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LaTeX (KaTeX) templates
Mathematical typesetting: equations, derivations, matrices, commutative diagrams. Renders LaTeX math in the browser.
Physics — Diffraction & Resolution
Grating equation, Rayleigh criterion and resolving power.
Chemistry & PhysicsPhysics — Pressure & Hydraulics
Pressure, Pascal's principle, hydrostatic pressure and buoyancy.
Chemistry & PhysicsChemistry — Yield & Atom Economy
Percentage yield, atom economy and theoretical mass.
Chemistry & PhysicsChemistry — Titration
Mole relations at the equivalence point.
Chemistry & PhysicsChemistry — Empirical & Molecular Formula
From mass percent to empirical and molecular formula.
Chemistry & PhysicsChemistry — Reaction Quotient & Le Chatelier
How Q vs K predicts the direction of shift.
Chemistry & PhysicsChemistry — Temperature Dependence of K
The van 't Hoff equation linking K to temperature.
Chemistry & PhysicsChemistry — Colligative Properties
Boiling-point elevation, freezing-point depression, osmotic pressure, Raoult.
Chemistry & PhysicsChemistry — Buffers
Henderson–Hasselbalch in acid and base forms; buffer range.
Chemistry & PhysicsMath — Derivatives of Trig Functions
Standard trigonometric and inverse-trig derivatives.
MathematicsMath — Integrals of Trig Functions
Common trigonometric antiderivatives.
MathematicsMath — Sum-to-Product Identities
Sum-to-product and product-to-sum trigonometric identities.
MathematicsMath — Convergence Tests
Ratio test, p-series and the n-th term test.
MathematicsMath — Counting & Combinatorics
Permutations, combinations, Pascal's rule and multinomials.
MathematicsMath — Scalar & Vector Triple Products
Scalar triple product as a determinant and the BAC–CAB rule.
MathematicsMath — Differential Equations
Linear first-order, exponential growth and the auxiliary equation.
MathematicsMath — Polar Coordinates
Conversions, polar area and arc length.
MathematicsBiology — Selection & Mean Fitness
Mean population fitness and the change in allele frequency.
Life SciencesBiology — Respiratory Quotient
RQ and typical values for the major respiratory substrates.
Life SciencesBiology — Cardiovascular Relations
Cardiac output, blood pressure and ejection fraction.
Life SciencesBiology — Photosynthesis & Limiting Factors
Light/inverse-square relation, net vs gross, overall equation.
Life SciencesPhysics — Moments of Inertia
Definition and standard moments of inertia for common bodies.
Chemistry & PhysicsPhysics — Thermodynamic Processes
Isothermal, adiabatic and isobaric relations plus isothermal work.
Chemistry & PhysicsPhysics — RC Charging & Discharging
Capacitor charge/discharge and the RC time constant.
Chemistry & PhysicsPhysics — Sound Intensity & Level
Intensity, the decibel scale and the inverse-square law.
Chemistry & PhysicsPhysics — Photon Properties
Photon energy, momentum, wavelength and photon flux.
Chemistry & PhysicsPhysics — Gravitational Fields
Field strength, potential and the field–potential gradient.
Chemistry & PhysicsPhysics — Drag & Terminal Velocity
Drag force and the terminal velocity of a falling body.
Chemistry & PhysicsChemistry — Electron Configuration
Orbital filling order, capacities and the governing principles.
Chemistry & PhysicsChemistry — Born–Haber & Lattice Energy
Born–Haber cycle, lattice-energy scaling and dissolution.
Chemistry & PhysicsChemistry — Formal Charge & Oxidation Number
Formal charge formula and the usual oxidation-state rules.
Chemistry & PhysicsChemistry — pH of Strong & Weak Acids
Hydrogen-ion concentration for strong vs weak acids.
Chemistry & PhysicsChemistry — Gas Density & Molar Mass
Linking gas density, molar mass and the ideal gas law.
Chemistry & PhysicsChemistry — Arrhenius & Temperature
Arrhenius equation in exponential, log and two-point forms.
Chemistry & PhysicsMath — Transformation Matrices
Rotation, reflection and scaling matrices in the plane.
MathematicsMath — Mean Value & Rolle's Theorem
MVT, Rolle's theorem and the mean value of an integral.
MathematicsMath — Arc Length, Surface & Volume
Arc length, surface of revolution and solids of revolution.
MathematicsMath — Double Integrals
Iterated and polar double integrals over a region.
MathematicsMath — Induction & Standard Sums
The closed forms commonly proved by induction.
MathematicsMath — Complex Number Forms
Cartesian form, modulus, argument and conjugate.
MathematicsBiology — Chi-square Test (Genetics)
Testing observed vs expected genetic ratios.
Life SciencesBiology — Ecological Productivity
Gross and net primary productivity and transfer efficiency.
Life SciencesBiology — Serial Dilution & Counting
Dilution factors and viable cell counts.
Life SciencesPhysics — Rolling Motion
Rolling without slipping and energy of a rolling body.
Chemistry & PhysicsPhysics — Energy in SHM
Energy, velocity and acceleration relations in SHM.
Chemistry & PhysicsPhysics — Elastic Moduli
Young's, shear and bulk moduli and strain energy.
Chemistry & PhysicsPhysics — Magnetic Flux & Induction
Flux, Faraday's law, motional EMF and inductor energy.
Chemistry & PhysicsPhysics — Relativistic Doppler & Redshift
Relativistic Doppler shift and cosmological redshift.
Chemistry & PhysicsPhysics — Bragg Diffraction
Bragg's law, crystal spacing and the de Broglie wavelength.
Chemistry & PhysicsChemistry — Redox & Cell Potential
Oxidation/reduction and standard cell potential.
Chemistry & PhysicsChemistry — Degree of Dissociation
Ostwald's dilution law for a weak electrolyte.
Chemistry & PhysicsChemistry — Osmotic Pressure
Van 't Hoff factor and osmotic pressure.
Chemistry & PhysicsChemistry — Henry's Law
Solubility of a gas in a liquid.
Chemistry & PhysicsMath — Binomial Series
The binomial expansion for any real index.
MathematicsMath — Sine & Cosine Rules
Solving non-right triangles and triangle area.
MathematicsMath — Parametric & Implicit Differentiation
Derivatives for parametric and implicit curves.
MathematicsMath — Vector Calculus Theorems
Green's, Stokes' and the divergence theorems.
MathematicsMath — Descriptive Statistics
Centre, spread and the interquartile range.
MathematicsBiology — Allele Frequencies from Counts
Computing allele frequencies from genotype counts.
Life SciencesBiology — Population Growth Rates
Per-capita rate, finite growth rate and doubling time.
Life SciencesBiology — Metabolic Energy
Kleiber's law and energy-balance relations.
Life SciencesPhysics — AC RMS Values & Power
Root-mean-square voltage/current and average AC power.
Chemistry & PhysicsPhysics — Thermal Conduction & U-values
Fourier conduction, thermal resistance and U-values.
Chemistry & PhysicsPhysics — Doppler Effect (Sound)
The full acoustic Doppler formula and sign conventions.
Chemistry & PhysicsPhysics — Rocket & Variable Mass
Thrust and the Tsiolkovsky rocket equation.
Chemistry & PhysicsPhysics — Magnetic Dipole
Magnetic moment, torque and potential energy of a current loop.
Chemistry & PhysicsPhysics — Wave Intensity & Power
Intensity, its dependence on amplitude and frequency, and level.
Chemistry & PhysicsChemistry — Rate Constant Units
Units of the rate constant for each overall order.
Chemistry & PhysicsChemistry — Entropy & Spontaneity
Entropy change, the second law and spontaneity criterion.
Chemistry & PhysicsChemistry — Kp & Partial Pressures
Equilibrium constant in terms of partial pressures.
Chemistry & PhysicsChemistry — Crystal Field & Colour
d-orbital splitting and the origin of transition-metal colour.
Chemistry & PhysicsMath — 3×3 Determinant & Cramer's Rule
Cofactor expansion, Cramer's rule and the inverse.
MathematicsMath — Geometric Series
The nth term, partial sums, common ratio and sum to infinity of a geometric series.
MathematicsMath — Logarithmic & Exponential Equations
Solving equations with logs and exponentials.
MathematicsMath — Integration Techniques
Parts, substitution, log trick and a standard arctan form.
MathematicsMath — Hypothesis Testing
z- and t-statistics, decision rule and a confidence interval.
MathematicsBiology — Genetic Cross Ratios
Expected phenotype ratios for common crosses.
Life SciencesBiology — ABO Blood Groups
Multiple-allele inheritance of the ABO system.
Life SciencesBiology — Lung Volumes
Respiratory volumes and capacities.
Life SciencesPhysics — Banked Curve with Friction
Maximum cornering speed on a banked road with friction.
Chemistry & PhysicsPhysics — Satellites & Geostationary Orbits
Orbital speed, period and the geostationary radius.
Chemistry & PhysicsPhysics — Photoelectric Effect
Einstein's photoelectric equation and the stopping voltage.
Chemistry & PhysicsPhysics — Physical Pendulum
Period of a rigid-body pendulum about a pivot.
Chemistry & PhysicsPhysics — Atomic Energy Levels
Hydrogen energy levels and emission/absorption transitions.
Chemistry & PhysicsPhysics — Electric Field Configurations
Fields of a point charge, a charged plane and parallel plates.
Chemistry & PhysicsChemistry — Calorimetry & Enthalpy of Combustion
Heat released and molar enthalpy of combustion.
Chemistry & PhysicsChemistry — ICE Table Equilibrium
Setting up equilibrium concentrations with an ICE table.
Chemistry & PhysicsChemistry — Percent Composition
Mass percent of an element and formula mass.
Chemistry & PhysicsChemistry — Electronegativity & Bond Polarity
Using electronegativity difference to classify bonds.
Chemistry & PhysicsMath — Types of Sequence
Arithmetic, geometric and recursive sequences.
MathematicsMath — Function Transformations
Translations, stretches and reflections of graphs.
MathematicsMath — Circle Theorems
Key angle theorems in circle geometry.
MathematicsMath — Stationary Points & Optimisation
Finding and classifying turning points.
MathematicsMath — Exponential Growth & Decay
Exponential models, half-life and compound interest.
MathematicsBiology — Energy & Biomass Pyramids
Ten-percent transfer rule between trophic levels.
Life SciencesBiology — Osmosis & Tonicity
Cell responses to solutions of different tonicity.
Life SciencesBiology — Enzyme Inhibition
Competitive vs non-competitive inhibition kinetics.
Life SciencesPhysics — Hubble's Law & Cosmology
Cosmic expansion, the Hubble time and redshift.
Chemistry & PhysicsPhysics — Stellar Luminosity & Magnitude
Luminosity, the distance modulus and the mass–luminosity relation.
Chemistry & PhysicsPhysics — Black Holes & Relativity
Schwarzschild radius, mass–energy and gravitational time dilation.
Chemistry & Physics