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Book · Beginner · 120+ hours
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.
46Chapters
353Sections
130hReading
10Parts
Part I·3 chapters · 30 sections
Foundations— Functions, limits, and continuity.
Mathematical Functions - The Building Blocks
Essential functions with interactive visualizations: exponential, logarithmic, trigonometric, and more
13 sections213 min read
- 01What is a Function? Visualizing Input-Output Relationships15m
- 02Linear Functions and Rate of Change Preview12m
- 03Polynomial Functions: Shapes and Behaviors18m
- 04Power Functions and Scaling Laws15m
- 05Exponential Functions: The Mathematics of Growth20m
- 06Logarithmic Functions: Inverting Exponentials18m
- 07Trigonometric Functions: Circular Motion25m
- 08Inverse Trigonometric Functions15m
- 09Hyperbolic Functions: The Other Trigonometry18m
- 10Parametric Representations15m
- 11Piecewise and Absolute Value Functions12m
- 12Function Transformations: Shift, Scale, Reflect15m
- 13Composition and Inverse Functions15m
Limits - Approaching the Infinite
The conceptual foundation for all of calculus
9 sections140 min read
- 01The Intuition of Limits: Getting Arbitrarily Close15m
- 02One-Sided Limits and Jump Discontinuities12m
- 03Limits at Infinity: Horizontal Asymptotes15m
- 04Infinite Limits: Vertical Asymptotes12m
- 05The Formal Definition: Epsilon-Delta20m
- 06Limit Laws and Computation Strategies18m
- 07The Squeeze Theorem15m
- 08Special Limits: sin(x)/x and (1+1/n)^n15m
- 09L'Hôpital's Rule Preview18m
Continuity - No Breaks, No Jumps
Connecting limits to function behavior
8 sections125 min read
Part II·4 chapters · 39 sections
Differential Calculus— Derivatives and applications.
The Derivative - Instantaneous Rate of Change
The central concept of differential calculus
10 sections169 min read
- 01From Average to Instantaneous: The Derivative Concept18m
- 02The Derivative as a Function15m
- 03The Formal Definition: Limit of Difference Quotient20m
- 04Differentiability and Corners15m
- 05Basic Derivative Rules: Power, Sum, Constant18m
- 06The Product Rule15m
- 07The Quotient Rule15m
- 08The Chain Rule: Derivatives of Compositions20m
- 09Implicit Differentiation18m
- 10Higher-Order Derivatives15m
Derivatives of Transcendental Functions
Exponential, logarithmic, and trigonometric derivatives
10 sections147 min read
- 01Derivative of e^x: The Special Exponential15m
- 02Derivatives of General Exponentials: a^x12m
- 03Derivative of ln(x)12m
- 04Derivatives of General Logarithms12m
- 05Logarithmic Differentiation15m
- 06Derivatives of Sine and Cosine18m
- 07Derivatives of Other Trig Functions15m
- 08Derivatives of Inverse Trig Functions18m
- 09Derivatives of Hyperbolic Functions15m
- 10Derivatives of Inverse Hyperbolic Functions15m
Applications of Differentiation
Using derivatives to understand function behavior
11 sections209 min read
- 01Related Rates: Connecting Changing Quantities20m
- 02Linear Approximation and Differentials18m
- 03Extrema: Maximum and Minimum Values18m
- 04The Mean Value Theorem20m
- 05First Derivative Test15m
- 06Concavity and the Second Derivative18m
- 07Second Derivative Test for Extrema15m
- 08Curve Sketching: Putting It All Together25m
- 09Optimization Problems25m
- 10Newton's Method20m
- 11Antiderivatives Introduction15m
Applications in Physics and Engineering
Real-world differential calculus applications
8 sections144 min read
Part III·4 chapters · 40 sections
Integral Calculus— Integration and applications.
The Definite Integral
From sums to areas
10 sections179 min read
- 01The Area Problem: Approximating with Rectangles18m
- 02Left, Right, and Midpoint Rules15m
- 03Sigma Notation and Summation15m
- 04The Definite Integral as a Limit20m
- 05Properties of Definite Integrals15m
- 06The Fundamental Theorem of Calculus (Part 1)25m
- 07The Fundamental Theorem of Calculus (Part 2)20m
- 08Average Value of a Function15m
- 09Numerical Integration: Trapezoidal Rule18m
- 10Simpson's Rule18m
The Indefinite Integral and Antiderivatives
Reversing differentiation
10 sections200 min read
- 01Antiderivatives and the Constant of Integration15m
- 02Basic Integration Rules18m
- 03Integration by Substitution (u-substitution)22m
- 04Integration by Parts20m
- 05Trigonometric Integrals22m
- 06Trigonometric Substitution25m
- 07Partial Fractions Decomposition25m
- 08Improper Integrals: Infinite Limits20m
- 09Improper Integrals: Discontinuous Integrands18m
- 10Comparison Tests for Improper Integrals15m
Applications of Integration
Using integrals to solve real problems
10 sections198 min read
- 01Area Between Curves18m
- 02Volumes by Slicing: Disk Method22m
- 03Volumes by Shells: Cylindrical Shell Method22m
- 04Volumes by Cross-Sections18m
- 05Arc Length18m
- 06Surface Area of Revolution20m
- 07Work and Energy20m
- 08Hydrostatic Force and Pressure18m
- 09Moments and Center of Mass22m
- 10Probability Density Functions20m
Applications in Physics and Engineering (Integration)
Real-world integral applications
10 sections187 min read
- 01Finding Position from Velocity15m
- 02Work Done by Variable Forces18m
- 03Fluid Dynamics: Flow Rate18m
- 04Electric Charge Distribution18m
- 05Heat Transfer and Thermal Energy18m
- 06Economic Surplus: Consumer and Producer15m
- 07Probability and Statistics Connection20m
- 08Signal Processing: Convolution Preview20m
- 09Impulse and Momentum20m
- 10Moment of Inertia and Rotational Dynamics25m
Part IV·3 chapters · 24 sections
Series— Sequences, series, and power series.
Sequences
Ordered lists approaching limits
5 sections81 min read
Infinite Series
Summing infinitely many terms
10 sections167 min read
Power Series
Functions as infinite polynomials
9 sections183 min read
Part V·5 chapters · 38 sections
Multivariable— Vectors and multivariable calculus.
Vectors and the Geometry of Space
Foundation for multivariable calculus
7 sections133 min read
Vector-Valued Functions
Curves in space
6 sections125 min read
Partial Derivatives
Differentiation in multiple dimensions
8 sections180 min read
- 01Functions of Several Variables18m
- 02Limits and Continuity in Multiple Dimensions20m
- 03Partial Derivatives20m
- 04Tangent Planes and Linear Approximations22m
- 05The Chain Rule for Multivariable Functions22m
- 06Directional Derivatives and the Gradient25m
- 07Maximum and Minimum Values25m
- 08Lagrange Multipliers28m
Multiple Integrals
Integration in higher dimensions
8 sections183 min read
- 01Double Integrals over Rectangles20m
- 02Double Integrals over General Regions22m
- 03Double Integrals in Polar Coordinates22m
- 04Applications of Double Integrals25m
- 05Triple Integrals25m
- 06Triple Integrals in Cylindrical Coordinates22m
- 07Triple Integrals in Spherical Coordinates22m
- 08Change of Variables: The Jacobian25m
Vector Calculus
Calculus of vector fields
9 sections205 min read
Part VI·5 chapters · 42 sections
ODEs— Ordinary differential equations.
Introduction to Differential Equations
Equations involving derivatives
5 sections98 min read
First-Order Differential Equations
Single derivative equations
8 sections158 min read
Second-Order Differential Equations
Two derivatives in play
11 sections261 min read
- 01Homogeneous Equations with Constant Coefficients22m
- 02Complex Roots and Oscillations22m
- 03Repeated Roots18m
- 04Nonhomogeneous Equations: Undetermined Coefficients25m
- 05Variation of Parameters22m
- 06Mechanical Vibrations25m
- 07Forced Oscillations and Resonance25m
- 08Electric Circuits: RLC22m
- 09Cauchy–Euler Equations: Variable Coefficients24m
- 10Laplace Transforms for Second-Order ODEs28m
- 11Boundary Value Problems and Eigenvalues28m
Systems of Differential Equations
Multiple coupled equations
10 sections227 min read
Laplace Transforms
Algebraic approach to DEs
8 sections180 min read
Part VII·9 chapters · 72 sections
PDEs— Partial differential equations.
Introduction to PDEs
Equations in multiple variables
4 sections83 min read
The Heat Equation
Diffusion and temperature
9 sections217 min read
- 01Derivation of the Heat Equation22m
- 02Heat Equation on a Finite Rod25m
- 03Fourier Series Solutions28m
- 04Steady-State Solutions20m
- 05Heat Equation in 2D25m
- 06Numerical Methods: Finite Differences28m
- 07Applications: Thermal Analysis22m
- 08Applications: Diffusion in Biology22m
- 09Applications: Financial Mathematics25m
The Wave Equation
Vibrations and propagation
8 sections194 min read
Laplace's Equation
Equilibrium and potential
8 sections180 min read
The Schrödinger Equation
Quantum mechanics and wave functions
8 sections229 min read
- 01Introduction to Quantum Mechanics25m
- 02The Time-Independent Schrödinger Equation28m
- 03The Time-Dependent Schrödinger Equation28m
- 04Particle in a Box: Infinite Square Well25m
- 05The Quantum Harmonic Oscillator30m
- 06Tunneling and Barrier Penetration28m
- 07The Hydrogen Atom35m
- 08Numerical Methods for Schrödinger Equation30m
The Navier-Stokes Equations
Fluid dynamics and turbulence
9 sections246 min read
- 01Introduction to Fluid Mechanics25m
- 02Derivation of the Navier-Stokes Equations30m
- 03The Continuity Equation22m
- 04Viscosity and the Stress Tensor28m
- 05Boundary Conditions in Fluid Flow25m
- 06Laminar vs Turbulent Flow28m
- 07The Millennium Prize Problem25m
- 08Numerical Methods: CFD Basics35m
- 09Applications: Aerodynamics and Weather28m
The Black-Scholes Equation
Mathematical finance and options pricing
9 sections270 min read
- 01Introduction to Financial Derivatives25m
- 02Stochastic Calculus: Brownian Motion30m
- 03Itô's Lemma and Stochastic Differential Equations32m
- 04Derivation of the Black-Scholes PDE35m
- 05The Black-Scholes Formula28m
- 06The Greeks: Delta, Gamma, Theta, Vega30m
- 07Implied Volatility and the Volatility Smile28m
- 08Monte Carlo Methods for Option Pricing32m
- 09Extensions: American Options and Exotic Derivatives30m
Maxwell's Equations
Electromagnetism and electromagnetic waves
9 sections246 min read
- 01Introduction to Electromagnetism25m
- 02Gauss's Law for Electric Fields28m
- 03Gauss's Law for Magnetism22m
- 04Faraday's Law of Induction28m
- 05Ampère-Maxwell Law28m
- 06Maxwell's Equations in Differential Form30m
- 07Electromagnetic Waves32m
- 08Energy and the Poynting Vector25m
- 09Applications: Antennas and Optics28m
Poisson's Equation
Sources, sinks, and potential theory
8 sections216 min read
- 01From Laplace to Poisson: Adding Sources22m
- 02Physical Interpretation and Applications25m
- 03Green's Functions for Poisson's Equation30m
- 04Poisson's Equation in Electrostatics28m
- 05Gravitational Potential and Newton's Law28m
- 06Image Processing: Poisson Blending30m
- 07Numerical Methods: Relaxation Techniques28m
- 08Applications in Machine Learning25m
Part VIII·4 chapters · 25 sections
Numerical Methods— Computational calculus.
Numerical Differentiation
Approximating derivatives
5 sections109 min read
Numerical Integration
Computing integrals numerically
6 sections137 min read
Numerical Solutions of ODEs
Solving differential equations computationally
8 sections192 min read
Numerical Solutions of PDEs
Solving partial differential equations
6 sections158 min read
Part IX·3 chapters · 19 sections
Advanced Topics— Special functions and transforms.
Calculus of Variations
Optimizing functionals
6 sections155 min read
Special Functions
Functions arising from applications
7 sections170 min read
Transform Methods
Beyond Laplace
6 sections161 min read
Part X·6 chapters · 24 sections
Applications— Real-world projects and simulations.
Physics Simulations
Calculus in action
4 sections145 min read
Engineering Applications
Practical engineering uses
4 sections130 min read
Economics and Finance
Calculus in business
4 sections123 min read
Biology and Medicine
Life science applications
4 sections121 min read
Computer Graphics
Visual computing with calculus
4 sections130 min read
Machine Learning Connections
Calculus meets AI
4 sections120 min read
The capstone
Where the book lands in practice.
353 sections. Begin with one.
Chapter 1 — Mathematical Functions - The Building Blocks — is where every reader starts.