x5.3Derivative of ln(x)5.4Derivatives of General Logarithms5.5Logarithmic Differentiation5.6Derivatives of Sine and Cosine5.7Derivatives of Other Trig Functions5.8Derivatives of Inverse Trig Functions5.9Derivatives of Hyperbolic Functions5.10Derivatives of Inverse Hyperbolic Functions
  3. 06Applications of Differentiation11 sections · 209m

    Using derivatives to understand function behavior

    6.1–6.11
    6.1Related Rates: Connecting Changing Quantities6.2Linear Approximation and Differentials6.3Extrema: Maximum and Minimum Values6.4The Mean Value Theorem6.5First Derivative Test6.6Concavity and the Second Derivative6.7Second Derivative Test for Extrema6.8Curve Sketching: Putting It All Together6.9Optimization Problems6.10Newton's Method6.11Antiderivatives Introduction
  4. 07Applications in Physics and Engineering8 sections · 144m

    Real-world differential calculus applications

    7.1–7.8
    7.1Position, Velocity, Acceleration7.2Projectile Motion Analysis7.3Simple Harmonic Motion7.4Electric Circuits: RC and RL Transients7.5Marginal Analysis in Economics7.6Population Dynamics7.7Chemical Reaction Rates7.8Computer Graphics: Curves and Normals

Part III

Integral Calculus

Integration and applications

  1. 08The Definite Integral10 sections · 179m

    From sums to areas

    8.1–8.10
    8.1The Area Problem: Approximating with Rectangles8.2Left, Right, and Midpoint Rules8.3Sigma Notation and Summation8.4The Definite Integral as a Limit8.5Properties of Definite Integrals8.6The Fundamental Theorem of Calculus (Part 1)8.7The Fundamental Theorem of Calculus (Part 2)8.8Average Value of a Function8.9Numerical Integration: Trapezoidal Rule8.10Simpson's Rule
  2. 09The Indefinite Integral and Antiderivatives10 sections · 200m

    Reversing differentiation

    9.1–9.10
    9.1Antiderivatives and the Constant of Integration9.2Basic Integration Rules9.3Integration by Substitution (u-substitution)9.4Integration by Parts9.5Trigonometric Integrals9.6Trigonometric Substitution9.7Partial Fractions Decomposition9.8Improper Integrals: Infinite Limits9.9Improper Integrals: Discontinuous Integrands9.10Comparison Tests for Improper Integrals
  3. 10Applications of Integration10 sections · 198m

    Using integrals to solve real problems

    10.1–10.10
    10.1Area Between Curves10.2Volumes by Slicing: Disk Method10.3Volumes by Shells: Cylindrical Shell Method10.4Volumes by Cross-Sections10.5Arc Length10.6Surface Area of Revolution10.7Work and Energy10.8Hydrostatic Force and Pressure10.9Moments and Center of Mass10.10Probability Density Functions
  4. 11Applications in Physics and Engineering (Integration)10 sections · 187m

    Real-world integral applications

    11.1–11.10
    11.1Finding Position from Velocity11.2Work Done by Variable Forces11.3Fluid Dynamics: Flow Rate11.4Electric Charge Distribution11.5Heat Transfer and Thermal Energy11.6Economic Surplus: Consumer and Producer11.7Probability and Statistics Connection11.8Signal Processing: Convolution Preview11.9Impulse and Momentum11.10Moment of Inertia and Rotational Dynamics

Part IV

Series

Sequences, series, and power series

  1. 12Sequences5 sections · 81m

    Ordered lists approaching limits

    12.1–12.5
    12.1Introduction to Sequences12.2Limit of a Sequence12.3Bounded and Monotonic Sequences12.4Recursive Sequences12.5Sequences in Computer Science
  2. 13Infinite Series10 sections · 167m

    Summing infinitely many terms

    13.1–13.10
    13.1Introduction to Infinite Series13.2Geometric Series13.3The Divergence Test13.4The Integral Test13.5p-Series and Harmonic Series13.6Comparison Tests13.7Alternating Series13.8Absolute vs Conditional Convergence13.9Ratio and Root Tests13.10Strategy for Testing Convergence
  3. 14Power Series9 sections · 183m

    Functions as infinite polynomials

    14.1–14.9
    14.1Power Series Definition14.2Radius and Interval of Convergence14.3Operations on Power Series14.4Taylor Series14.5Maclaurin Series14.6Taylor Series Error Bounds14.7Applications of Taylor Series14.8Binomial Series14.9Fourier Series Introduction

Part V

Multivariable

Vectors and multivariable calculus

  1. 15Vectors and the Geometry of Space7 sections · 133m

    Foundation for multivariable calculus

    15.1–15.7
    15.1Three-Dimensional Coordinate Systems15.2Vectors in 2D and 3D15.3The Dot Product15.4The Cross Product15.5Lines and Planes in Space15.6Quadric Surfaces15.7Cylindrical and Spherical Coordinates
  2. 16Vector-Valued Functions6 sections · 125m

    Curves in space

    16.1–16.6
    16.1Vector Functions and Space Curves16.2Derivatives of Vector Functions16.3Arc Length and Curvature16.4Motion in Space16.5The TNB Frame16.6Planetary Motion and Kepler's Laws
  3. 17Partial Derivatives8 sections · 180m

    Differentiation in multiple dimensions

    17.1–17.8
    17.1Functions of Several Variables17.2Limits and Continuity in Multiple Dimensions17.3Partial Derivatives17.4Tangent Planes and Linear Approximations17.5The Chain Rule for Multivariable Functions17.6Directional Derivatives and the Gradient17.7Maximum and Minimum Values17.8Lagrange Multipliers
  4. 18Multiple Integrals8 sections · 183m

    Integration in higher dimensions

    18.1–18.8
    18.1Double Integrals over Rectangles18.2Double Integrals over General Regions18.3Double Integrals in Polar Coordinates18.4Applications of Double Integrals18.5Triple Integrals18.6Triple Integrals in Cylindrical Coordinates18.7Triple Integrals in Spherical Coordinates18.8Change of Variables: The Jacobian
  5. 19Vector Calculus9 sections · 205m

    Calculus of vector fields

    19.1–19.9
    19.1Vector Fields19.2Line Integrals19.3The Fundamental Theorem for Line Integrals19.4Green's Theorem19.5Curl and Divergence19.6Parametric Surfaces19.7Surface Integrals19.8Stokes' Theorem19.9The Divergence Theorem

Part VI

ODEs

Ordinary differential equations

  1. 20Introduction to Differential Equations5 sections · 98m

    Equations involving derivatives

    20.1–20.5
    20.1What is a Differential Equation?20.2Direction Fields and Solution Curves20.3Separable Equations20.4Euler's Method20.5Existence and Uniqueness
  2. 21First-Order Differential Equations8 sections · 158m

    Single derivative equations

    21.1–21.8
    21.1Linear First-Order Equations21.2Exact Equations21.3Substitution Methods21.4Applications: Growth and Decay21.5Applications: Newton's Law of Cooling21.6Applications: Mixing Problems21.7Applications: Population Models21.8Applications: Circuits
  3. 22Second-Order Differential Equations11 sections · 261m

    Two derivatives in play

    22.1–22.11
    22.1Homogeneous Equations with Constant Coefficients22.2Complex Roots and Oscillations22.3Repeated Roots22.4Nonhomogeneous Equations: Undetermined Coefficients22.5Variation of Parameters22.6Mechanical Vibrations22.7Forced Oscillations and Resonance22.8Electric Circuits: RLC22.9Cauchy–Euler Equations: Variable Coefficients22.10Laplace Transforms for Second-Order ODEs22.11Boundary Value Problems and Eigenvalues
  4. 23Systems of Differential Equations10 sections · 227m

    Multiple coupled equations

    23.1–23.10
    23.1Introduction to Systems23.2The Eigenvalue Method23.3Phase Portraits23.4Real Distinct Eigenvalues23.5Complex Eigenvalues23.6Repeated Eigenvalues23.7Nonlinear Systems and Linearization23.8Predator-Prey Models23.9Competing Species23.10The Lorenz System and Chaos
  5. 24Laplace Transforms8 sections · 180m

    Algebraic approach to DEs

    24.1–24.8
    24.1The Laplace Transform24.2Properties of Laplace Transforms24.3Inverse Laplace Transforms24.4Solving Initial Value Problems24.5Step Functions and Discontinuous Forcing24.6Impulse Functions and the Delta Function24.7Convolution24.8Transfer Functions

Part VII

PDEs

Partial differential equations

  1. 25Introduction to PDEs4 sections · 83m

    Equations in multiple variables

    25.1What are Partial Differential Equations?25.2Classification: Elliptic, Parabolic, Hyperbolic25.3Boundary and Initial Conditions25.4Separation of Variables Method
  2. 26The Heat Equation9 sections · 217m

    Diffusion and temperature

    26.1–26.9
    26.1Derivation of the Heat Equation26.2Heat Equation on a Finite Rod26.3Fourier Series Solutions26.4Steady-State Solutions26.5Heat Equation in 2D26.6Numerical Methods: Finite Differences26.7Applications: Thermal Analysis26.8Applications: Diffusion in Biology26.9Applications: Financial Mathematics
  3. 27The Wave Equation8 sections · 194m

    Vibrations and propagation

    27.1–27.8
    27.1Derivation of the Wave Equation27.2D'Alembert's Solution27.3Standing Waves and Normal Modes27.4Wave Equation in 2D: Membranes27.5Numerical Methods for Waves27.6Applications: Musical Instruments27.7Applications: Electromagnetic Waves27.8Applications: Seismic Waves
  4. 28Laplace's Equation8 sections · 180m

    Equilibrium and potential

    28.1–28.8
    28.1Laplace's Equation and Harmonic Functions28.2Boundary Value Problems28.3Laplace's Equation in a Rectangle28.4Laplace's Equation in a Circle28.5Poisson's Equation Preview28.6Applications: Electrostatics28.7Applications: Fluid Flow28.8Applications: Steady-State Heat
  5. 29The Schrödinger Equation8 sections · 229m

    Quantum mechanics and wave functions

    29.1–29.8
    29.1Introduction to Quantum Mechanics29.2The Time-Independent Schrödinger Equation29.3The Time-Dependent Schrödinger Equation29.4Particle in a Box: Infinite Square Well29.5The Quantum Harmonic Oscillator29.6Tunneling and Barrier Penetration29.7The Hydrogen Atom29.8Numerical Methods for Schrödinger Equation
  6. 30The Navier-Stokes Equations9 sections · 246m

    Fluid dynamics and turbulence

    30.1–30.9
    30.1Introduction to Fluid Mechanics30.2Derivation of the Navier-Stokes Equations30.3The Continuity Equation30.4Viscosity and the Stress Tensor30.5Boundary Conditions in Fluid Flow30.6Laminar vs Turbulent Flow30.7The Millennium Prize Problem30.8Numerical Methods: CFD Basics30.9Applications: Aerodynamics and Weather
  7. 31The Black-Scholes Equation9 sections · 270m

    Mathematical finance and options pricing

    31.1–31.9
    31.1Introduction to Financial Derivatives31.2Stochastic Calculus: Brownian Motion31.3Itô's Lemma and Stochastic Differential Equations31.4Derivation of the Black-Scholes PDE31.5The Black-Scholes Formula31.6The Greeks: Delta, Gamma, Theta, Vega31.7Implied Volatility and the Volatility Smile31.8Monte Carlo Methods for Option Pricing31.9Extensions: American Options and Exotic Derivatives
  8. 32Maxwell's Equations9 sections · 246m

    Electromagnetism and electromagnetic waves

    32.1–32.9
    32.1Introduction to Electromagnetism32.2Gauss's Law for Electric Fields32.3Gauss's Law for Magnetism32.4Faraday's Law of Induction32.5Ampère-Maxwell Law32.6Maxwell's Equations in Differential Form32.7Electromagnetic Waves32.8Energy and the Poynting Vector32.9Applications: Antennas and Optics
  9. 33Poisson's Equation8 sections · 216m

    Sources, sinks, and potential theory

    33.1–33.8
    33.1From Laplace to Poisson: Adding Sources33.2Physical Interpretation and Applications33.3Green's Functions for Poisson's Equation33.4Poisson's Equation in Electrostatics33.5Gravitational Potential and Newton's Law33.6Image Processing: Poisson Blending33.7Numerical Methods: Relaxation Techniques33.8Applications in Machine Learning

Part VIII

Numerical Methods

Computational calculus

  1. 34Numerical Differentiation5 sections · 109m

    Approximating derivatives

    34.1–34.5
    34.1Forward, Backward, and Central Differences34.2Higher-Order Finite Difference Formulas34.3Error Analysis34.4Richardson Extrapolation34.5Applications: Image Edge Detection
  2. 35Numerical Integration6 sections · 137m

    Computing integrals numerically

    35.1–35.6
    35.1Review: Riemann Sums and Trapezoidal Rule35.2Simpson's Rule and Higher-Order Methods35.3Gaussian Quadrature35.4Adaptive Integration35.5Monte Carlo Integration35.6Multidimensional Integration
  3. 36Numerical Solutions of ODEs8 sections · 192m

    Solving differential equations computationally

    36.1–36.8
    36.1Euler's Method Revisited36.2Improved Euler (Heun's) Method36.3Runge-Kutta Methods (RK4)36.4Adaptive Step Size Control36.5Stiff Equations36.6Systems of ODEs36.7Higher-Order ODEs36.8Boundary Value Problems
  4. 37Numerical Solutions of PDEs6 sections · 158m

    Solving partial differential equations

    37.1–37.6
    37.1Finite Difference Methods for PDEs37.2Explicit vs Implicit Methods37.3Crank-Nicolson Method37.4Solving Elliptic PDEs37.5The Finite Element Method Introduction37.6Stability and Convergence Analysis

Part IX

Advanced Topics

Special functions and transforms

  1. 38Calculus of Variations6 sections · 155m

    Optimizing functionals

    38.1–38.6
    38.1Introduction to Functionals38.2The Euler-Lagrange Equation38.3The Brachistochrone Problem38.4Geodesics on Surfaces38.5Isoperimetric Problems38.6Applications in Physics: Lagrangian Mechanics
  2. 39Special Functions7 sections · 170m

    Functions arising from applications

    39.1–39.7
    39.1The Gamma Function39.2The Beta Function39.3Bessel Functions39.4Legendre Polynomials39.5Spherical Harmonics39.6Error Function and Gaussian Integral39.7Elliptic Integrals
  3. 40Transform Methods6 sections · 161m

    Beyond Laplace

    40.1–40.6
    40.1Fourier Transform40.2Discrete Fourier Transform and FFT40.3Applications: Signal Processing40.4Applications: Image Processing40.5Z-Transform Introduction40.6Wavelet Transform Introduction

Part X

Applications

Real-world projects and simulations

  1. 41Physics Simulations4 sections · 145m

    Calculus in action

    41.1N-Body Gravitational Simulation41.2Electromagnetic Field Simulator41.3Fluid Dynamics Visualization41.4Quantum Mechanics Simulator
  2. 42Engineering Applications4 sections · 130m

    Practical engineering uses

    42.1Structural Analysis42.2Control Systems42.3Signal Processing Pipeline42.4Heat Exchanger Design
  3. 43Economics and Finance4 sections · 123m

    Calculus in business

    43.1Advanced Option Pricing43.2Economic Growth Models43.3Portfolio Optimization43.4Supply Chain Optimization
  4. 44Biology and Medicine4 sections · 121m

    Life science applications

    44.1Epidemic Modeling: SIR Model44.2Pharmacokinetics44.3Neural Firing: Hodgkin-Huxley44.4Population Ecology
  5. 45Computer Graphics4 sections · 130m

    Visual computing with calculus

    45.1Bezier and B-Spline Curves45.2Surface Modeling45.3Ray Tracing Fundamentals45.4Physics-Based Animation
  6. 46Machine Learning Connections4 sections · 120m

    Calculus meets AI

    46.1Gradient Descent Optimization46.2Backpropagation and the Chain Rule46.3Regularization and Calculus46.4Neural Network as Function Approximator
The capstone

Where the book lands in practice.

Chapter 41·4 sections

Physics Simulations

Calculus in action

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Chapter 42·4 sections

Engineering Applications

Practical engineering uses

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Chapter 43·4 sections

Economics and Finance

Calculus in business

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Chapter 44·4 sections

Biology and Medicine

Life science applications

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Chapter 45·4 sections

Computer Graphics

Visual computing with calculus

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Chapter 46·4 sections

Machine Learning Connections

Calculus meets AI

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402 sections. Begin with one.

Chapter 1 — Mathematical Functions - The Building Blocks — is where every reader starts.

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