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SI and CGS Conversion Relations
N is Newton, dyn is dyne, J is Joule, and erg is erg.
Physical conversions for force (Newton to dyne) and energy (Joules to ergs) between SI and CGS systems.
General Dimensional Representation
[Q] represents the dimensions of physical quantity Q; M,L,T,I,θ,N,J are the seven fundamental dimensions.
Expressing any physical quantity in terms of base dimensions (Mass, Length, Time, etc.).
Mean Absolute Error
ai is the i-th measurement, amean is the arithmetic mean, and Δamean is the mean absolute error.
The mean of the absolute values of the differences between individual measurements and the true mean value.
Relative and Percentage Error
er represents relative error, and ep represents percentage error.
Relative error is the fractional value of absolute error relative to the mean, and percentage error is its value expressed as a percentage.
Propagation of Errors in Calculations
A,B,C are measured quantities with errors ΔA,ΔB,ΔC; Z is the resulting calculated quantity.
Formulas to calculate the maximum absolute and relative errors propagated through arithmetic combinations.
Average Speed and Velocity Inequality
vavg is average speed, vavg is average velocity vector, Δr is displacement vector, and Δt is total elapsed time.
Defines average speed and average velocity vector, establishing that the magnitude of average velocity is always less than or equal to average speed.
Average and Instantaneous Velocity
vavg is the average velocity, vinst is the instantaneous velocity, x represents position, and t represents time.
Average velocity is displacement divided by time interval, while instantaneous velocity is the time derivative of position.
Equations of Motion for Uniform Acceleration
u is initial velocity, v is final velocity, a is constant acceleration, t is elapsed time, s is displacement, and sn is the displacement in the n-th second.
Relationships between velocity, displacement, time, and acceleration for constant acceleration.
Relative Velocity in One Dimension
vAB is the velocity of object A relative to B, vA is the velocity of A, and vB is the velocity of B.
The velocity of an object A as observed from the reference frame of an object B.
Parallelogram Law of Vector Addition
R is magnitude of resultant vector, A,B are magnitudes of the individual vectors, θ is angle between them, and α is angle of resultant with vector A.
Calculates the magnitude and direction of the resultant vector of two vectors added at angle θ.
Dot and Cross Products of Vectors
A and B are vectors, A and B are their magnitudes, and θ is the angle between them.
Mathematical representations of scalar (dot) and vector (cross) products of two vectors.
Formulas for Projectile Motion under Gravity
u is the launch speed, θ is the launch angle with the horizontal, T is time of flight, H is max height, R is range, and g is acceleration due to gravity.
Time of flight, maximum height, horizontal range, and the equation of the trajectory for a projectile launched from ground level.
Centripetal Acceleration and Angular Velocity Relations
v is linear speed, ω is angular velocity, r is circular radius, f is frequency, T is time period, and ac is centripetal acceleration.
Relations linking linear speed, angular velocity, frequency, and centripetal acceleration in uniform circular motion.
Newton's Second Law
F is force, p is linear momentum, m is mass, and a is acceleration.
Definition of force as the rate of change of linear momentum.
Momentum and Impulse
p is linear momentum, m is mass, v is velocity, J is impulse, and Δp is the change in momentum.
Definition of momentum and impulse as the change in momentum.
Static and Kinetic Friction Limits
fs,max is limiting static friction, fk is kinetic friction, μs and μk are coefficients of static and kinetic friction respectively, and N is normal force.
Formulas to calculate maximum static friction and kinetic friction.
Critical Speeds on Level and Banked Circular Roads
r is circular radius, g is acceleration due to gravity, μs is coefficient of static friction, θ is banking angle, vopt is friction-free optimum speed, and vmax is maximum safe speed.
Formulas for maximum safe speed on flat and banked roads.
Work Done by Forces
W is work done, F is force, d is constant displacement, and dr is differential displacement.
Definition of work done by constant and variable forces.
Work-Energy Theorem and Power
Wnet is net work done, ΔK is change in kinetic energy, P is power, F is force, and v is velocity.
The work-energy theorem (net work equals change in kinetic energy) and instantaneous power.
Relation Between Force and Potential Energy
Fx is the conservative force along the x-axis, and U is the potential energy as a function of position x.
Formula relating a conservative force to the spatial gradient of its potential energy.
Potential Energy of a Hookean Spring
U is the stored potential energy, k is the spring constant, and x is the extension or compression.
Potential energy stored in a spring stretched or compressed by an displacement x.
Critical Speeds in Vertical Circular Motion
v is the critical speed, T is the tension in the string, R is the radius of the circle, m is the mass, and g is the acceleration due to gravity.
Minimum speeds required at the lowest and highest points for a mass on a string to complete a vertical circle.
Coefficient of Restitution and Collision Speeds
e is the coefficient of restitution, u1,u2 are velocities before collision, v1,v2 are velocities after collision, and m1,m2 are the colliding masses.
Definition of coefficient of restitution (e) and final velocities after a 1D elastic or inelastic collision.
Position of Centre of Mass
rcm is the center of mass position vector, and mi and ri are the mass and position of the i-th particle.
Weighted average position of a discrete system of particles.
Centre of Mass for Continuous Media
rcm is the center of mass position vector, M is total mass, and dm is a differential mass element.
Weighted average position for a continuous mass distribution.
Torque and Angular Acceleration Relation
τ is torque, r is position vector, F is force vector, I is moment of inertia, and α is angular acceleration.
Definition of torque and the relation linking net torque to angular acceleration.
Angular Momentum and Net Torque
L is angular momentum, r is position, p is linear momentum, I is moment of inertia, ω is angular velocity, and τnet is net torque.
Definition of angular momentum and the relation linking net torque to the rate of change of angular momentum.
Moment of Inertia and Radius of Gyration
I is moment of inertia, r represents distance from axis of rotation, M is total mass, and K is the radius of gyration.
General definition of moment of inertia and its relation to the radius of gyration.
Parallel and Perpendicular Axes Theorems
Iz is moment of inertia about the target axis, Icm is moment of inertia about parallel center-of-mass axis, M is total mass, d is perpendicular distance between axes, and Ix,Iy are moments of inertia about perpendicular axes in the plane of a laminar sheet.
Mathematical expressions of the parallel axes theorem and perpendicular axes theorem for moments of inertia.
Universal Gravitational Force and Kepler's Third Law
F is gravitational force, G is gravitational constant, m1,m2 are masses, r is orbital radius, T is orbital period, and Ms is mass of the central star/Sun.
Newton's force equation for gravity and the mathematical relation for Kepler's third law of planetary motion.
Variation of g with Altitude, Depth, and Rotation
g is gravity at sea level, gh and gd are gravity values at height h and depth d, R is Earth's radius, ω is Earth's rotational speed, and ϕ is the latitude.
Formulas calculating how earth's gravitational acceleration changes at heights (h≪R), depths (d), and latitudes (ϕ).
Gravitational Potential Energy and Potential
U is gravitational potential energy, V is gravitational potential, M,m are masses, and r is the separation distance.
Potential energy of a two-mass system and the potential due to a point mass.
Escape Velocity and Satellite Orbital Velocity
ve is escape velocity, vo is orbital velocity, M is Earth's mass, R is Earth's radius, and g is acceleration due to gravity.
Formulas to calculate escape speed from Earth and the speed required for circular orbit close to Earth.
Hooke's Law of Elasticity
F is applied deforming force, A is cross-sectional area, E is modulus of elasticity, ΔL is elongation, and L0 is original length.
Fundamental linear relationship of elasticity indicating stress is directly proportional to strain within the elastic limit.
Young's and Bulk Moduli
Y is Young's Modulus, B is Bulk Modulus, F is stretching force, L is length, A is area, ΔL is extension, ΔP is pressure change, and ΔV is volume change.
Definitions of Young's and Bulk Moduli of elasticity.
Poisson's Ratio and Elastic Constants Relations
σ is Poisson's ratio (theoretical limits: −1≤σ≤0.5), Y is Young's modulus, B is Bulk modulus, and η is Shear modulus (rigidity modulus).
Defines Poisson's ratio and the fundamental mathematical relations between Young's, Bulk, and Shear moduli.
Elastic Strain Energy Density
uenergy is energy density, Stress represents applied force per unit area, and Strain is fractional deformation.
The elastic strain energy stored per unit volume of a stretched body.
Hydrostatic Pressure variation
P is total pressure at depth h, P0 is atmospheric pressure at surface, ρ is fluid density, and g is acceleration due to gravity.
Formula for pressure at depth h in a fluid of density ρ.
Buoyant Force (Archimedes' Principle)
FB is the buoyant force, ρf is the density of the fluid, Vsub is the volume of the submerged part of the body, and g is the acceleration due to gravity.
Formula to calculate the upward buoyant force acting on a body fully or partially submerged in a fluid.
Stokes' Law and Terminal Velocity
Fd is viscous drag, η is coefficient of viscosity, r is sphere radius, v is speed, vt is terminal velocity, ρbody is density of sphere, and ρfluid is fluid density.
Viscous drag force on a sphere and its ultimate terminal speed through a viscous medium.
Reynolds Number Flow Regime
Re is Reynolds number, ρ is fluid density, v is flow velocity, d is diameter of tube, and η is coefficient of viscosity.
Dimensionless quantity determining whether fluid flow is laminar (streamline) or turbulent.
Equation of Continuity and Bernoulli's Theorem
A represents cross-sectional area, v represents flow velocity, P is pressure, h is height, and ρ is fluid density.
Conservation of mass and mechanical energy principles in streamline fluid flow.
Relation Between Surface Tension and Surface Energy
W is the work done (stored as surface energy), T is the surface tension of the liquid, and ΔA is the increase in surface area (taking both surfaces into account for a film).
Formula for the work done in increasing the surface area of a liquid film.
Excess Pressure in Drops/Bubbles
T is surface tension, R is radius of curved interface, and ΔP is the excess pressure.
Formulas calculating excess internal pressure in curved interfaces.
Height of Capillary Rise
T is surface tension, h is capillary height, θ is contact angle, ρ is density of liquid, g is acceleration due to gravity, and r is tube radius.
Formula calculating height of capillary liquid column.
Temperature Scale Conversions
C is temperature in degrees Celsius, F is in degrees Fahrenheit, and K is in Kelvin.
Formula to convert temperatures between Celsius, Fahrenheit, and Kelvin scales.
Linear Expansion
L is final length, L0 is initial length, α is linear expansion coefficient, and ΔT is temperature change.
Formula for linear dimension changes with temperature.
Specific Heat Capacity and Latent Heat
Q is heat exchanged, m is mass, s is specific heat, L is latent heat, and ΔT is temperature change.
Basic heat transfer calculations for temperature and phase changes.
Conduction Rate
k is thermal conductivity, A is area, L is length, and T is temperature.
Formula for conductive heat current through a material.
Wien's Displacement Law
λmax is wavelength of maximum radiation intensity, T is absolute temperature, and b is Wien's displacement constant (2.898×10−3 m K).
Relates the peak wavelength of blackbody radiation to the absolute temperature of the body.
Stefan's Law and Wien's Law
E is radiative power, σ is Stefan-Boltzmann constant, e is emissivity, A is area, T is temperature, λm is peak wavelength, and b is Wien's constant.
Formulas for total radiative power of a blackbody and wavelength of maximum emission.
Newton's Law of Cooling
T is body temperature, Ts is surrounding temperature, K is positive constant, and t is time.
Approximate rate of temperature fall of a hot body to its surroundings.
Temperature Scale Conversions
TC is Celsius temperature, TF is Fahrenheit temperature, and TK is Kelvin (absolute) temperature.
Conversion equations among Celsius, Fahrenheit, and Kelvin temperature scales.
First Law Equation
ΔQ is heat added, ΔU=nCvΔT is internal energy change, and W is work.
Energy conservation equation for thermodynamic systems.
Work in Isothermal & Adiabatic Processes
W is work, n is mole count, R is gas constant, T is temperature, Vi,Vf are volumes, Pi,Pf are pressures, and γ is adiabatic index.
Work formulas for key thermodynamic paths.
Adiabatic State Equations (Poisson's Relations)
P is pressure, V is volume, T is absolute temperature, and γ=Cp/Cv is the adiabatic index (ratio of specific heats).
Governing state relations for a quasi-static adiabatic process of an ideal gas.
Refrigerator Coefficient of Performance (COP)
β is coefficient of performance, QC is heat extracted from cold reservoir, W is work input, TC is cold temperature, and TH is hot temperature.
Measures the efficiency (COP) of an ideal Carnot refrigerator.
Carnot Engine and Refrigerator performance
η is efficiency, βCOP is coefficient of performance, TC is absolute cold reservoir temperature, and TH is absolute hot reservoir temperature.
Efficiency of a Carnot cycle engine and the coefficient of performance of a refrigerator.
Equation of State and Compression Work
P is pressure, V is volume, n is moles, T is temperature, R is gas constant, and W is work done on the gas.
Relation linking pressure, volume, temperature, and moles, and the general work integral for compression.
Kinetic Model Pressure
P is pressure, ρ is density, and vrms is RMS molecular speed.
Concept of pressure based on kinetic theory of gases.
Molecular Speeds and Specific Heats
vrms is RMS speed, M is molar mass, m is single molecule mass, kB is Boltzmann constant, Cv,Cp are specific heats, f is degrees of freedom, and R is gas constant.
Expressions for RMS velocity and molar heat capacities based on degrees of freedom.
Mean Free Path
λ is mean free path, d is collision diameter, and n is number density.
The average distance traveled by a moving gas molecule between successive collisions.
Displacement, Velocity and Acceleration in SHM
x is displacement, A is amplitude, ω is angular frequency, ϕ is initial phase, v is velocity, and a is acceleration.
Kinematic equations describing standard simple harmonic motion.
Energy in SHM
Etotal is energy, m is mass, ω is angular frequency, and A is amplitude.
Total mechanical energy of SHM.
Periods of Standard Oscillators
T is time period, k is spring constant, m is mass, L is pendulum length, and g is acceleration due to gravity.
Time period formulas for springs and pendulums.
Amplitude Decay in Damped SHM
A(t) is damped amplitude, A0 is initial amplitude, b is damping constant, m is mass, and ω′ is the damped angular frequency.
Expression showing exponential decay of amplitude over time in a weakly damped system.
Speed of Wave on String and in Gas
T is tension, μ is linear mass density, γ is adiabatic index, R is gas constant, T is temperature, and M is molar mass.
Determines the velocity of transverse waves on a stretched string, and longitudinal sound waves in a gas.
Progressive Wave Equation and String/Gas Wave Speeds
y is displacement, A is amplitude, k=2π/λ is wave number, ω=2πf is angular frequency, v represents speed, T is tension, μ is linear mass density, and γ is adiabatic index of gas.
Equation of a harmonic progressive wave and formulas for transverse wave speed on a string and longitudinal wave in gases.
Equation of a Standing Wave
y is the displacement at position x and time t, A is the amplitude of individual waves, k=2π/λ is the wave number, and ω=2πf is the angular frequency.
Mathematical representation of a stationary wave formed by the superposition of two identical travelling waves in opposite directions.
Harmonics in Organ Pipes
f represents frequency, v is sound velocity, and L is length of pipe.
Fundamental frequencies of open and closed organ pipes.
Beat Frequency
fbeat is beat frequency, and f1,f2 are source frequencies.
Formula to calculate the beat frequency from two close source frequencies.
General Doppler Shift Formula in Sound
f′ is observed frequency, f is source frequency, v is speed of sound in medium, vo is observer speed, and vs is source speed.
Formula for observed frequency when observer and source are in relative motion along line of sight.
Quantization of Electric Charge
q is the net charge, n is any integer (positive or negative), and e≈1.6×10−19 C is the elementary charge of an electron.
Formula expressing that the total charge on a body is an integral multiple of the basic unit of charge.
Coulomb's Law of Electrostatic Force
F is force, q1,q2 are point charges, r is separation distance, r^ is unit vector along separation line, and ε0 is vacuum permittivity.
Electrostatic force vector acting between two point charges separated in vacuum.
Electric Field of a Point Charge
E is electric field intensity, q is charge magnitude, r is distance, and ε0 is permittivity of free space.
Calculates the magnitude of the electrostatic field produced by a point charge in vacuum.
Electric Dipole Fields and Torque
p=q(2a) is electric dipole moment magnitude, r is distance from dipole center (r≫a), E is electric field, and τ is torque vector in field E.
Electric fields at axial/equatorial positions and torque experienced in a uniform external field.
Gauss's Law and Electrostatic Field Applications
ΦE is electric flux, qen is enclosed net charge, λ is linear charge density, σ is surface charge density, and r is radial distance.
Integral definition of flux and fields due to standard charge distributions.
Electric Potential
V is electric potential, q is charge, and r is separation distance.
Potential of a point charge in electrostatic space.
Electric Potential Energy
U is electrostatic potential energy, q1,q2 are charges, and r is separation distance.
Potential energy of a two-charge configuration.
Induced Charge on a Dielectric Slab
qp is the induced (polarization) charge, q is the free charge on the capacitor plates, and K is the dielectric constant of the slab.
Formula for the induced charge appearing on the faces of a dielectric slab placed in an external electric field.
Capacitance of Parallel Plate Capacitors
C is capacitance, K is dielectric constant (K=1 in vacuum), A is plate area, and d is separation.
Parallel plate capacitance (with dielectric K).
Series/Parallel Capacitance Formulas
Cparallel is parallel equivalent capacitance, and Cseries is series equivalent.
Combinations equivalent formulas.
Stored Energy
V is voltage, Q is charge, and Ustored is stored potential energy.
Energy stored in a capacitor.
Drift Velocity and Current
vd is drift velocity, e is electronic charge, E is electric field, τ is relaxation time, m is electron mass, I is current, n is free charge density, and A is area.
Microscopic model of current carrying conductors.
Ohm's Law and Electrical Resistance
V is potential difference, I is current, R is resistance, ρ is electrical resistivity, L is length, and A is cross-sectional area.
Formulas for voltage-current relationship and the dependence of resistance on geometry and resistivity.
Temperature effect on Resistance
R(T) is final resistance, R0 is initial resistance, α is temperature coefficient of resistance, and ΔT is temperature change.
Effect of temperature on resistance values.
EMF and Internal Resistance
V is terminal voltage, E is cell electromotive force (EMF), I is current, and r is internal resistance.
Terminal potential difference of cells.
Equivalent EMF for Parallel Cells
Ei and ri are the EMF and internal resistance values of individual cells, and Eeq,parallel is equivalent EMF.
Equivalent values for parallel groups of cells.
Kirchhoff's Current and Voltage Laws
Iin,Iout are currents entering/leaving junctions, and ΔV represents potential differences around a closed loop.
Conservation rules for charge (junction law) and energy (loop law) in circuits.
Balanced Wheatstone Bridge and Meter Bridge
P,Q,R,S are bridge resistances, and l is the balance length in centimeters along the 1-meter wire.
Symmetry conditions for zero current in the central galvanometer of a Wheatstone bridge and its meter bridge application.
Galvanometer to Ammeter and Voltmeter Conversion
S is parallel shunt resistance, R is series multiplier resistance, Rg is galvanometer coil resistance, Ig is full-scale deflection current, I is ammeter range, and V is voltmeter range.
Formulas to convert a basic galvanometer into a high-range ammeter or voltmeter.
Potentiometer Equations
E1,E2 are EMFs of cells, l1,l2 are balancing lengths, r is internal resistance, and R is standard resistance box resistance.
Formulas to compare cell EMFs and measure internal resistance.
Biot-Savart Law and Circular Loop Field
dB is differential magnetic field, μ0 is vacuum permeability, I is current, dl is length vector, r is position vector, R is circular loop radius, and x is axial distance.
Magnetic field differential vector and field value on the axis of a circular loop.
Magnetic Field of Finite Wire and Circular Arc
B is magnetic field, I is current, d is perpendicular distance, ϕ1,ϕ2 are angles subtended by wire ends, R is circular arc radius, and θ is arc angle in radians.
Magnetic fields at a point due to a straight wire segment and at the center of a circular current arc.
Ampere's Circuital Law and Straight Wire Field
Ien is enclosed net current, Bwire is magnetic field strength at distance r from wire, and dl is differential length.
Line integral relation for magnetic fields and field due to a long straight current conductor.
Lorentz Force and Circular Orbit Radius
F is Lorentz force, q is charge, E is electric field, v is velocity, B is magnetic field, rorbit is radius of orbit, and m is mass of charge.
Force vector on moving charges and orbital parameters in uniform magnetic fields.
Magnetic Force on Current Wire
I is current, L is wire length vector, and B is field vector.
Force vector on a current-carrying wire.
Force per Unit Length between Parallel Wires
I1,I2 are currents, d is separation distance, and F/L is force per unit length.
Force per unit length between two parallel wires.
Torque on Loop, Galvanometer sensitivities
τ is torque, M is magnetic dipole moment, B is field, N is turn count, A is area, C is torsional restoring constant, and Rg is galvanometer coil resistance.
Torque vector on current loop and equations for galvanometer sensitivities.
Magnetic Moment of a Revolving Electron
Melectron is magnetic moment, e is electronic charge, v is orbital speed, r is orbital radius, m is electron mass, and L is orbital angular momentum.
Orbital magnetic dipole moment (Bohr Magneton) of a revolving hydrogenic electron.
Magnetic Field of a Bar Magnet
B is the magnetic field, M is the magnetic dipole moment, r is the distance from the magnet center (r≫ magnet length), and μ0 is the permeability of free space.
Magnetic field formulas at axial and equatorial points of a short bar magnet.
Magnetic Susceptibility and Curie's Law
μr is relative magnetic permeability, χm is magnetic susceptibility, C is Curie's constant, and T is temperature in Kelvin.
Formula for relative permeability and Curie's temperature dependence of paramagnetic susceptibility.
Faraday's and Lenz's Law of Induction
E is induced EMF, ΦB is magnetic flux, and B is magnetic field vector.
Mathematical expression relating induced EMF to time rate of change of magnetic flux.
Self and Mutual Inductance Definition
L is self-inductance coefficient, M is mutual inductance coefficient, I,I2 are currents, and Es,E1 are induced EMFs.
Relationships linking magnetic flux linkage to current through coils.
Peak and RMS values of AC
Irms,Vrms are RMS current and voltage, and I0,V0 are peak values.
Formulas calculating Root Mean Square values for sinusoidal AC.
Impedance in LCR Series Circuits
Z is impedance, R is resistance, XL=ωL is inductive reactance, and XC=1/(ωC) is capacitive reactance.
Impedance magnitude of LCR series circuits.
Resonant Frequency and Q-Factor
ω0 is resonant angular frequency, Q is quality factor, L is inductance, C is capacitance, and R is resistance.
Resonant frequency and Q-factor of LCR series circuits.
Average Power and Power Factor in AC
Pavg is average power, cosϕ is power factor, and ϕ is the phase difference between current and voltage.
Expression for average active power dissipation in AC circuits.
Transformer Voltage and Current Relations
Vp,Vs are primary/secondary voltages, Np,Ns are primary/secondary turns, and Ip,Is are primary/secondary currents.
Voltage ratio and current ratio based on turns ratio in ideal transformers.
Maxwell's Displacement Current
Id is displacement current, ε0 is vacuum permittivity, and ΦE is electric flux.
Current defined in terms of time rate of change of electric flux.
Velocity of Light and Fields Ratio
c is speed of light, E0 is electric field amplitude, B0 is magnetic field amplitude, μ0 is permeability of free space, and ε0 is permittivity of free space.
Relationship relating the speed of electromagnetic waves in a vacuum to the ratio of electric and magnetic field amplitudes, and vacuum constants.
Mirror Formula
u,v are object/image distances, and f is mirror focal length.
Fundamental equations describing reflection in spherical mirrors.
Snell's Law and Critical Angle
θ1,θ2 are angles, n1,n2 are refractive indices, and θc is critical angle.
Equations describing refraction and the limit of total internal reflection.
Refraction at Spherical Surfaces and Lens Maker's Formula
u is object distance, v is image distance, μi are refractive indices, Ri are curvature radii, and f is focal length.
Refraction relation at a single curved boundary and the formula to determine the focal length of a thin lens.
Lens Equations
u,v are object/image distances, f is focal length, R1,R2 are spherical radii, and n represents refractive indices.
Thin lens formula and lensmaker's equation.
Prism Formula
nprism is the refractive index, A is prism apex angle, and δm is minimum deviation angle.
Refractive index formula for a prism at minimum deviation.
Magnification of Microscope and Telescope
fo and fe are focal lengths of objective and eyepiece respectively, L is tube length, and D=25 cm is the least distance of distinct vision.
Magnifying power formulas for compound microscope and astronomical telescope at normal adjustment.
Relation Between Phase and Path Difference
Δϕ is phase difference, Δx is path difference, and λ is wavelength of light.
Converts spatial path difference of coherent waves into their temporal phase difference.
Interference Intensities and Fringe Width in YDSE
Ires is resulting intensity, I1,I2 are slit source intensities, ϕ is phase difference, βfringe is fringe width, λ is wavelength, D is slit-screen distance, and d is slit separation distance.
Mathematical description of resulting intensity and bright/dark fringe width in Young's Double Slit Experiment.
Central Maximum Width
wlinear is linear width of central maximum, a is slit width, and D is distance to screen.
Formula for angular/linear width of central maximum in single slit diffraction.
Brewster's Law
n is refractive index of medium, and ip is polarizing angle (Brewster's angle).
Brewster's polarization angle for reflecting surfaces.
Resolving Power of Microscope and Telescope
μ is the refractive index of the medium between the object and the objective, θ is the semi-vertical angle of the cone of light, λ is the wavelength, and D is the diameter of the objective lens (aperture).
Formulas for the limit of resolution and resolving power of optical instruments.
Einstein's Photoelectric Equation
Kmax is maximum kinetic energy of electrons, hν is incident light energy, ϕ is material work function, e is electronic charge, and V0 is stopping potential.
Relation between incident photon energy, work function, and maximum kinetic energy of emitted photoelectrons.
de Broglie Wavelength of Matter Waves
λ is de Broglie wavelength, h is Planck's constant, p is momentum, m is mass, K is kinetic energy, q is charge, and V is accelerating potential.
Wavelength associated with a moving particle of momentum p or kinetic energy K.
Distance of Closest Approach
r0 is the distance of closest approach, Z is the atomic number of the target nucleus, e is elementary charge, Kα is kinetic energy of the incoming alpha particle.
Formula to find the minimum distance an alpha particle reaches before being repelled by a nucleus.
Bohr Model Radii and Energy Levels
rn is the n-th orbit radius, a0 is Bohr radius, En is n-th level energy, Z is atomic number, n,n1,n2 are principal quantum numbers, and R is Rydberg constant.
Quantized orbit radius and energy levels of hydrogen-like atoms.
Nuclear Radius and Mass Number Relation
R is nuclear radius, R0≈1.2 fm, and A is mass number.
Empirical formula relating nuclear radius to mass number.
Mass Defect and Binding Energy
Δm is mass defect, Z is proton count, A is mass number, mp,mn are proton/neutron masses, Mnuc is nuclear mass, and Eb is binding energy.
Difference in mass of components and energy binding the nucleus together.
Nuclear Reaction Q-Value
Q is the reaction energy, Mreactants is the total mass of the reactants, Mproducts is the total mass of the products, and c is the speed of light.
The amount of energy released or absorbed during a nuclear fission or fusion reaction.
Radioactive Decay Law and Half-Life
N(t) is remaining active nuclei at time t, N0 is initial count, λ is decay constant, T1/2 is half-life, and τmean is mean life.
Equations describing exponential decay rate and half-life duration of radioactive elements.
Bandgap Energy
Eg is the bandgap energy, h is Planck's constant, c is the speed of light, and λ is the photon wavelength.
Energy difference between the valence band and conduction band, which determines the optical properties of semiconductors.
Law of Mass Action in Semiconductors
ne is free electron density, nh is hole density, and ni is intrinsic carrier concentration.
Thermal equilibrium concentration product relation for electrons and holes.
Maximum Efficiency of Half-Wave and Full-Wave Rectifiers
η is rectifier efficiency, Rf is forward diode resistance, and RL is load resistance.
Maximum theoretical conversion efficiency of AC power to DC power for half-wave and full-wave rectifiers.
Zener Diode as a Shunt Voltage Regulator
IS,IZ,IL are series, Zener, and load currents; Vin is input voltage, VZ is Zener breakdown voltage, RS is series resistance, and RL is load resistance.
Circuit equations calculating series resistor and Zener currents to maintain constant load voltage.
Boolean Logic Gate Outputs
A,B are binary inputs (0 or 1), and Y is the output.
Mathematical outputs for basic logic gates (AND, OR, NAND, NOR).
Least Count of Vernier Calliper and Screw Gauge
LC represents Least Count, MSD represents Main Scale Division, VSD represents Vernier Scale Division, and Pitch is the linear advancement per full rotation of the screw gauge thimble.
Expressions defining the limits of measurement resolution for callipers and gauges.
Mole Calculations
w is mass of substance, M is molar mass, N is count of particles, NA≈6.022×1023 is Avogadro's number, and VSTP is volume of gas at standard temperature and pressure.
Basic relationships to calculate the number of moles (n) from mass, number of particles, or gas volume at STP.
Molarity, Molality and Mole Fraction
M is molarity, m is molality, xA is mole fraction of component A, n represents moles, V represents volume, and W represents solvent mass.
Formulas to express concentration of solute in a solution.
Rydberg Formula for Spectral Lines
νˉ is wave number, λ is wavelength, RH≈1.09677×107 m−1 is Rydberg constant, Z is atomic number, and n1,n2 are principal quantum numbers (n2>n1).
Formula to calculate the wavelength or wave number of spectral lines emitted during electronic transitions in hydrogen-like atoms.
de Broglie Wavelength and Heisenberg Uncertainty Principle
λ is de Broglie wavelength, h is Planck's constant, m is mass, v is velocity, p is momentum, Δx is position uncertainty, and Δp is momentum uncertainty.
Mathematical statements of wave-particle duality and the fundamental limit on simultaneous measurement precision.
Steric Number and Hybridization
V is the number of valence electrons on the central atom, M is the number of monovalent atoms bonded to it, C is the charge of the cation, and A is the charge of the anion.
Formula to calculate the steric number (SN) of a central atom to determine its hybridization and geometry.
Bond Order (Molecular Orbital Theory)
Nb is the number of electrons in bonding molecular orbitals, and Na is the number of electrons in antibonding molecular orbitals.
Formulas to calculate bond order, predicting stability and bond length comparison.
Dipole Moment
μ is dipole moment, q is magnitude of charge, and d is separation distance.
Mathematical definition of molecular dipole moment for a charge separation.
Ideal Gas Equation
P is pressure, V is volume, n is moles, T is temperature, and R is gas constant.
Equation of state for an ideal gas.
van der Waals Equation
P is pressure, V is volume, n is moles, T is temperature, R is gas constant, a is molecular attraction factor, and b is excluded volume factor.
Real gas equation incorporating molecular attraction and volume corrections.
Dalton's Law
Pi is partial pressure of gas i, xi is its mole fraction, and Ptotal is total pressure.
Partial pressures in gas mixtures.
Graham's Law of Diffusion
r1,r2 are diffusion rates, M1,M2 are molar masses, and d1,d2 are densities.
Relative rates of diffusion/effusion.
First Law and Expansion Work
ΔU is change in internal energy, q is heat exchanged, w is work, n is moles, R is gas constant, T is temperature, and V1,V2 are volumes.
Energy conservation equation and reversible expansion work formulas for gas systems.
Enthalpy and Internal Energy Relation
ΔH is change in enthalpy, ΔU is change in internal energy, Δng is change in gaseous moles, R is gas constant, and T is temperature.
Enthalpy change relation for gaseous reactions.
Entropy Change and Second Law of Thermodynamics
ΔS is the change in entropy, qrev is the heat exchanged reversibly, and T is the absolute temperature in Kelvin.
Definition of entropy change and the total entropy criterion for a spontaneous process.
Gibbs Free Energy and Spontaneity
ΔG is Gibbs free energy change, ΔG∘ is standard Gibbs energy change, ΔH is enthalpy change, T is absolute temperature, ΔS is entropy change, and Keq is equilibrium constant.
Definition of Gibbs free energy change and standard Gibbs energy relation.
Relationship Between Kp and Kc
Kp,Kc are equilibrium constants, R is gas constant, T is temperature, and Δng is change in moles of gaseous components.
Mathematical formula linking pressure-based and concentration-based equilibrium constants.
Reaction Quotient Shift Direction
Qc is the reaction quotient, and Kc is the equilibrium constant.
Determines the shift direction of equilibrium by comparing the reaction quotient Qc and the equilibrium constant Kc.
Van 't Hoff Reaction Isochore
K1,K2 are equilibrium constants at temperatures T1,T2, ΔH∘ is standard enthalpy of reaction, and R is gas constant.
Relates the equilibrium constants at two different temperatures to the standard enthalpy of reaction.
pH Definition
[H+] is the concentration of hydronium ions.
Mathematical definition of the pH scale.
Henderson-Hasselbalch Buffer Equations
pKa=−logKa, pKb=−logKb, and brackets denote molar concentrations.
Formulas for calculating pH of acidic/basic buffer solutions.
Solubility Product Constant
Ksp is solubility product, S is molar solubility of salt, and x,y are stoichiometric coefficients of cations and anions.
Relationship between solubility (S) and solubility product (Ksp) for standard electrolyte salts.
Sum of Oxidation States
ni is the number of atoms of species i, O.S.i is their oxidation state, and Net Charge is the overall charge.
Rule stating the sum of oxidation numbers of all atoms in a molecule/ion equals its total electrical charge.
Equivalent Weight of Redox Agent
E is equivalent weight, Molar Mass is the molecular weight, and n-factor is the number of electrons gained or lost per mole of reactant.
Definition of equivalent mass of an oxidizing or reducing agent in a redox reaction.
Conductivity, Molar Conductivity and Kohlrausch's Law
Λm is molar conductivity, κ is specific conductivity, C is molar concentration, Λm∘ is limiting molar conductivity, and ν+,ν− are ion counts with limiting conductances λ+∘,λ−∘.
Expressions for molar conductivity and its value at infinite dilution.
Debye-Hückel-Onsager Equation
Λm is molar conductivity, Λm∘ is limiting molar conductivity at infinite dilution, C is electrolyte concentration, and A is a constant depending on solvent, charge type, and temperature.
Describes the concentration dependence of molar conductivity for strong electrolytes.
Nernst Equation
Ecell is cell potential, Ecell∘ is standard potential, n is number of electrons transferred, and Q is reaction quotient.
Calculates cell potential under non-standard conditions.
Gibbs Free Energy and EMF
ΔG is Gibbs free energy change, n is electrons transferred, F≈96500 C mol−1 is Faraday constant, and Ecell is cell potential.
Links cell potential to Gibbs free energy change.
Faraday's Laws of Electrolysis
w is mass deposited, Z is electrochemical equivalent, I is current, t is time, Eeq is equivalent weight of substance, and F is Faraday constant.
Quantitative mass yield of substances discharged at electrodes during electrolysis.
Henry's Law and Raoult's Law
p,Ptotal are vapor pressures, KH is Henry's constant, x,xA,xB are mole fractions, and PA∘,PB∘ are pure component vapor pressures.
Equations showing vapor pressures of solute gases and volatile liquid solutions.
Vapor Pressure and Osmotic Pressure Equations
P∘,P are vapor pressures, i is van 't Hoff factor, xsolute is mole fraction of solute, π is osmotic pressure, C is molarity, R is gas constant, and T is temperature.
Formulas to calculate vapor pressure lowering and osmotic pressure.
van 't Hoff Factor and Degree of Dissociation/Association
i is the van 't Hoff factor, α is the degree of dissociation, β is the degree of association, and n is the number of ions/molecules formed or associated per reactant unit.
Relates the van 't Hoff factor (i) to the degree of electrolyte dissociation (α) and association (β).
Boiling and Freezing Point Shifts
ΔTb,ΔTf are temperature shifts, i is van 't Hoff factor, Kb,Kf are molal constants, and m is molality of solution.
Formulas to calculate boiling point elevation and freezing point depression.
Integrated Rate Laws
[A]0 is initial concentration, [A]t is concentration at time t, and k is rate constant.
Rate constant relations for zero-order and first-order chemical reactions.
Half-Lives
[A]0 is initial concentration, k is rate constant, and t1/2 is half-life.
Half-life relations for zero-order and first-order chemical reactions.
Arrhenius Equation for Temperature Dependence
k,k1,k2 are rate constants, A is frequency factor, Ea is activation energy, R is gas constant, and T,T1,T2 are absolute temperatures.
Shows how chemical rate constant varies with temperature based on activation energy.
Crystal Density and Packing Efficiency
ρ is the density, Z is the number of atoms per unit cell, M is the molar mass, a is the edge length, NA is Avogadro's number, and r is the atomic radius.
Formulas to calculate unit cell density and packing fraction in cubic crystal lattices.
Density of a Cubic Unit Cell
ρ is density, z is number of atoms per unit cell, M is molar mass, NA is Avogadro's number, and a is edge length of the cubic cell.
Theoretical density of a crystalline solid based on lattice parameters.
Freundlich Adsorption Isotherm
x is mass of adsorbate, m is mass of adsorbent, P is pressure, and k,n are temperature-dependent constants.
Empirical relationship for the adsorption of gas on solid adsorbent surfaces.
Coagulation Value
Coagulation Value is expressed in millimoles per litre of colloid.
Minimum concentration of an electrolyte required to cause coagulation of a sol.
Gold Number
Gold Number is measured in milligrams.
Defines the protective power of a lyophilic colloid. A lower gold number indicates better protection.
Effective Nuclear Charge
Zeff is the effective nuclear charge, Z is atomic number, and σ is the shielding constant.
Slater's rule equation for calculating the effective nuclear charge felt by valence electrons.
Pauling Electronegativity Difference
χA,χB are electronegativities, and Ed(A−B) represents the bond dissociation energy of A−B bond.
Estimates the electronegativity difference between two bonded atoms using bond dissociation energies in kJ/mol.
Volume Strength of Hydrogen Peroxide
Volume strength is the volume of O2 (in L) released by 1 L of H2O2 at STP; Molarity and Normality represent solution concentrations.
Relations between volume strength, molarity, and normality of H2O2 solution.
Solvay Process Net Reaction
Reactants are sodium chloride and calcium carbonate; products are sodium carbonate and calcium chloride.
Net overall reaction representing the manufacturing of sodium carbonate via the Solvay process.
Plaster of Paris Preparation
Reactant is Gypsum, product is Plaster of Paris and steam.
Thermal decomposition of Gypsum to Plaster of Paris.
Borax Bead Thermal Decomposition
NaBO2 is sodium metaborate and B2O3 is boric anhydride (boron trioxide).
Thermal swelling and decomposition of borax to form sodium metaborate and boric anhydride.
Xenon Tetrafluoride Hydrolysis
XeF4 is xenon tetrafluoride, and XeO3 is explosive xenon trioxide.
Complete hydrolysis reaction of xenon tetrafluoride producing xenon gas and xenon trioxide.
Chromate-Dichromate pH-Dependent Equilibrium
CrO42− is yellow chromate ion (stable in alkaline solution), and Cr2O72− is orange dichromate ion (stable in acidic solution).
Reversible pH-dependent equilibrium between yellow chromate and orange dichromate ions.
Dichromate Reduction in Acidic Medium
Cr3+ is green chromic ion, and n-factor of Cr2O72− is 6.
Half-reaction for the reduction of dichromate ion in acidic solution, where chromium is reduced from +6 to +3.
Spin-Only Magnetic Moment Formula
μspin is spin-only magnetic moment in Bohr Magnetons (B.M.), and n is number of unpaired electrons.
Calculates magnetic moment of transition metal ions based on unpaired d-electrons.
Shielding Effectiveness Trend
σ is shielding constant of respective subshells.
Shielding constant comparison showing the poor shielding power of f-orbitals, which leads to lanthanoid contraction.
General Lanthanoid Configuration
[Xe] is xenon noble gas core, 4f is filling inner f-orbital, 5d is transition d-orbital, and 6s is valence s-orbital.
General valence shell electronic configuration of the lanthanoid series elements.
Effective Atomic Number (EAN) Rule
Z is atomic number of metal, O.S. is oxidation state of metal, and C.N. is coordination number (number of coordinate bonds).
Calculates the effective atomic number of a central metal atom/ion in a coordination complex.
CFSE for Octahedral Complexes
CFSE is Crystal Field Stabilization Energy, nt2g is number of electrons in t2g orbitals, neg is number of electrons in eg orbitals, Δo is octahedral splitting parameter, P is pairing energy, and m is number of paired electron configurations.
Stabilization energy calculation of d-electrons in octahedral crystal fields.
Ellingham Diagram Thermodynamic Condition
ΔG∘ is standard Gibbs free energy change, ΔH∘ is standard enthalpy change, T is temperature, and ΔS∘ is standard entropy change.
Thermodynamic spontaneity condition governing the reduction of metal oxides using carbon or other metals.
Mond Process for Nickel Refining
Ni(CO)4 is volatile nickel tetracarbonyl complex.
Two-step gaseous carbonyl metallurgy process used for the purification of nickel metal.
Steam Distillation Mass Ratio
w1,w2 are masses of organic compound and water, p1,p2 are vapor pressures at distillation temperature, and M1,M2 are molecular weights.
Formula relating the masses of the distilled organic liquid and water to their vapor pressures and molecular weights.
Chromatography Retention Factor
Rf is a dimensionless fraction, representing chromatographic mobility.
Defines the retardation factor (Rf) of a compound in thin-layer or paper chromatography.
Nitrogen Percentage by Kjeldahl's Method
N is normality of acid, V is volume of acid consumed by ammonia (in mL), and w is mass of organic sample analyzed (in grams).
Quantitative estimation of nitrogen percentage in organic samples.
Nitrogen Percentage by Dumas' Method
VSTP is volume of nitrogen collected at STP in mL, and w is mass of organic compound in grams.
Quantitative estimation of nitrogen by measuring volume of nitrogen collected at STP.
Halogen Percentage by Carius Method
mAgCl,mAgBr,mAgI are the masses of the respective silver halide precipitates formed (in grams), and w is the mass of the organic compound analyzed.
Quantitative estimation of halogens (Chlorine, Bromine, Iodine) from silver halide precipitates.
Sulfur Percentage by Carius Method
mBaSO4 is the mass of the barium sulfate precipitate formed (in grams), and w is the mass of the organic compound analyzed.
Quantitative estimation of sulfur by converting it to barium sulfate precipitate.
Phosphorus Percentage Estimation
mMg2P2O7 is the mass of magnesium pyrophosphate precipitate formed (in grams), and w is the mass of the organic compound analyzed.
Quantitative estimation of phosphorus by precipitating it as magnesium pyrophosphate.
Specific Optical Rotation
[α]λT is specific rotation, αobserved is observed rotation in degrees, l is tube path length in decimeters, and c is concentration in g/mL.
Calculates specific optical rotation of an optically active substance in solution.
Stereoisomer Count for Unsymmetrical Molecules
N is total number of stereoisomers, and n is the number of chiral centers.
Calculates total number of stereoisomers (enantiomers + diastereomers) for an organic molecule with n unsymmetric chiral centers.
Acid Dissociation Constant and pKa Relation
Ka is acid dissociation constant, and pKa is the acid index (smaller pKa means stronger acid).
Relates the acid dissociation constant (Ka) to its logarithmic index (pKa), indicating relative organic acidity.
Hyperconjugation Alpha-Hydrogen Stability
Nα-H is the number of hydrogen atoms attached to carbons directly adjacent to the sp2 carbocation/radical center.
Relates the stability of carbocations and carbon free radicals to the number of hyperconjugable α-hydrogens.
Double Bond Equivalent (Degree of Unsaturation)
C is number of carbon atoms, H is hydrogen atoms, X is halogen atoms, and N is nitrogen atoms in the molecular formula.
Formula to calculate the number of rings or pi bonds in an organic molecule.
SN1 Reaction Kinetic Rate Law
k is the rate constant, and [R-X] is the concentration of the alkyl halide substrate.
First-order rate equation governing unimolecular nucleophilic substitution (SN1) reactions.
SN2 Reaction Kinetic Rate Law
[R-X] is substrate concentration, and [Nu−] is nucleophile concentration.
Second-order rate equation governing bimolecular nucleophilic substitution (SN2) reactions.
Reimer-Tiemann Phenol Formylation
Reactants are phenol, chloroform, and sodium hydroxide; product is o-salicylaldehyde.
Formylation of phenol to salicylaldehyde using chloroform and sodium hydroxide.
Hydroboration-Oxidation Net Reaction
Reactants are alkene and borane, giving a primary alcohol and boric acid.
Anti-Markovnikov hydration of an alkene to form a primary alcohol via hydroboration and oxidation.
Aldol Self-Condensation Reaction
Reactant is acetaldehyde; product is crotonaldehyde (but-2-enal).
Self-condensation of acetaldehyde in dilute base followed by dehydration to form crotonaldehyde (\alpha,\beta-unsaturated aldehyde).
Cannizzaro Reaction of Benzaldehyde
Reactant is benzaldehyde; products are sodium benzoate and benzyl alcohol.
Redox disproportional of non-enolizable aldehydes (e.g. benzaldehyde) in concentrated base.
Hoffmann Bromamide Degradation
R-CONH2 is primary amide, and R-NH2 is primary amine product.
Degradation of an amide to a primary amine with one fewer carbon atom using bromine and base.
Carbylamine Primary Amine Test
R-NH2 is primary amine, and R-NC is alkyl/aryl isocyanide.
Diagnostic test for primary amines (both aliphatic and aromatic) forming an offensively smelling isocyanide.
Sucrose Acid Hydrolysis (Inversion of Cane Sugar)
Specific rotation changes from +66.5∘ (sucrose) to a net laevorotatory mixture of products.
Hydrolysis of sucrose to form D-glucose and D-fructose, resulting in inversion of optical rotation.
Isoelectric Point of Simple Amino Acids
pI is isoelectric point, pKa1 is logarithmic acid dissociation constant of carboxylic group, and pKa2 is that of the ammonium group.
Calculates the isoelectric point (pI) of an amino acid lacking an ionizable side chain.
Nylon-6,6 Preparation
Reactants are adipic acid and hexamethylenediamine; product is the Nylon-6,6 repeat unit.
Condensation copolymerization of adipic acid and hexamethylenediamine to form Nylon-6,6 polyamide.
Buna-S Synthetic Rubber Synthesis
Reactants are butadiene and styrene; product is butadiene-styrene copolymer.
Addition copolymerization of buta-1,3-diene and styrene in 3:1 ratio to form Buna-S elastomer.
Saponification of Tristearin (Soap Formation)
C17H35COONa is sodium stearate (soap) and C3H5(OH)3 is glycerol.
Alkali hydrolysis of tristearin fat using sodium hydroxide to yield sodium stearate soap and glycerol.
Aspirin Preparation (Acetylation of Salicylic Acid)
Reactants are salicylic acid and acetic anhydride; products are aspirin and acetic acid.
Acetylation of salicylic acid's phenolic hydroxyl group using acetic anhydride to form acetylsalicylic acid (Aspirin).
Acid Rain Sulfuric Acid Formation
Reactants are sulfur dioxide gas, oxygen, and atmospheric water vapor; product is sulfuric acid.
Atmospheric oxidation of sulfur dioxide in water droplets leading to sulfuric acid formation in acid rain.
Biochemical Oxygen Demand (BOD)
BOD is measured in parts per million (ppm) or mg/L.
Measure of water pollution denoting the amount of dissolved oxygen needed by aerobic biological organisms to break down organic material.
Prussian Blue Complex Formation (Lassaigne's Test)
Fe4[Fe(CN)6]3 is ferric ferrocyanide (Prussian blue precipitate).
Confirmatory test reaction for nitrogen in organic compounds, yielding a characteristic Prussian blue precipitate.
Halogen Percentage by Carius Method
X is halogen (Cl, Br, I), AgX is silver halide precipitate, and masses are measured in grams.
Quantitative formula for the estimation of halogens in organic compounds using the Carius method.
De Morgan's Laws of Sets
A′ and B′ represent the complements of sets A and B, while ∪ and ∩ represent union and intersection.
Fundamental set identities relating union, intersection, and complements.
Cardinality of Union of Sets
n(S) represents the number of elements in set S; ∪ and ∩ represent union and intersection respectively.
Formula to count elements in the union of two or three finite sets.
Relation Domain and Range Definitions
R is relation, A is domain set, B is co-domain set, and (a,b) represents ordered pairs in R.
Formally defines the domain and range sets of a binary relation R⊆A×B.
Number of Relations on a Set
n(S) represents the number of elements in set S, and n is the cardinality of set A.
Formulas to calculate total, reflexive, and symmetric relations on a finite set.
Fundamental Domain Conditions
g(x) is a real-valued sub-function, and ⟹ represents constraints for mathematical definedness.
Mathematical domain conditions for real-valued rational, radical, and logarithmic functions.
Symmetry and Periodicity of Functions
f(x) is a real-valued function, and T>0 is the smallest positive constant satisfying the relation.
Mathematical definitions of even/odd functions and periodic functions.
Injective (One-to-One) Functions Count
a=n(A) is the size of the domain, b=n(B) is the size of the co-domain, and P(b,a) represents permutations.
Formula to calculate the number of injective functions from set A to set B.
Composition and Inverse of Functions
g∘f is the composite function of f and g, and f−1 is the inverse function of f.
Mathematical definitions of function composition and the condition for the existence of an inverse.
Properties of Conjugate and Modulus
z1,z2 are complex numbers, z represents the complex conjugate of z, and ∣z∣ represents the modulus.
Key algebraic relationships for the conjugate and modulus of complex numbers.
Trigonometric and Euler Representations
r is the modulus, θ is the argument (amplitude), and i=−1 is the imaginary unit.
Polar representation of a complex number z=x+iy.
Square Root of a Complex Number
∣z∣=a2+b2, and sgn(b) is 1 if b≥0 and −1 if b<0.
Algebraic formula to find the square root of z=a+ib.
Triangle Inequalities for Complex Numbers
z1 and z2 are complex numbers, and ∣z∣ represents the modulus of z.
Bound limits for the modulus of the sum and difference of two complex numbers.
Roots Formula and Vieta's Relations
a,b,c are coefficients, D is the discriminant, and α,β are the roots.
Formula to solve ax2+bx+c=0 and the sum/product relationships of its roots α,β.
Formation of Quadratic Equation from Roots
x is the variable, S is the sum of the roots, P is the product of the roots, and α,β are the roots.
Standard equation constructed using the sum and product of its roots α,β.
Condition for One Common Root
a1,b1,c1 and a2,b2,c2 are the coefficients of the two quadratic equations respectively.
Algebraic condition for two quadratic equations to share exactly one root.
Symmetric, Skew-Symmetric and Orthogonal Matrices
AT is the transpose of square matrix A, I is the identity matrix, and −A is the negative matrix of A.
Definitions based on the transpose of a square matrix A.
Inverse of a Square Matrix
A−1 is the inverse of matrix A, ∣A∣ is the determinant of A, and adj(A) is the adjoint matrix of A.
Relation linking the inverse of a square matrix to its adjoint and determinant.
Evaluation of 2x2 and 3x3 Determinants
a,b,c,d are elements of a 2×2 matrix; ai,bi,ci are elements of a 3×3 matrix expanding along the first row.
Formulas to compute the determinant value for second and third-order square matrices.
Fundamental Properties of Determinants
AT is the transpose, k is a scalar constant, n is the dimension of the square matrices, and ∣A∣ is the determinant of A.
Standard properties relating matrix transpose, scalar multiplication, and matrix products to determinants.
Cramer's Rule for Linear Systems
Δ is the coefficient determinant, and Δx,Δy,Δz are determinants obtained by replacing columns with the constant terms.
Method to solve a system of three simultaneous linear equations using determinants.
Multiplication and Addition Principles
m is the number of ways task 1 can occur, and n is the number of ways task 2 can occur.
Core combinatorial rules for compound tasks occurring sequentially (AND) or mutually exclusively (OR).
Permutations and Combinations Formulas
n is the total number of items, and r is the number of items to arrange or select.
Formulas to calculate arrangements of r objects out of n (P(n,r)) and selections of r objects out of n (C(n,r)).
Geometric Combinatorics Formulas
n is the number of vertices/points, and p is the number of collinear points among them.
Formulas to find the number of diagonals, lines, and triangles formed by n coplanar points.
Binomial Theorem and General Term
n is a positive integer, Tr+1 represents the (r+1)-th term in the expansion, and C(n,r) is the binomial coefficient.
Mathematical expansion of (x+y)n and its (r+1)-th term.
Binomial Coefficient Sum Identities
Cr represents the binomial coefficient C(n,r) for a positive integral index n.
Formulas for sums of binomial coefficients derived from (1+x)n expansions.
Multinomial Theorem Expansion
n is the positive power index, k is the number of variables inside the brackets, and ri are non-negative integer powers.
General term and total terms in the expansion of a multinomial expression.
Sums of Arithmetic and Geometric Progressions
a is the first term, d is the common difference, r is the common ratio, and Sn represents the sum of n terms.
Formulas for sum of n terms in AP and GP, and the sum of an infinite GP (∣r∣<1).
Arithmetic, Geometric, and Harmonic Means Relation
a,b are positive real numbers.
Inequality relation for positive real numbers.
Harmonic and Arithmetico-Geometric Progressions
a is first term of AP/AGP, d is common difference, r is common ratio of GP part, and S∞ is sum to infinity.
General term of HP and the sum to infinity of an AGP series.
Standard Limit Evaluations
x is a real variable approaching 0, and e is Euler's number.
Fundamental trigonometric and exponential limit identities.
Mathematical Criterion for Continuity
f(x) is the function, c is the point of interest, and the left-hand and right-hand limits must match the value f(c).
Condition for a function f(x) to be continuous at a specific point x=c.
Derivative from First Principles
f′(x) is the derivative, and h is an infinitesimally small change in the input variable x.
Limit definition of the derivative of a real-valued function f(x).
Rolle's and Lagrange's Mean Value Theorems
f(x) is continuous on [a,b] and differentiable on (a,b), and c is a point inside the interval.
Core theorems of differential calculus for continuous and differentiable functions on [a,b].
Rules of Differentiation
u,v,y are differentiable functions of a real variable x, and u is a function of x.
Formulas for differentiating products, quotients, and compositions of functions.
Derivatives of Standard Functions
x is a real variable within the valid domain of each corresponding function.
Standard formulas for differentiation of basic trigonometric, logarithmic and exponential functions.
Related Rates of Change
y is a function of x, and both are differentiable functions of time t.
Formula linking the rate of change of dependent variables using chain rule.
Monotonicity and Tangent Slope
f(x) is a differentiable function, and mtangent is the slope of the tangent at x0.
Conditions using the first derivative to determine increasing/decreasing nature, and the slope of a tangent.
First and Second Derivative Tests
f(x) is a twice-differentiable function, and c is a critical point inside its domain.
Calculus tests to locate and classify local extrema of a function.
Equations of Tangent and Normal to a Curve
(x1,y1) is the point on the curve, and f′(x1) is the slope of the tangent.
Formulas to find the lines tangent and normal to y=f(x) at (x1,y1).
Standard Indefinite Integrals
n is a constant exponent, x is a real variable, and C is the integration constant.
Standard integration formulas for power and reciprocal functions.
Integration by Parts Formula
u,v,f(x),g(x) are differentiable functions of x.
Formula derived from product rule to integrate the product of two functions.
Integration by Partial Fractions
p,q,a,b are constants, and A,B are solved numerical coefficients.
Decomposition template of a rational function into simpler fractions for integration.
Fundamental Theorem of Calculus and Leibniz Rule
f is continuous, F is antiderivative, and ϕ(x),ψ(x) are differentiable boundary functions.
FTC evaluation formula and Leibniz rule for differentiation under the integral sign.
Properties of Definite Integrals (King's Property)
f(x) is an integrable function on the specified intervals, and a,b are real limits.
Fundamental symmetry properties of definite integrals, including the King's Property.
Area Under a Single Curve
A is the area of the region, and x=a to x=b are boundary lines.
Integral formula calculating the area of a region bounded by y=f(x) and the x-axis.
Area Bounded Between Two Curves
A is the bounded area, and f(x),g(x) are curves intersecting or bounded at x=a and x=b.
Integral formula calculating the area bounded between curves y=f(x) and y=g(x) from x=a to x=b.
Formation of Ordinary Differential Equations
x is independent variable, y is dependent variable, and y(n) is the n-th derivative of y.
General algebraic ODE expression representing a family of curves.
Separable and Homogeneous Differential Equations
v is a temporary helper variable, x,y are coordinate variables, and f,g are functions.
Formulas for variables-separable integration and substitution for homogeneous differential equations.
First-Order Linear Differential Equation
P and Q are functions of x only, I.F. is the integrating factor, and C is the integration constant.
Standard form and solution using an Integrating Factor (I.F.).
Standard Forms of Straight Line Equations
m is the slope, c is the y-intercept, a,b are axes intercepts, and (xi,yi) are points on the line.
Standard Cartesian formulations for straight lines in coordinate geometry.
Distance of a Point from a Line
d is the perpendicular distance, (x1,y1) is the coordinates of the point, and a,b,c are the coefficients of the line.
Perpendicular distance from a point (x1,y1) to the line ax+by+c=0.
Angle Between Lines and Parallel Distance
m1,m2 are slopes of the intersecting lines, dparallel is distance between lines ax+by+c1=0 and ax+by+c2=0.
Calculates the acute angle between two intersecting lines, and the perpendicular distance between two parallel lines.
Foot of Perpendicular and Mirror Image of a Point
(x1,y1) is the given point, (xp,yp) is the foot of the perpendicular, and (xi,yi) is the mirror image point.
Formulas to find coordinate positions of foot of perpendicular and mirror image of point (x1,y1) with respect to line ax+by+c=0.
Family of Lines and Angle Bisectors
L1,L2 are lines, λ is a parameter, and ai,bi,ci are line coefficients.
Equations for family of intersecting lines and the bisectors of angles between two lines.
Standard and General Equations of a Circle
(h,k) is the center, r is the radius, and g,f,c are general equation parameters.
Mathematical equations defining a circle in coordinate space.
Equation of Tangent to a Circle (Slope Form)
m is the slope of the tangent, and a is the radius of the circle.
Slope form tangent equation for x2+y2=a2.
Chord of Contact and Director Circle
(x1,y1) is an external point, and a is the radius of the circle.
Equations of chord of contact and director circle for a standard circle.
Standard Equations of Parabola, Ellipse and Hyperbola
a,b are semi-axes lengths, and e represents the eccentricity of the ellipse or hyperbola.
Standard mathematical equations for the three main conic sections.
Equations of Tangents in Slope Form
m is the slope of the tangent line, and a,b are conic parameters.
Tangent equations in terms of slope m for standard parabola, ellipse, and hyperbola.
Direction Cosines and Ratios
l,m,n are direction cosines; a,b,c are direction ratios.
Mathematical relations linking direction cosines and direction ratios of a vector in 3D.
Line Equations in 3D Space
a (or (x1,y1,z1)) is a point on the line, b (or bx,by,bz) is the direction vector, and λ is a scalar parameter.
Vector and symmetric Cartesian equations of a line in 3D space.
Shortest Distance Between Skew Lines
dshortest is the shortest distance; the lines are defined by r=a1+λb1 and r=a2+μb2.
Formula for the shortest distance between two non-parallel, non-intersecting lines in 3D space.
Plane Equation and Angle Between Planes
a,b,c are normal coefficients, and θ is the angle between planes with coefficients ai,bi,ci.
General Cartesian equation of a plane and angle between two planes in 3D.
Distance of a Point from a Plane
d is the perpendicular distance, and a,b,c,d are coefficients of the plane equation.
Perpendicular distance from a point (x1,y1,z1) to the plane ax+by+cz+d=0.
Angle Between Line and Plane
b is the direction vector of the line, n is the normal vector of the plane, and θ is the angle between them.
Formula to calculate the angle between a line and a plane in 3D space.
Resultant of Vector Addition
R is the magnitude of resultant, a,b are magnitudes, θ is the angle between vectors, and α is the angle of resultant with vector a.
Formulas to calculate the magnitude and direction of the sum of two vectors.
Dot and Cross Products of Vectors
a,b are magnitudes of vectors a,b, θ is the angle between them, and n^ is the unit normal vector perpendicular to both.
Mathematical definitions of scalar (dot) and vector (cross) products.
Scalar and Vector Triple Products
a,b,c are vectors, and [a b c] represents the scalar triple product.
Triple products of vectors and their algebraic properties.
Empirical Relation of Central Tendency
Mean, Median, and Mode represent standard measures of central tendency in statistics.
Empirical formula connecting mean, median, and mode for moderately asymmetrical distributions.
Mean, Variance and Standard Deviation
xˉ is the arithmetic mean, σ2 is variance, σ is standard deviation, and n is total count of observations.
Mathematical calculations for statistical dispersion of data.
Probability Addition and Multiplication Rules
P(S) represents the probability of event S, ∪ is union, and ∩ is intersection.
Formulas for union probability and multiplication rule for independent events.
Conditional Probability and Bayes' Theorem
P(A∣B) is the probability of event A given B has occurred, and Ai represent mutually exclusive partition events.
Formulas for conditional probability and the probability of causes (Bayes' Theorem).
Expectation and Variance of Random Variables
xi represent values of random variable X, pi is probability of xi, μ is mean, and σ2 is variance.
Formulas to compute mean (expectation) and variance of a discrete probability distribution.
Binomial Probability Distribution
p is probability of success, q=1−p is probability of failure, n is total trials, and r is target successes.
Probability of obtaining exactly r successes in n independent Bernoulli trials.
Compound and Double Angle Formulas
A,B,θ are angles in radians.
Core trigonometric expansion formulas for compound and double angles.
Trigonometric Sum to Product (C-D) Formulas
C,D are angles in radians.
Identities converting sums and differences of sine and cosine functions into products.
Trigonometric Product to Sum Formulas
A,B are angles in radians.
Identities converting products of sine and cosine functions into sums or differences.
Triple Angle Formulas
θ is the angle in radians.
Multiple angle identities expanding functions of 3θ.
Trigonometric Products and Series Sums
θ,α are starting angles, β is the common difference angle of the series, and n is the number of terms.
Products of cosine series and sums of sine/cosine series in arithmetic progressions.
General Solutions of Trigonometric Equations
θ is the unknown angle, α is the principal value, and n is any integer.
General values for variables solving standard trigonometric equations.
Principal Value Branches of ITF
x is the input variable belonging to the appropriate domain of each inverse function.
Standard ranges (principal value branches) of inverse trigonometric functions.
Inverse Trigonometric Sum Identities
x represents the input variable within its valid domain.
Standard value identities of inverse trigonometric relations.
Inverse Trig Sum, Difference and Double Angle Conversions
x,y are input variables within their appropriate domains.
Formulas for sum and difference of inverse tangents and conversions of 2tan−1x.
Sine Rule and Cosine Rule of Triangles
a,b,c are lengths of sides opposite to angles A,B,C respectively, and R is the circumradius of the triangle.
Relationships between the sides and angles of a triangle.
Projection Rule and Napier's Analogy
a,b,c are side lengths opposite to angles A,B,C respectively.
Projection of side lengths in triangles and tangent-based angle/side relationships.
Circumradius and Inradius of Triangles
a,b,c are side lengths, Δ is area of the triangle, s is the semi-perimeter, and A is the opposite angle.
Formulas connecting circumradius (R) and inradius (r) to the sides and area of a triangle.
Half-Angle and Area Formulas
a,b,c are side lengths, s=2a+b+c is the semi-perimeter, Δ is the area of the triangle, r is the inradius, and R is the circumradius.
Sine, cosine, and tangent half-angle formulas in terms of semi-perimeter (s), and Heron's area formula.
Ex-Radii of Triangles
r1,r2,r3 are ex-radii opposite to vertices A,B,C respectively, s is the semi-perimeter, Δ is the area, and R is the circumradius.
Formulas for circumradius and inradius values of escribed (outer) circles tangent to side extensions.