Articles

A Survival Strategy for Software Engineers in the AI Era: Estimating the Ceiling of AutomationWhere LLM code generation speeds up development and where it does not, analysed quantitatively with Amdahl's law, the generate-verify cycle, an information-theoretic bound on specifications, and Rice's theorem.Computer ScienceSurviving the AI EraUndergraduatesoftware engineering automation Amdahl's law computability problem framing~25 min
LLMs and Programming: The Break-Even Point of Delegation and How to Spot Plausible ErrorsReduces the decision to delegate work to an LLM to one inequality between success rate and verification cost, explains why splitting tasks helps, and shows how to catch plausible errors.Computer ScienceSurvival Strategies for the AI EraUndergraduateLLM software development code review debugging verification~11 min
Relearning Mathematics: Three Languages for Reading AIReads the inside of an LLM in three languages: linear algebra (embeddings, low-rank approximation), calculus (gradient descent, learning rate) and probability (cross-entropy, Bayes).Computer ScienceSurvival Strategy in the AI EraUndergraduatelinear algebra calculus probability machine learning abstraction~23 min
Complexity and Big-O Notation: Measuring Speed as a Function of Input SizeDefines time and space complexity from a machine model, states O, Ω and Θ as sets of functions, proves the growth hierarchy, and compares classes from O(1) to O(2^n) numerically.Computer ScienceAlgorithms and Data StructuresUndergraduatecomplexity big-O notation asymptotic analysis master theorem polynomial time~24 min
Fundamental Data Structures: Arrays, Linked Lists, Stacks and QueuesDerives the cost of search, insertion and deletion from memory layout, proves the amortized O(1) bound for dynamic arrays, and analyses stacks and queues as abstract data types.Computer ScienceAlgorithms and Data StructuresUndergraduatedata structures arrays linked lists amortized analysis abstract data types~36 min
Sorting Algorithms: Bubble Sort, Merge Sort, Quicksort and the Quadratic WallWhy bubble sort is quadratic, explained through inversions; a proof that the divide-and-conquer structure of merge sort achieves n log n; and how the gap between quicksort's average and worst case grows out of the choice of pivot.Computer ScienceAlgorithms and Data StructuresUndergraduatesorting divide and conquer complexity quicksort lower bounds~22 min
Computer Architecture and the Structure of a CPU: From Transistors to RISC-VBuilding up from the switching behaviour of transistors through logic gates, adders and clocked circuits to the five functional units, the von Neumann architecture, the CPU datapath and RISC-V instruction encoding, with proofs and numbers throughout.Computer ScienceFoundations of Computer ScienceUndergraduatecomputer architecture logic circuits CPU von Neumann architecture RISC-V~28 min
What an Operating System Does: Abstraction, Scheduling and Virtual MemoryStarting from what goes wrong on bare hardware, this article treats privileged mode and system calls, CPU scheduling and the optimality of SJF, paging and virtual memory (multi-level page tables, the TLB, the stack property of LRU), and the necessary conditions for deadlock, with proofs.Computer ScienceFoundations of Computer ScienceUndergraduateoperating systems processes scheduling virtual memory paging~12 min
Programming Language Theory: From Machine Code to Type Systems and ParadigmsThe layers of machine code, assembly and high-level languages, checked against the actual RISC-V encoding; progress and preservation proved over the operational semantics of a tiny language; and a comparison of static with dynamic typing and of functional with object-oriented design.Computer ScienceFoundations of Computer ScienceUndergraduateprogramming languages type systems compilers operational semantics paradigms~27 min
Why Machine Learning Needs Mathematics: Rewriting Learning as Loss MinimisationRegression and classification both reduce to a single optimisation problem, empirical risk minimisation. We show where linear algebra, calculus and probability enter, with complete proofs for least squares and gradient descent.Computer ScienceMathematics for Machine LearningUndergraduatemachine learning empirical risk minimisation least squares gradient descent maximum likelihood~23 min
Linear Regression and Least Squares: Reading the Normal Equations as an Orthogonal ProjectionDerives the normal equations X^T X w = X^T y from least squares both algebraically and by calculus, shows the fit is an orthogonal projection onto the column space, and treats rank and conditioning.Computer ScienceMathematics for Machine LearningUndergraduateLinear Regression Least Squares Normal Equations Orthogonal Projection Hat Matrix~38 min
Logistic Regression: Deriving the Sigmoid and the Cross Entropy from Maximum LikelihoodThe sigmoid and the logit turn a linear score into a probability, and Bernoulli maximum likelihood turns into the cross entropy error; the gradient and Hessian show the loss is convex but has no closed form.Computer ScienceMathematics for Machine LearningUndergraduatelogistic regression sigmoid function cross entropy maximum likelihood convex optimisation~28 min
Git: History as a Merkle DAG and the Collaborative WorkflowGit's content-addressable store and commit DAG, with proofs that one hash fixes an entire history, that fast-forward is an ancestry condition, and that merge bases are not unique.Computer ScienceSoftware EngineeringUndergraduateversion control Git hash functions directed acyclic graphs collaboration~37 min
Docker and Kubernetes: Why Containers Are Light and Clusters Heal ThemselvesNamespaces and cgroups as the real content of a container, why the build cache hits only on a prefix, and how the reconciliation loop of Kubernetes makes a cluster heal itself.Computer ScienceSoftware EngineeringUndergraduateDocker Kubernetes containers orchestration availability~38 min
Database Design: Reading the Relational Model, SQL and Normalization as the Placement of FactsDefines tables, keys and foreign keys, follows SQL on concrete data, and derives the first three normal forms with proofs from functional dependencies and attribute closure.Computer ScienceSoftware EngineeringUndergraduaterelational databases SQL normalization functional dependencies NoSQL~35 min
Introduction to Group Theory: The Axioms, and a Language for Computing with SymmetryStarting from three axioms — associativity, an identity, inverses — we build up to the multiplication table of S3, the order of an element, and cyclic groups, with full proofs.MathematicsAlgebra: Groups, Rings and FieldsUndergraduateAlgebra Groups Symmetric group Cyclic groups Order of an element~31 min
Subgroups and Cosets: Why Lagrange's Theorem Produces DivisibilityFrom the one-step subgroup test to Lagrange's theorem: left cosets split a group into blocks of equal size, giving Euler's theorem and a counterexample to the converse.MathematicsAlgebra: Groups, Rings and FieldsUndergraduateGroup theory Subgroups Cosets Lagrange's theorem Finite groups~34 min
Normal Subgroups and Quotient Groups: The Condition for Multiplying CosetsFrom the wish to multiply cosets by their representatives we derive normality, prove that G/N is a group with a natural surjection onto it, and work out Z/nZ in full detail.MathematicsAlgebra: Groups, Rings and FieldsUndergraduateGroup theory Normal subgroups Quotient groups Cosets Congruences~32 min
Limits and Continuity: Reading ε-δ as a Contract on ErrorDirichlet's function shows why 'approaches indefinitely' is no definition. From ε-δ alone we prove uniqueness, the algebra of limits, squeezing, continuity and uniform continuity.MathematicsCalculusUndergraduateLimits Continuity Epsilon-delta Uniform continuity~14 min
The Derivative: From Difference Quotients to the Chain RuleWe define the derivative as a limit of difference quotients, prove it equivalent to linear approximation, and derive the product, quotient, chain and inverse-function rules.MathematicsCalculusUndergraduateDerivative Chain rule Product rule Linear approximation~24 min
Mean Value Theorems and Taylor's Theorem: Recovering a Function from Its DerivativesFrom the extreme value theorem we prove the mean value theorems of Rolle, Lagrange and Cauchy, then derive l'Hôpital's rule and Taylor's theorem with explicit error bounds.MathematicsCalculusUndergraduateMean value theorem Taylor expansion L'Hopital's rule Remainder term Analysis~26 min
Complex Numbers and the Complex Plane: Why Imaginary Numbers Became Necessary, and Why They Are RotationsFrom the cubic equations that made imaginary numbers unavoidable to the complex field, Euler's formula, De Moivre's theorem, the n-th roots of unity, and the topology of domains.MathematicsComplex AnalysisUndergraduateComplex numbers Complex plane Euler's formula De Moivre's theorem Domains~29 min
Holomorphic Functions and the Cauchy-Riemann Equations: Why Complex Differentiability Is So StrongComplex differentiability asks for one limit from every direction. We derive the Cauchy-Riemann equations, show partial derivatives alone fail, and prove exp, sin and Log holomorphic.MathematicsComplex AnalysisUndergraduateComplex analysis Holomorphic functions Cauchy-Riemann equations Total differentiability Harmonic functions~21 min
The Grammar of Mathematics: Reading and Writing Sets and LogicA precise treatment of elements, subsets, power sets, unions, intersections and complements, together with truth tables, the negation rules for quantifiers, and necessary versus sufficient conditions, organized around proofs of De Morgan's laws.MathematicsFoundations of MathematicsUndergraduateSet theory Propositional logic Predicate logic De Morgan's laws Quantifiers~28 min
What Is a Number? From the Naturals to the Reals, and Why 1 = 0.999… Is TrueFollows the extension from the naturals to the reals through unsolvable equations, proves that the rationals have gaps, and derives 1 = 0.999… three ways from the completeness axiom.MathematicsFoundations of MathematicsUndergraduateReal numbers Rational numbers Irrational numbers Completeness Infinite decimals~15 min
Techniques of Proof: Why Induction and Contradiction Are ValidWhy induction is valid, how it breaks without a base case, how proof by contradiction differs from contraposition, and two proofs that no rational number squares to 2.MathematicsFoundations of MathematicsUndergraduateMathematical induction Proof by contradiction Contraposition Peano axioms Irrational numbers~29 min
Vector Spaces and Linear Maps: From the Eight Axioms to the Rank-Nullity TheoremStarting from the eight axioms, we build subspaces, bases and dimension, prove the exchange lemma and the rank-nullity theorem, and test them on projections, differentiation and PCA.MathematicsLinear AlgebraUndergraduateLinear Algebra Vector Spaces Basis and Dimension Linear Maps Rank-Nullity Theorem~44 min
Matrices and Linear Systems: Coordinate Representations of Linear Maps and the Structure of SolutionsIntroduces matrices as representations of linear maps, derives the product from composition, and shows the solutions of Ax = b form a particular solution plus the kernel.MathematicsLinear AlgebraUndergraduateLinear Algebra Matrices Representation Matrix Gaussian Elimination Rank~48 min
Determinants and Their Properties: det as Signed VolumeDefines the determinant both by the Leibniz formula with permutation signs and by cofactor expansion, characterizes it through multilinearity and alternation, proves det(AB)=det(A)det(B) and the invertibility criterion, and reads det as signed volume.MathematicsLinear AlgebraUndergraduatelinear algebra determinant multilinearity cofactor expansion invertible matrices~38 min
Differentiable Manifolds: Charts and Atlases That Bring Calculus to Curved SpacesWe introduce charts, transition maps and maximal atlases, build explicit atlases on the sphere, projective space and the torus, and define C^k functions and maps.MathematicsManifold TheoryGraduateManifolds Charts Atlases Differentiable structure Projective space~30 min
Tangent Spaces and the Tangent Bundle: Three Ways to Define Velocity on a Curved SpaceWe build the tangent space three ways — curves, coordinate components, derivations — prove them naturally isomorphic via Hadamard's lemma, and construct the tangent bundle.MathematicsManifold TheoryGraduateManifolds Tangent space Tangent bundle Differential Derivations~36 min
Is 1 Equal to 0.999…? Fixing the Meaning of an Infinite Decimal FirstWe examine 0.999… = 1 by first defining what an infinite decimal means, then carry the three classical arguments to completion and explain why intuition rebels.MathematicsMathematical ColumnsHigh school / generalReal numbers Infinite decimals Limits Geometric series Density~11 min
Why Mathematics Is Hard: Abstraction, Logic, and the Cumulative WallMathematics is hard for three structural reasons — abstraction, strict logic, and cumulative dependence — tested here on domino tilings and the birthday problem, together with why everyday intuition fails and what happens the moment we understand.MathematicsMathematics ColumnsHigh school / generalmathematics education abstraction proof intuition~25 min
Why You Cannot Divide by Zero: Granting 0 a Reciprocal Collapses the Number System to a PointThe school rule 'never divide by zero' explained from the definition of division as the inverse of multiplication. 5÷0 has no solution, 0÷0 has too many, and granting 0 a reciprocal forces 1=0, wiping out every number.MathematicsMathematical ColumnsHigh school / generalDivision by zero Division Reciprocal Limits~21 min
The Collatz Conjecture: A Problem You Can State in Three Lines and Nobody Can SolveHalve the evens, triple the odds and add one. We prove what can be proved — no short cycles, density 3/4 — and show why the 3n-1 counterexample defeats the probabilistic heuristic.MathematicsMath ColumnsHigh school / generalCollatz conjecture Open problems Number theory Dynamical systems~8 min
The Monty Hall Problem: Why Switching Doors Doubles Your Chance of WinningYou pick one of three doors and the host opens a goat door. Enumeration and Bayes' theorem both push the win probability from 1/3 to 2/3, and we go on to biased hosts and n doors.MathematicsMathematical ColumnsHigh school / generalprobability conditional probability Bayes theorem counterintuitive results~28 min
The Four Color Theorem: Why Maps Need Four Colors, and the First Proof Written by a ComputerFour colors suffice for every planar map. We prove the six- and five-color theorems from Euler's formula, then ask what the 1976 computer-assisted proof left mathematics with.MathematicsMathematics ColumnsHigh school / generalGraph theory Four color theorem Planar graphs Graph coloring Computer-assisted proof~32 min
Euler and Ramanujan: The Giant of Calculation and the Man of IntuitionWe derive Euler's identity from power series, then treat the Basel problem, the polyhedron formula, the Königsberg bridges, and Ramanujan's π series and partition congruences.MathematicsMath ColumnsHigh school / generalHistory of mathematics Euler Ramanujan Pi Infinite series~9 min
Motivating Noncommutative Geometry: Gelfand Duality and the Slogan 'Space = Algebra of Functions'We prove the Gelfand–Naimark theorem matching commutative C*-algebras with locally compact spaces, then motivate noncommutative spaces via quantum commutators and bad quotients.MathematicsNoncommutative GeometryResearchC*-algebras Gelfand duality operator algebras noncommutative torus~34 min
C*-Algebras: Spectral Theory and the Gelfand RepresentationSpectra in Banach algebras are nonempty and compact; from this we derive the spectral radius formula, identify commutative C*-algebras with C0(X), and build the functional calculus.MathematicsNoncommutative GeometryResearchoperator algebras C*-algebras spectral theory Gelfand representation~37 min
Primes and the Prime Number Theorem: From Infinitude and Unique Factorization to the Law of DistributionFrom divisibility to Euclid's theorem and unique factorization, then Chebyshev-type bounds from binomial coefficients, the prime number theorem, and the open problems beyond it.MathematicsNumber TheoryUndergraduateNumber theory Prime numbers Prime number theorem Unique factorization Distribution of primes~7 min
Congruences and Fermat's Little Theorem: Computing in a World of RemaindersCongruence as an equivalence relation, the ring Z/nZ and its group of units, proofs of Fermat and Euler theorems, and the correctness of RSA via the Chinese remainder theorem.MathematicsNumber TheoryUndergraduateCongruences Modular arithmetic Fermat's little theorem Euler's totient RSA~6 min
Probability Spaces and Kolmogorov's Axioms: Probability as a Measure of Total Mass OneFrom Bertrand's paradox to Kolmogorov's three axioms: sample spaces, σ-algebras, probability measures, and the continuity, Bayes and Borel–Cantelli results derived from them.MathematicsProbability TheoryUndergraduateprobability measure theory sigma-algebra Bayes' theorem independence~31 min
Random Variables and Expectation: Measurable Functions and the Lebesgue IntegralWe define random variables as measurable functions, build the expectation as a Lebesgue integral, and prove the transfer formula, the covariance rules and Chebyshev's inequality.MathematicsProbability TheoryUndergraduaterandom variables expectation variance covariance Chebyshev's inequality~20 min
The Law of Large Numbers and the Central Limit Theorem: Where the Sample Mean Goes, and How FastFour notions of convergence and how they imply one another, the weak and strong laws of large numbers, and a proof of the central limit theorem by characteristic functions.MathematicsProbability TheoryUndergraduateLaw of large numbers Central limit theorem Convergence in probability Characteristic functions Confidence intervals~29 min
Completeness of the Real Numbers and Cauchy Sequences: The Absence of GapsFrom the axiom that every bounded increasing sequence converges we derive the Archimedean property, suprema, density, Bolzano-Weierstrass and Cauchy completeness, omitting no steps.MathematicsReal AnalysisUndergraduatecompleteness Cauchy sequences supremum Bolzano-Weierstrass theorem~17 min
Continuous Functions and Uniform Continuity: What It Means for δ Not to Depend on the PointFrom Bolzano-Weierstrass and nested intervals we prove that a continuous function on a closed bounded interval is bounded, attains its extrema and takes intermediate values, then define uniform continuity and prove the Heine-Cantor theorem.MathematicsReal AnalysisUndergraduateUniform continuity Continuous functions Intermediate value theorem Extreme value theorem Completeness~13 min
Topological Spaces: What Remains of Nearness When the Metric Is DiscardedOpen sets in a metric space satisfy three properties; taking them as axioms we define topological spaces, build the discrete, order and Zariski topologies, and prove the basic laws of closure.MathematicsGeneral TopologyGraduatetopological space open sets metric space closure neighbourhood~24 min
Continuous Maps and Homeomorphisms: Continuity Recast through Preimages of Open SetsContinuity is defined by requiring preimages of open sets to be open; we prove this agrees with ε-δ continuity on metric spaces, and compute in full why R is homeomorphic to (0,1) but not to [0,1].MathematicsGeneral TopologyGraduatetopological spaces continuous maps homeomorphism topological invariants~26 min
Why Is the Night Sky Dark? Olbers' Paradox and the Finite Age of the UniverseIf the universe were infinite and eternal, the night sky would shine as brightly as the Sun's surface. We quantify the contradiction with solid angles and the inverse-square law, refute the absorbing-dust escape, and arrive at finite light speed, finite age, and a shortage of energy.PhysicsCosmologyHigh school / generalOlbers' paradox Cosmic microwave background Redshift Inverse-square law Surface brightness~30 min
Was There Really a Big Bang? Expansion, Background Radiation, and the Ratio of the ElementsThree pieces of evidence for a hot, dense early universe — the Hubble–Lemaître law, the 2.7 K microwave background, and the primordial helium ratio — checked down to the numbers.PhysicsCosmologyHigh school / generalBig Bang Hubble's law Cosmic microwave background Big Bang nucleosynthesis Primordial helium~32 min
How We Know the Distance to a Star: Climbing the Cosmic Distance Ladder One Rung at a TimeParallax, Cepheids and Type Ia supernovae form a three-rung ruler. We build each rung from the inverse-square law, find the distance to δ Cephei, and track how errors accumulate.PhysicsCosmologyHigh school / generalcosmology distance measurement parallax Cepheid variables supernovae~30 min
What Lies Beyond a Black Hole: Event Horizons, Spaghettification, and EvaporationHow can an object exist that even light cannot leave? We derive the Schwarzschild radius, then follow tidal forces, Hawking evaporation, and why wormholes stay shut.PhysicsCosmologyHigh school / generalBlack holes Event horizon Hawking radiation Tidal forces Wormholes~19 min
Dark Matter and Dark Energy: 95% of the Universe Has Only a Name So FarGalaxy rotation curves and cluster virial masses force dark matter upon us; Type Ia supernovae force dark energy. We check the 4.9 / 26.4 / 68.5 percent cosmic budget by hand.PhysicsCosmologyHigh school / generalDark matter Dark energy Cosmological constant Galaxy rotation curves Accelerating expansion~18 min
How Many Aliens Are There? The Drake Equation and the Fermi ParadoxWe put real numbers into the Drake equation for the number of civilizations in the Galaxy, then sharpen the Fermi paradox using two figures: the time needed to colonize the Galaxy and the range at which a radio beacon can be heard.PhysicsCosmologyHigh school / generalDrake equation Fermi paradox Extraterrestrial intelligence Great Filter SETI~34 min
The Edge and the Age of the Universe: Where 13.8 Billion Years and 46.5 Billion Light-Years Come FromHow the reciprocal of the Hubble constant yields 13.8 billion years, why the observable universe reaches 46.5 billion light-years, and whether space itself has an edge.PhysicsCosmologyHigh school / generalCosmic expansion Hubble constant Age of the universe Event horizon Observable universe~28 min
Electrostatic Fields and Gauss's Law: From Coulomb's Law to the Poisson EquationFrom Coulomb's law we define the electric field, prove Gauss's law in integral form by a solid-angle argument, derive div E = ρ/ε₀, and reach the Poisson equation.PhysicsElectromagnetismUndergraduateElectrostatics Gauss's law Coulomb's law Poisson equation Vector calculus~29 min
Steady Currents and Magnetostatic Fields: From the Biot–Savart Law to curl B = μ0 JFrom current density and charge conservation, via the Biot–Savart law, to the two equations of magnetostatics: div B = 0 and curl B = μ0 J, the curl being circulation per unit area.PhysicsElectromagnetismUndergraduateMagnetostatics Current density Biot-Savart law Ampère's law Vector calculus~28 min
Foundations of Newtonian Mechanics: From the Three Laws to Momentum and Energy ConservationThe three laws restated as the existence of inertial frames, force versus acceleration, and action-reaction; then one-dimensional motion, momentum and energy conservation.PhysicsMechanics: Classical and AnalyticalUndergraduateNewtonian mechanics Equation of motion Conservation laws Harmonic oscillator~27 min
Planetary Motion and Central Forces: From Conservation of Angular Momentum to Kepler's Three LawsReducing the two-body problem to one body of reduced mass, we derive Kepler's three laws from angular momentum and Binet's equation, and treat the Runge–Lenz vector.PhysicsClassical and Analytical MechanicsUndergraduateMechanics Central force Angular momentum Kepler's laws Two-body problem~29 min
Lagrangian Mechanics: From the Principle of Least Action to the Euler-Lagrange EquationsFrom Hamilton's principle we derive the Euler-Lagrange equations, show that L = T - U reproduces Newton's laws, and prove covariance under any change of generalized coordinates.PhysicsMechanics: Classical and Analytical MechanicsUndergraduateAnalytical mechanics Principle of least action Euler-Lagrange equations Generalized coordinates Calculus of variations~21 min
Laplace's Demon and Determinism: Is the Future Already Fixed?Laplace's predictable universe, recast as a uniqueness theorem for initial value problems, and how chaos and the uncertainty relation dismantled it, with explicit computations.PhysicsPhysics ColumnsHigh school / generaldeterminism chaos uncertainty principle history of science~24 min
Schrödinger's Cat: Where Does Superposition Stop?Unpacks the thought experiment of a cat in a box suspended between life and death using polarizer experiments and elementary vector algebra, up to the differences between the Copenhagen interpretation, many-worlds, and decoherence.PhysicsPhysics ColumnsHigh school / generalquantum mechanics superposition measurement problem thought experiment decoherence~21 min
Maxwell's Demon: Can a Machine Run on Information?Can a demon sorting fast molecules beat the second law? Szilard's engine reduces the question to one molecule, and Landauer's principle settles it: erasing a bit costs kT ln 2 of heat.PhysicsPhysics ColumnsHigh school / generalthermodynamics entropy information Landauer's principle thought experiments~25 min
Einstein and Feynman: The Man Who Bent Spacetime and the Man Who Counted LightHow Einstein dismantled our intuitions about time and space, traced through the light clock and E=mc², followed by Feynman's quantum electrodynamics and the scientific ethic visible in his safecracking and O-ring stories.PhysicsPhysics ColumnsHigh school / generalRelativity Quantum electrodynamics Einstein Feynman History of physics~24 min
Unsolved Problems in Physics: 95% of the Cosmos Still Has Only a NameDark matter, dark energy, inflation and quantum gravity: four open problems of modern physics, traced through the arithmetic of rotation curves and the critical density.PhysicsPhysics ColumnsHigh school / generalDark matter Dark energy Inflation Quantum gravity Cosmology~30 min
Physics in Everyday Life: The Quantum Mechanics and Relativity Running Your Phone, GPS and MRIWorks out the numbers behind three everyday machines: the band gap that sets an LED's color, the 38 microseconds a day GPS clocks gain, and the 5 ppm spin polarization MRI detects.PhysicsPhysics ColumnsHigh school / generalsemiconductors relativity nuclear magnetic resonance quantum mechanics band gap~26 min
Why We Need Quantum Field Theory: One-Particle Quantum Mechanics Is Incompatible with RelativityRelativity grafted onto one-particle quantum mechanics yields negative energies, a negative probability density and acausal propagation. Quantizing the field is the only way out.PhysicsQuantum Field TheoryGraduateQuantum field theory Second quantization Klein-Gordon equation Antiparticles Causality~37 min
Classical Field Theory and the Lagrangian: From a Variational Principle for Infinitely Many Degrees of Freedom to Noether's TheoremThe Lagrangian density from the continuum limit of coupled oscillators, the Euler–Lagrange equations from the variational principle, and Noether's theorem for the Klein–Gordon field.PhysicsQuantum Field TheoryGraduateClassical field theory Lagrangian density Klein-Gordon equation Noether's theorem Conserved current~34 min
The Birth of Quantum Mechanics: How Black-Body Radiation, the Photoelectric Effect and Matter Waves Broke the Classical PictureThe Rayleigh-Jeans divergence shows why black-body radiation defeats classical physics; then come Planck's quantum hypothesis, Einstein's light quanta and de Broglie matter waves.PhysicsQuantum MechanicsUndergraduateQuantum theory Black-body radiation Photoelectric effect Matter waves Planck constant~29 min
The Schrödinger Equation and the Wave Function: From the Born Rule to the Evolution of Expectation ValuesWe introduce the time-dependent and time-independent Schrödinger equations and prove the Born rule, the conservation of probability, the momentum operator and Ehrenfest's theorem.PhysicsQuantum MechanicsUndergraduatequantum mechanics Schrödinger equation wave function Born rule expectation values~22 min
Operators and Observables: From Hermitian Operators and Commutators to the Uncertainty RelationWhy observables are represented by Hermitian operators, derived from the reality of expectation values; the meaning of the canonical commutation relation, a proof of Heisenberg's principle via Robertson's inequality, and measurement as projection.PhysicsQuantum MechanicsUndergraduatequantum mechanics Hermitian operators commutation relations uncertainty principle measurement~33 min
The Principles of Special Relativity: From Galilean Relativity to Einstein's Two PostulatesWe verify by computation that Maxwell's equations change form under a Galilean transformation, see what the Michelson-Morley experiment actually ruled out, and follow the logic by which Einstein's two postulates force us to abandon absolute time.PhysicsRelativityUndergraduatespecial relativity Galilean transformation invariance of light speed Michelson-Morley experiment relativity of simultaneity~30 min
Lorentz Transformations: Time Dilation and Length Contraction from the Light ClockFrom the constancy of the speed of light and the light-clock thought experiment we derive time dilation, Lorentz contraction and the relativity of simultaneity, then assemble them into the Lorentz transformation and the invariant spacetime interval of Minkowski spacetime.PhysicsRelativityUndergraduateSpecial relativity Lorentz transformation Time dilation Relativity of simultaneity Minkowski spacetime~24 min
Relativistic Mechanics: What the Four-Momentum Says About E = mc²Newtonian momentum conservation fails between inertial frames. Building proper time, four-velocity and four-momentum repairs it, yielding E = γmc², E² = (pc)² + (mc²)², and non-additive mass.PhysicsRelativityUndergraduatespecial relativity four-momentum rest energy mass-energy equivalence mass defect~35 min
The Laws of Thermodynamics: Where Temperature, Internal Energy and Entropy Come FromEmpirical temperature is built from the transitivity of thermal equilibrium, internal energy and heat from the path independence of adiabatic work, and entropy from the equivalence of the Thomson and Clausius principles, Carnot's theorem and the Clausius inequality.PhysicsThermodynamics and Statistical MechanicsUndergraduateThermodynamics Entropy Second law Carnot cycle Absolute temperature~30 min
Free Energy and Thermodynamic Potentials: Trading Variables You Cannot Fix for Ones You CanLegendre transformation replaces the natural variables (S,V,N) of internal energy by the controllable (T,p,μ), giving F, H, G, Ω, maximum work, and the Maxwell relations.PhysicsThermodynamics and Statistical MechanicsUndergraduatethermodynamics free energy Legendre transformation Maxwell relations response functions~23 min

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