Calculus×Computing
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Syllabus

The complete syllabus

Every topic in the portal, area by area: the calculus syllabus first, then the computing domains it feeds.

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Calculus and its neighbours

ƒFoundations

Numbers, functions and the families of functions that the rest of calculus works with — and how a computer represents them.

  1. Real numbers
  2. Complex numbers
  3. Intervals
  4. Absolute value
  5. Inequalities
  6. Functions
  7. Domain and range
  8. Composition
  9. Inverse functions
  10. Polynomial functions
  11. Rational functions
  12. Exponential functions
  13. Logarithmic functions
  14. Trigonometric functions
  15. Hyperbolic functions
  16. Transformations of functions

v⃗Linear algebra (bridge)

The minimum of linear algebra that multivariable calculus needs: vectors, the dot product and matrices. The full story lives in the linear algebra chapter of Math of AI.

  1. Vectors
  2. Dot product
  3. Matrices and linear maps

limSequences and limits

What it means to approach a value — the idea everything else in calculus is built on — and its computing twin, asymptotic analysis.

  1. Sequences
  2. Convergence of sequences
  3. Monotone and bounded sequences
  4. Subsequences
  5. Limit of a function
  6. One-sided limits
  7. Infinite limits
  8. Limits at infinity
  9. Infinitesimals and equivalences
  10. Orders of growth
  11. Asymptotic notation (O, o, Ω, Θ)

εContinuity

Functions without jumps, and the theorems that guarantee roots, maxima and minima — the existence results behind bisection, optimization and interpolation.

  1. Continuity
  2. Discontinuities
  3. Uniform continuity
  4. Lipschitz continuity
  5. Bolzano's theorem
  6. Intermediate value theorem
  7. Weierstrass extreme value theorem

f′Derivatives

The instantaneous rate of change: slope, velocity, sensitivity. The single most used idea of calculus in computing.

  1. Derivative◐
  2. Differentiability and one-sided derivatives
  3. Differentiation rules
  4. Derivatives of elementary functions
  5. Chain rule
  6. Implicit differentiation
  7. Higher-order derivatives
  8. Differential and linear approximation

∎Fundamental theorems

Rolle, the mean value theorem, Cauchy, L'Hôpital and the fundamental theorem of calculus: the results that turn local information (derivatives) into global guarantees — error bounds, limits, areas.

  1. Rolle's theorem
  2. Mean value theorem
  3. Cauchy's mean value theorem
  4. L'Hôpital's rule
  5. Fundamental theorem of calculus

minExtrema and optimization

Finding the best: maxima, minima, convexity and optimization under constraints. The mathematics behind training models, planning routes and allocating resources.

  1. Maxima and minima
  2. Critical points and derivative tests
  3. Convexity and concavity
  4. Constrained optimization
  5. Lagrange multipliers
  6. KKT conditions

≈Numerical methods

Calculus as an algorithm: finding roots, interpolating, differentiating and integrating with finite arithmetic — and knowing how wrong the answer can be.

  1. Absolute and relative error
  2. Conditioning
  3. Numerical stability
  4. Order of convergence
  5. Bisection method
  6. Newton's method◐
  7. Fixed-point iteration
  8. Interpolation
  9. Numerical differentiation
  10. Numerical integration (quadrature)

TₙTaylor series

Approximating any smooth function by polynomials built from its derivatives at one point — the most used approximation in science and engineering.

  1. Taylor polynomial◐
  2. Taylor's theorem and the remainder
  3. Power series
  4. Radius of convergence
  5. Taylor and Maclaurin series

∫Integrals

Accumulation: areas, totals, averages and probabilities. The other half of calculus, and the mathematics of Monte Carlo, rendering and probabilistic machine learning.

  1. Antiderivatives and indefinite integrals
  2. Riemann sums◐
  3. Definite integral
  4. Integration by substitution
  5. Integration by parts
  6. Trigonometric integrals
  7. Partial fractions
  8. Improper integrals

ΣSeries

Infinite sums and when they make sense: the tests for convergence, and their use in analysing algorithms, discounting rewards and summing numbers safely in floating point.

  1. Numerical series
  2. Geometric series
  3. Comparison tests
  4. Integral test
  5. Ratio test
  6. Root test
  7. Alternating series
  8. Absolute and conditional convergence

∇Multivariable calculus

Functions of many variables: partial derivatives, the gradient, Jacobians and Hessians. The language in which a neural network with a billion parameters is trained.

  1. Functions of several variables
  2. Surfaces and level sets
  3. Limits and continuity in several variables
  4. Partial derivatives
  5. Gradient
  6. Directional derivative
  7. Total differential and linearization
  8. Multivariable chain rule
  9. Jacobian matrix
  10. Hessian matrix
  11. Extrema in several variables
  12. Multiple integrals and change of variables
  13. Curvature (basic differential geometry)

∮Vector calculus

Fields in space and the operators that describe them — gradient, divergence, curl, Laplacian — plus the integral theorems of Green, Gauss and Stokes. The mathematics of fluids, electromagnetism and physically based graphics.

  1. Scalar fields
  2. Vector fields
  3. Divergence
  4. Curl
  5. Laplacian
  6. Line integrals
  7. Surface integrals and flux
  8. Green's theorem
  9. Divergence theorem (Gauss)
  10. Stokes' theorem

ẏDifferential equations

Laws written as rates of change, and how to solve them — exactly when possible, numerically always. Every simulation, from a game engine to a climate model, is a differential equation being integrated.

  1. Ordinary differential equations
  2. Initial value problems: existence and uniqueness
  3. Separable equations
  4. First-order linear equations
  5. Second-order linear equations (oscillations)
  6. Systems of ODEs
  7. Euler's method
  8. Runge–Kutta methods
  9. Stiffness and implicit methods
  10. Partial differential equations

φDynamical systems and chaos

The long-term behaviour of systems that evolve: equilibria, stability, attractors, bifurcations and chaos — and the limits they impose on prediction.

  1. Dynamical systems
  2. Phase space
  3. Equilibria and stability
  4. Attractors
  5. Bifurcations
  6. Chaos and sensitivity to initial conditions
  7. Fractals

ℱTransforms

Fourier, Laplace and Z: changing the point of view from time to frequency, where convolutions become products and differential equations become algebra. The mathematics of audio, images, compression and telecommunications.

  1. Fourier series
  2. Fourier transform
  3. Convolution
  4. Laplace transform
  5. Z-transform
  6. Wavelets

ℙProbability and calculus

Continuous probability is calculus: densities are integrated to get probabilities, expectations are integrals, and maximum likelihood is setting a derivative to zero. The mathematical core of statistics and probabilistic AI.

  1. Continuous random variables
  2. Probability density function
  3. Cumulative distribution function
  4. Expectation
  5. Variance
  6. Continuous distributions
  7. Maximum likelihood estimation

Computing domains

λScientific computing and algorithms

How computers represent numbers, how fast algorithms grow, how they manipulate formulas and how they estimate what cannot be computed exactly.

  1. Floating point (IEEE 754)
  2. Algorithm analysis and complexity
  3. Symbolic computation (CAS)
  4. Scientific computing
  5. Monte Carlo methods

ℒAI and machine learning

The calculus behind modern AI: loss functions, gradients, backpropagation, optimizers and probabilistic models.

  1. Linear regression
  2. Loss function
  3. Gradient descent◐
  4. Learning rate
  5. Logistic regression
  6. Activation functions
  7. Neural networks
  8. Automatic differentiation
  9. Backpropagation◐
  10. Stochastic gradient descent (SGD)
  11. Momentum and Adam
  12. Regularization
  13. Loss landscape
  14. Second-order (Hessian-based) optimization
  15. Deep learning
  16. Convolutional networks (CNNs)
  17. Support vector machines
  18. Reinforcement learning
  19. Bayesian inference
  20. Generative models
  21. Neural ODEs

3DComputer graphics

Normals, lighting, curves, ray tracing and rendering: derivatives orient surfaces, interpolation fills triangles and integrals gather light.

  1. Surface normals
  2. Lighting and shading
  3. Bézier curves and splines
  4. Procedural generation and noise
  5. Ray tracing
  6. The rendering equation
  7. Signed distance fields and ray marching
  8. Mesh processing (discrete differential geometry)

⚙Robotics and control

Motion is derivatives: velocity, acceleration, Jacobians from joints to hands, dynamics as ODEs, and feedback control that keeps it all stable.

  1. Kinematics: position, velocity, acceleration
  2. Robot Jacobian (velocity kinematics)
  3. Inverse kinematics
  4. Robot dynamics
  5. PID control
  6. Control theory
  7. Kalman filter
  8. Trajectory optimization and MPC

⚛Physics and simulation

Mechanics, gravity, heat, waves, fluids and electromagnetism written as differential equations — and the numerical methods that turn them into games, films, forecasts and engineering software.

  1. Classical mechanics
  2. Physics engines
  3. N-body gravitational simulation
  4. Heat equation and diffusion
  5. Wave equation
  6. Electromagnetism (Maxwell's equations)
  7. Fluid dynamics and CFD
  8. Finite element method
  9. Weather and climate modelling
  10. Population and epidemic models

∿Signals, media and vision

Audio, images, video and radio as functions to be sampled, transformed, filtered and compressed — Fourier analysis and derivatives at industrial scale.

  1. Signal processing
  2. Sampling theorem (Nyquist–Shannon)
  3. Fast Fourier transform (FFT)
  4. Digital filters
  5. Media compression (JPEG, MP3, video)
  6. Image processing and computer vision
  7. Telecommunications (modulation, OFDM)

⇄Optimization and systems

Planning, logistics, recommendations and the performance of computer systems: optimization and probability applied to decisions at scale.

  1. Operations research and logistics
  2. Recommender systems
  3. Queueing theory and performance

⊕Cryptography and security

An honest note: classical cryptography rests on algebra, number theory and discrete structures, not on calculus. The connections that do exist are real but indirect.

  1. Elliptic curve cryptography
  2. Side-channel analysis

ψQuantum computing and physics

Quantum mechanics combines linear algebra, calculus, complex numbers and probability: wave functions, the Schrödinger equation, Fourier duality and unitary evolution.

  1. Wave function
  2. Schrödinger equation
  3. Uncertainty principle
  4. Quantum computing
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