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Every unit and every derivation, theorem, result and concept across the physics2u network — 99 units, 849 pages — linked directly. Use ⌘K search for a faster lookup by name.

Physics · 437 Maths · 144 Chemistry · 135 Biology · 133 ★ New Pages →

Physics2u · 24 units · 437 derivations

Physics2u home → · full curriculum → · vault index →

Phase 1 · PU-103 18 derivations

Vibrations & Waves

Shm from linear restoring force Shm energy conservation Damped harmonic oscillator Quality factor and energy decay Driven oscillator resonance Resonance lorentzian lineshape Coupled oscillators normal modes Normal modes general diagonalization Wave equation from loaded string Dalembert solution Standing waves normal modes string Fourier series mode decomposition Energy density and power in waves Reflection transmission impedance Sound waves speed in gas Dispersion phase group velocity Wave packet group velocity Doppler effect and shock

Phase 1 · PU-104 17 derivations

Mathematical Methods I — Linear Algebra & Operators

Vector space basis dimension invariance Gram schmidt orthonormalization Inner product cauchy schwarz Adjoint operator existence uniqueness Riesz representation finite dim Hermitian operator properties Determinant multilinear characterization Characteristic polynomial eigenvalues Diagonalizability eigenbasis criterion Spectral theorem self adjoint Unitary operators preserve inner product Simultaneous diagonalization commuting operators Generalized eigenvalue normal modes Rayleigh quotient variational eigenvalues Singular value decomposition Matrix exponential generator flow Projection operators orthogonal decomposition

Phase 1 · PU-105 18 derivations

Special Relativity

Lorentz transformation from postulates Invariance of spacetime interval Relativity of simultaneity Time dilation from lorentz Length contraction from lorentz Relativistic velocity addition Rapidity and additivity of boosts Proper time and four velocity Relativistic momentum from conservation Mass energy equivalence Energy momentum four vector Relativistic doppler effect Four wavevector of light Relativistic force and acceleration Electromagnetic field tensor Transformation of em fields Covariant maxwell equations Four current and charge conservation

Phase 1 · PU-106 17 derivations

Experimental Physics & Data Analysis I

Moments expectation variance Variance of linear combinations Error propagation first order Error propagation covariance matrix Standard error of the mean Bessel correction unbiased variance Inverse variance weighted mean Binomial distribution from bernoulli Poisson from binomial limit Characteristic functions and moments Central limit theorem Maximum likelihood estimation Least squares from gaussian likelihood Linear least squares normal equations Parameter covariance from fisher information Chi squared distribution from gaussians Chi squared goodness of fit and dof

Phase 2 · PU-201 18 derivations

Analytical Mechanics

Dalembert principle from newton Euler lagrange equation Lagrangian equals t minus v Lagrange multipliers constraint forces Noether time translation Noether continuous symmetry general Cyclic coordinates conserved momenta Central force orbit equation Central force effective potential Kepler problem closed orbits Small oscillations eigenproblem Normal coordinates decoupling Legendre transform hamiltonian Hamilton equations of motion Poisson brackets Liouville theorem phase space Canonical transformations generating functions Hamilton jacobi equation

Phase 2 · PU-205 18 derivations

Mathematical Methods II — Complex Analysis & Asymptotics

Cauchy riemann equations Cauchy goursat theorem Cauchy integral formula Taylor series analytic Laurent series expansion Residue theorem Contour integration real integrals Argument principle rouche Branch points riemann surfaces Analytic continuation uniqueness Gamma function continuation reflection Conformal mapping potential theory Asymptotic series optimal truncation Watsons lemma Laplace method integrals Method of steepest descent Stirling saddle point Stationary phase approximation

Phase 3 · PU-301 18 derivations

Quantum Mechanics II

Spectral theorem hermitian observables Generalized uncertainty relation Angular momentum ladder spectrum Spherical harmonics from orbital angular momentum Separation central potential radial equation Hydrogen atom bound spectrum Spin half pauli algebra Clebsch gordan addition angular momentum Nondegenerate perturbation theory Degenerate perturbation theory Fine structure hydrogen Variational principle ground state Wkb quantization Time dependent perturbation transition amplitude Fermi golden rule Dipole radiation selection rules Identical particles exchange symmetry Born approximation scattering

Phase 3 · PU-302 18 derivations

Statistical Mechanics

Boltzmann entropy from microstate counting Temperature from entropy maximization Boltzmann distribution from reservoir Canonical partition function and free energy Gibbs entropy and maximum entropy ensembles Grand canonical ensemble and fugacity Energy fluctuations and heat capacity Equivalence of ensembles in thermodynamic limit Equipartition theorem Gibbs paradox and indistinguishability Ideal gas from canonical ensemble Quantum occupation statistics bose fermi Planck blackbody law from photon gas Debye model of solid heat capacity Degenerate fermi gas and sommerfeld expansion Bose einstein condensation Ising model mean field and phase transition Critical exponents and landau theory

Phase 3 · PU-305 18 derivations

Classical Electrodynamics

Laplacian of inverse r delta Gauss law from coulomb Scalar potential and curl free e Electrostatic energy density Multipole expansion potential Polarization bound charge Ampere law from biot savart Vector potential and magnetization Faraday law and induced emf Displacement current and maxwell completion Poynting theorem energy conservation Maxwell stress tensor momentum Wave equation from maxwell Fresnel equations reflection Retarded potentials and gauge Larmor radiation formula Lorentz covariance of maxwell Field transformation under boost

Phase 3 · PU-308 18 derivations

Astrophysics & Cosmology

Hydrostatic equilibrium stellar Virial theorem self gravitating Lane emden polytrope Chandrasekhar mass limit Tov equation relativistic stars Saha ionization equation Frw metric cosmological principle Friedmann equations from gr Cosmological redshift scale factor Distance measures cosmology Particle horizon flatness problem Inflation slow roll dynamics Cmb blackbody temperature scaling Recombination saha decoupling Big bang nucleosynthesis helium Jeans instability criterion Linear growth density perturbations Flat rotation curves dark matter

Phase 4 · PU-401 18 derivations

Advanced Quantum Theory

Interaction picture dyson series Fermi golden rule Semiclassical radiation selection rules Einstein coefficients relation Clebsch gordan addition Wigner eckart theorem Variational upper bound Wkb connection formulas Gamow tunneling rate Adiabatic theorem Berry phase geometric Lippmann schwinger equation Born approximation cross section Partial wave phase shifts Optical theorem Identical particles exchange symmetry Density matrix von neumann Path integral propagator

Phase 4 · PU-402 18 derivations

Lagrangian Field Theory

Euler lagrange field equations Noether first theorem conserved current Canonical stress energy tensor Belinfante rosenfeld symmetric tensor Klein gordon from scalar lagrangian Lorentz lie algebra representations Dirac equation from lagrangian Maxwell from gauge lagrangian Gauge invariance implies charge conservation Proca massive vector degrees of freedom Hamiltonian field formulation Canonical commutation field quantization Scalar field mode expansion fock space Feynman propagator scalar field Noether charge generates symmetry Spin statistics connection Goldstone theorem massless bosons Abelian higgs mechanism

Phase 4 · PU-403 18 derivations

General Relativity

Equivalence principle to metric gravity Geodesic equation from extremal proper time Covariant derivative and christoffel symbols Riemann tensor from commutator of covariant derivatives Parallel transport and geodesic deviation Contracted bianchi identity and einstein tensor Stress energy tensor from noether currents Einstein field equations from einstein hilbert action Newtonian limit and fixing einstein constant Schwarzschild solution from vacuum equations Perihelion precession of mercury Light deflection and gravitational redshift Event horizon and kruskal extension Linearized gravity and wave equation Quadrupole formula for gravitational radiation Friedmann equations from flrw metric Cosmological redshift and hubble law Raychaudhuri equation and focusing

Phase 4 · PU-404 18 derivations

Quantum Information & Computation

No cloning theorem Reduced density matrix partial trace Schmidt decomposition Kraus stinespring representation Chsh tsirelson bound Quantum teleportation protocol Superdense coding protocol Universal gate set cnot su2 Solovay kitaev theorem Deutsch jozsa oracle separation Grover search optimality Quantum fourier transform circuit Quantum phase estimation Shor order finding Von neumann entropy properties Holevo bound Landauer erasure principle Stabilizer codes error discretization

Maths2u · 25 units · 144 theorems

Maths2u home → · full curriculum → · vault index →

Phase 1 · MU-101 9 theorems

Foundations & Mathematical Proof

Principle of induction Well ordering principle Pigeonhole principle Irrationality of sqrt2 Euclid infinitude of primes Cantor uncountability reals Cantor theorem power set Division algorithm Equivalence relations and partitions

Phase 1 · MU-102 9 theorems

Calculus I

Chain rule Rolles theorem Mean value theorem Extreme value theorem Fundamental theorem of calculus Taylors theorem Lhopitals rule Substitution rule for integrals Fermats interior extremum theorem

Phase 1 · MU-103 8 theorems

Linear Algebra I

Rank nullity theorem Invertible matrix theorem Determinant multiplicative Cauchy schwarz inequality Gram schmidt process Cramers rule Row rank equals column rank Grassmann dimension formula

Phase 1 · MU-104 9 theorems

Analysis I — Sequences & Series

Algebra of limits Squeeze theorem Monotone convergence sequences Bolzano weierstrass Cauchy completeness reals Ratio test Alternating series test Riemann rearrangement theorem Cauchy hadamard theorem

Phase 1 · MU-105 6 theorems

Discrete Mathematics

Binomial theorem Inclusion exclusion Handshaking lemma Fermats little theorem Chinese remainder theorem Eulerian circuit theorem

Phase 1 · MU-106 6 theorems

Probability I

Law of total probability Bayes theorem Linearity of expectation Markov inequality Chebyshev inequality Weak law large numbers

Phase 2 · MU-201 7 theorems

Real Analysis

Intermediate value theorem Heine borel theorem Uniform continuity on compacts Riemann integrability criterion Uniform limit continuous Weierstrass m test Lagrange theorem for groups

Phase 2 · MU-202 6 theorems

Linear Algebra II

Dimension well defined Spectral theorem Cayley hamilton Jordan normal form Singular value decomposition Orthogonal projection theorem

Phase 2 · MU-203 5 theorems

Multivariable & Vector Calculus

Change of variables jacobian Lagrange multipliers Greens theorem Stokes theorem Divergence theorem

Phase 2 · MU-204 6 theorems

Group Theory

Lagranges theorem groups Orbit stabiliser theorem First isomorphism theorem Cauchys theorem groups Sylow theorems Fundamental theorem abelian groups

Phase 2 · MU-205 5 theorems

Ordinary Differential Equations

Picard lindelof Linear superposition Wronskian independence Variation of parameters Linear stability phase plane

Phase 2 · MU-206 6 theorems

Complex Analysis

Cauchy riemann equations Cauchy integral theorem Cauchy integral formula Liouvilles theorem Fundamental theorem of algebra Residue theorem

Phase 2 · MU-207 5 theorems

Number Theory

Euclidean algorithm bezout Fundamental theorem arithmetic Eulers theorem totient Wilsons theorem Quadratic reciprocity

Phase 3 · MU-301 5 theorems

Metric & Topological Spaces

Continuity via preimages Compactness open cover sequential Banach fixed point Baire category theorem Tychonoff theorem

Phase 3 · MU-302 5 theorems

Measure & Integration

Caratheodory extension Monotone convergence lebesgue Fatou lemma Dominated convergence Fubini tonelli

Phase 3 · MU-303 6 theorems

Rings, Fields & Galois Theory

Pid implies ufd Eisenstein criterion Tower law field extensions Galois correspondence Classification finite fields Insolvability of the quintic

Phase 3 · MU-304 5 theorems

Functional Analysis

Hahn banach theorem Open mapping theorem Closed graph theorem Uniform boundedness principle Riesz representation theorem

Phase 3 · MU-305 5 theorems

Partial Differential Equations

Separation of variables Dalembert wave solution Maximum principle harmonic Fourier series convergence Method of characteristics

Phase 3 · MU-306 4 theorems

Differential Geometry

Frenet serret formulas First fundamental form Theorema egregium Gauss bonnet theorem

Phase 3 · MU-307 5 theorems

Combinatorics & Graph Theory

Ramseys theorem Halls marriage theorem Max flow min cut Kuratowski theorem Turans theorem

Phase 3 · MU-308 4 theorems

Numerical Analysis

Fixed point iteration convergence Newton method convergence Lagrange interpolation error Gaussian quadrature

Phase 4 · MU-401 4 theorems

Algebraic Topology

Fundamental group Seifert van kampen Brouwer fixed point Euler characteristic invariance

Phase 4 · MU-402 4 theorems

Representation Theory

Maschke theorem Schur lemma Character orthogonality Burnside counting lemma

Phase 4 · MU-403 5 theorems

Logic & Set Theory

Zorns lemma Compactness theorem logic Lowenheim skolem Godel completeness Godel incompleteness

Phase 4 · MU-404 5 theorems

Measure-Theoretic Probability

Borel cantelli lemmas Kolmogorov strong law Central limit theorem Conditional expectation projection Martingale convergence

Chemistry2u · 25 units · 135 results

Chemistry2u home → · full curriculum → · vault index →

Phase 1 · CU-101 8 results

Atomic Structure & Periodicity

Bohr model hydrogen Rydberg formula Quantum numbers orbitals Aufbau hund pauli Periodic trends Photoelectric ionisation Hess law and enthalpy cycles Born haber cycle

Phase 1 · CU-102 7 results

Chemical Bonding

Ionic born haber Lewis structures octet Vsepr geometry Orbital hybridisation Molecular orbital theory Electronegativity polarity Le chatelier principle quantified

Phase 1 · CU-103 7 results

Stoichiometry & the Mole

Mole avogadro Empirical molecular formula Limiting reagent Gas stoichiometry avogadro Concentration dilution Solubility product and common ion effect Henderson hasselbalch equation

Phase 1 · CU-104 6 results

States of Matter & Gases

Ideal gas law Kinetic theory pressure Maxwell boltzmann speeds Van der waals equation Grahams law effusion Arrhenius equation and activation energy

Phase 1 · CU-105 5 results

Introduction to Organic Chemistry

Functional groups nomenclature Isomerism Chirality optical activity Conformational analysis Degree of unsaturation

Phase 1 · CU-106 4 results

Chemical Energetics

Hess law Enthalpy of formation Bond enthalpy estimation Calorimetry

Phase 2 · CU-201 7 results

Chemical Thermodynamics

First law chemistry Entropy second law Gibbs free energy Gibbs equilibrium constant Clausius clapeyron Entropy of mixing Gibbs free energy and cell potential

Phase 2 · CU-202 7 results

Chemical Kinetics

Rate law order Integrated rate laws Arrhenius equation Collision theory Transition state theory Steady state approximation Raoults law and ideal solutions

Phase 2 · CU-203 6 results

Chemical Equilibrium

Law of mass action Le chatelier Reaction quotient Kp kc relation Solubility product Van t hoff equation

Phase 2 · CU-204 5 results

Acids, Bases & Buffers

Bronsted lowry Water autoionisation ph Henderson hasselbalch Titration curves Polyprotic acids

Phase 2 · CU-205 5 results

Electrochemistry

Galvanic cell emf Nernst equation Faraday electrolysis Gibbs emf relation Electrochemical series

Phase 2 · CU-206 6 results

Organic Reaction Mechanisms

Sn1 sn2 E1 e2 elimination Electrophilic addition markovnikov Electrophilic aromatic substitution Nucleophilic carbonyl addition Reaction energy profiles

Phase 2 · CU-207 5 results

Coordination Chemistry

Crystal field splitting Spectrochemical colour Chelate effect Complex isomerism Coordination geometry

Phase 3 · CU-301 6 results

Quantum Chemistry

Particle in a box Hydrogen atom solution Variational principle Huckel theory Born oppenheimer Lcao molecular orbitals

Phase 3 · CU-302 6 results

Spectroscopy

Beer lambert law Rotational spectroscopy Vibrational ir spectroscopy Nmr chemical shift Uv vis electronic Mass spectrometry

Phase 3 · CU-303 5 results

Statistical Thermodynamics

Molecular partition function Boltzmann populations Equipartition theorem Statistical entropy Heat capacity from levels

Phase 3 · CU-304 5 results

Solid State & Materials Chemistry

Crystal lattices unit cell Bragg law Close packing efficiency Band theory Born lande lattice energy

Phase 3 · CU-305 5 results

Advanced Organic Synthesis

Retrosynthetic analysis Diels alder Protecting groups Stereoselective synthesis Cross coupling

Phase 3 · CU-306 5 results

Main Group & Inorganic Chemistry

P block trends Hsab principle Frost diagrams Vsepr main group Oxide acidity trends

Phase 3 · CU-307 4 results

Electrochemistry & Interfaces

Debye huckel Electrical double layer Butler volmer Kohlrausch conductivity

Phase 3 · CU-308 4 results

Analytical Chemistry

Gravimetric analysis Chromatography partition Error propagation Calibration standard addition

Phase 4 · CU-401 4 results

Computational & Theoretical Chemistry

Hartree fock Density functional theory Basis sets Potential energy surfaces

Phase 4 · CU-402 5 results

Chemical Biology & Biochemistry

Michaelis menten Enzyme inhibition Protein folding thermodynamics Dna base pairing Atp free energy coupling

Phase 4 · CU-403 4 results

Catalysis & Reaction Dynamics

Catalysis activation energy Langmuir isotherm Enzyme catalysis mechanism Heterogeneous catalysis

Phase 4 · CU-404 4 results

Polymer & Soft Matter Chemistry

Degree of polymerisation Step vs chain growth Glass transition Flory huggins

Biology2u · 25 units · 133 concepts

Biology2u home → · full curriculum → · vault index →

Phase 1 · BU-101 7 concepts

Cell Biology

Cell theory Prokaryote eukaryote Organelle structure function Fluid mosaic membrane Cell cycle mitosis Endosymbiotic theory Lineweaver burk linearisation

Phase 1 · BU-102 7 concepts

Biological Molecules

Protein structure levels Enzyme lower activation energy Dna double helix Lipids amphipathic assembly Carbohydrates polysaccharides Selection coefficient and allele frequency change Hardy weinberg equilibrium

Phase 1 · BU-103 7 concepts

Genetics

Law of segregation Independent assortment Dihybrid cross Linkage recombination Sex linkage Nernst potential for a single ion Goldman hodgkin katz equation

Phase 1 · BU-104 6 concepts

Evolution & Natural Selection

Natural selection Fitness adaptation Speciation Evidence common descent Sexual selection Lotka volterra predator prey

Phase 1 · BU-105 5 concepts

Ecology

Trophic energy flow Logistic population growth Competitive exclusion Lotka volterra predation Biogeochemical cycles

Phase 1 · BU-106 4 concepts

Organismal Physiology

Homeostasis negative feedback Diffusion osmosis Gas exchange surfaces Mass transport principle

Phase 2 · BU-201 7 concepts

Molecular Biology

Central dogma Semiconservative replication Transcription Translation genetic code Mutation types Fick law of diffusion in tissue Starling forces and capillary exchange

Phase 2 · BU-202 7 concepts

Cell Signalling & Membranes

Membrane transport Signal transduction Receptor ligand binding Second messengers Resting membrane potential Oxygen haemoglobin dissociation curve Renal clearance and glomerular filtration rate

Phase 2 · BU-203 5 concepts

Microbiology

Bacterial growth curve Antibiotic resistance Horizontal gene transfer Viral replication cycles Microbial metabolic diversity

Phase 2 · BU-204 5 concepts

Population & Molecular Genetics

Hardy weinberg Genetic drift Gene flow Mutation selection balance Inbreeding heterozygosity

Phase 2 · BU-205 5 concepts

Animal Physiology

Action potential Oxygen dissociation curve Countercurrent kidney Sliding filament Hormonal regulation

Phase 2 · BU-206 5 concepts

Plant Biology

Light reactions Calvin cycle Cohesion tension transpiration Plant hormones tropisms Stomatal regulation

Phase 2 · BU-207 5 concepts

Developmental Biology

Differential gene expression Morphogen gradients Hox genes body plan Embryonic induction Apoptosis in development

Phase 3 · BU-301 5 concepts

Immunology

Innate adaptive immunity Clonal selection Antibody diversity Mhc antigen presentation Immunological memory

Phase 3 · BU-302 5 concepts

Neuroscience

Synaptic transmission Neural integration Long term potentiation Sensory transduction Neural coding

Phase 3 · BU-303 6 concepts

Molecular Genetics & Genomics

Operon model Eukaryotic gene regulation Epigenetic inheritance Polymerase chain reaction Dna sequencing Crispr cas9

Phase 3 · BU-304 5 concepts

Evolutionary Biology

Phylogenetics Molecular clock Kin selection Coevolution Neutral theory

Phase 3 · BU-305 5 concepts

Ecology & Population Dynamics

Metapopulation dynamics Island biogeography Ecological succession Keystone species R k selection

Phase 3 · BU-306 5 concepts

Structural Biology & Biophysics

Protein folding X ray crystallography Michaelis menten kinetics Allosteric regulation Molecular motors

Phase 3 · BU-307 5 concepts

Cancer Biology

Oncogenes tumour suppressors Hallmarks of cancer Cell cycle checkpoints Angiogenesis metastasis Clonal evolution of tumours

Phase 3 · BU-308 4 concepts

Systems Biology

Gene regulatory networks Biological feedback loops Metabolic flux analysis Emergent properties

Phase 4 · BU-401 5 concepts

Bioinformatics & Computational Biology

Sequence alignment Blast database search Phylogenetic tree construction Hidden markov models Expression clustering

Phase 4 · BU-402 4 concepts

Synthetic Biology & Genetic Engineering

Recombinant dna Cloning vectors Engineered genetic circuits Gene therapy

Phase 4 · BU-403 5 concepts

Epidemiology & Disease Ecology

Sir model Basic reproduction number Herd immunity Transmission dynamics Vaccination strategy

Phase 4 · BU-404 4 concepts

Molecular Medicine

Pharmacokinetics Dose response Receptor pharmacology Genomic medicine