PhysicsKinematics

Average speed

v̄ = total distance / total time

PhysicsKinematics

Average speed · form 2

distance = v̄ × time

PhysicsKinematics

Average speed · form 3

time = distance / v̄

PhysicsKinematics

Average velocity

v̄ = displacement / time

PhysicsKinematics

Average velocity · form 2

displacement = v̄ × time

PhysicsKinematics

Average velocity · form 3

time = displacement / v̄

PhysicsKinematics

Acceleration

a = (v − u)/t

PhysicsKinematics

Acceleration · form 2

v = u + at

PhysicsKinematics

Acceleration · form 3

t = (v − u)/a

PhysicsKinematics

Uniform acceleration displacement

s = ut + ½at²

PhysicsKinematics

Uniform acceleration displacement · form 2

s = (u + v)t/2

PhysicsKinematics

Uniform acceleration displacement · form 3

v² = u² + 2as

PhysicsKinematics

Projectile horizontal range

R = u² sin(2θ)/g

PhysicsKinematics

Projectile horizontal range · form 2

Rmax = u²/g

PhysicsKinematics

Projectile horizontal range · form 3

u = √(Rg/sin(2θ))

PhysicsKinematics

Projectile maximum height

H = u² sin²θ/(2g)

PhysicsKinematics

Projectile maximum height · form 2

uᵧ = u sinθ

PhysicsKinematics

Projectile maximum height · form 3

H = uᵧ²/(2g)

PhysicsKinematics

Projectile flight time

T = 2u sinθ/g

PhysicsKinematics

Projectile flight time · form 2

uᵧ = gT/2

PhysicsKinematics

Projectile flight time · form 3

g = 2u sinθ/T

PhysicsDynamics

Newton's second law

Fnet = ma

PhysicsDynamics

Newton's second law · form 2

a = Fnet/m

PhysicsDynamics

Newton's second law · form 3

m = Fnet/a

PhysicsDynamics

Linear momentum

p = mv

PhysicsDynamics

Linear momentum · form 2

v = p/m

PhysicsDynamics

Linear momentum · form 3

m = p/v

PhysicsDynamics

Impulse

J = FΔt

PhysicsDynamics

Impulse · form 2

J = Δp

PhysicsDynamics

Impulse · form 3

Favg = Δp/Δt

PhysicsDynamics

Limiting friction

fmax = μsN

PhysicsDynamics

Limiting friction · form 2

μs = fmax/N

PhysicsDynamics

Limiting friction · form 3

N = fmax/μs

PhysicsDynamics

Centripetal acceleration

ac = v²/r

PhysicsDynamics

Centripetal acceleration · form 2

ac = ω²r

PhysicsDynamics

Centripetal acceleration · form 3

v = ωr

PhysicsDynamics

Centripetal force

Fc = mv²/r

PhysicsDynamics

Centripetal force · form 2

Fc = mω²r

PhysicsDynamics

Centripetal force · form 3

r = mv²/Fc

PhysicsEnergy

Work

W = Fs cosθ

PhysicsEnergy

Work · form 2

F = W/(s cosθ)

PhysicsEnergy

Work · form 3

s = W/(F cosθ)

PhysicsEnergy

Kinetic energy

K = ½mv²

PhysicsEnergy

Kinetic energy · form 2

v = √(2K/m)

PhysicsEnergy

Kinetic energy · form 3

m = 2K/v²

PhysicsEnergy

Gravitational potential energy

U = mgh

PhysicsEnergy

Gravitational potential energy · form 2

h = U/(mg)

PhysicsEnergy

Gravitational potential energy · form 3

m = U/(gh)

PhysicsEnergy

Power

P = W/t

PhysicsEnergy

Power · form 2

W = Pt

PhysicsEnergy

Power · form 3

P = Fv cosθ

PhysicsRotation

Torque

τ = rF sinθ

PhysicsRotation

Torque · form 2

F = τ/(r sinθ)

PhysicsRotation

Torque · form 3

τ = Iα

PhysicsRotation

Angular momentum

L = Iω

PhysicsRotation

Angular momentum · form 2

ω = L/I

PhysicsRotation

Angular momentum · form 3

I = L/ω

PhysicsRotation

Rotational kinetic energy

Krot = ½Iω²

PhysicsRotation

Rotational kinetic energy · form 2

I = 2Krot/ω²

PhysicsRotation

Rotational kinetic energy · form 3

ω = √(2Krot/I)

PhysicsRotation

Rolling without slipping

v = ωR

PhysicsRotation

Rolling without slipping · form 2

ω = v/R

PhysicsRotation

Rolling without slipping · form 3

R = v/ω

PhysicsGravitation

Newton's gravitation

F = Gm₁m₂/r²

PhysicsGravitation

Newton's gravitation · form 2

G = Fr²/(m₁m₂)

PhysicsGravitation

Newton's gravitation · form 3

r = √(Gm₁m₂/F)

PhysicsGravitation

Gravitational acceleration

g = GM/r²

PhysicsGravitation

Gravitational acceleration · form 2

M = gr²/G

PhysicsGravitation

Gravitational acceleration · form 3

r = √(GM/g)

PhysicsGravitation

Escape speed

vesc = √(2GM/R)

PhysicsGravitation

Escape speed · form 2

vesc = √(2gR)

PhysicsGravitation

Escape speed · form 3

g = vesc²/(2R)

PhysicsGravitation

Orbital speed

vorbit = √(GM/r)

PhysicsGravitation

Orbital speed · form 2

vorbit² = GM/r

PhysicsGravitation

Orbital speed · form 3

r = GM/vorbit²

PhysicsFluids

Hydrostatic pressure

P = ρgh

PhysicsFluids

Hydrostatic pressure · form 2

h = P/(ρg)

PhysicsFluids

Hydrostatic pressure · form 3

ρ = P/(gh)

PhysicsFluids

Buoyancy

Fb = ρfluid gVdisplaced

PhysicsFluids

Buoyancy · form 2

Vdisplaced = Fb/(ρfluid g)

PhysicsFluids

Buoyancy · form 3

ρfluid = Fb/(gVdisplaced)

PhysicsFluids

Continuity equation

A₁v₁ = A₂v₂

PhysicsFluids

Continuity equation · form 2

v₂ = A₁v₁/A₂

PhysicsFluids

Continuity equation · form 3

A₂ = A₁v₁/v₂

PhysicsThermal physics

Heat capacity

Q = mcΔT

PhysicsThermal physics

Heat capacity · form 2

c = Q/(mΔT)

PhysicsThermal physics

Heat capacity · form 3

ΔT = Q/(mc)

PhysicsThermal physics

Latent heat

Q = mL

PhysicsThermal physics

Latent heat · form 2

L = Q/m

PhysicsThermal physics

Latent heat · form 3

m = Q/L

PhysicsThermal physics

Ideal gas law

PV = nRT

PhysicsThermal physics

Ideal gas law · form 2

P = nRT/V

PhysicsThermal physics

Ideal gas law · form 3

T = PV/(nR)

PhysicsThermal physics

First law

ΔU = Q − Wby

PhysicsThermal physics

First law · form 2

Q = ΔU + Wby

PhysicsThermal physics

First law · form 3

Wby = Q − ΔU

PhysicsThermal physics

Carnot efficiency

η = 1 − Tc/Th

PhysicsThermal physics

Carnot efficiency · form 2

Tc = Th(1 − η)

PhysicsThermal physics

Carnot efficiency · form 3

Th = Tc/(1 − η)

PhysicsOscillations

Spring period

T = 2π√(m/k)

PhysicsOscillations

Spring period · form 2

f = (1/2π)√(k/m)

PhysicsOscillations

Spring period · form 3

k = 4π²m/T²

PhysicsOscillations

Pendulum period

T = 2π√(L/g)

PhysicsOscillations

Pendulum period · form 2

L = gT²/(4π²)

PhysicsOscillations

Pendulum period · form 3

g = 4π²L/T²

PhysicsWaves

Wave speed

v = fλ

PhysicsWaves

Wave speed · form 2

λ = v/f

PhysicsWaves

Wave speed · form 3

f = v/λ

PhysicsWaves

String wave speed

v = √(T/μ)

PhysicsWaves

String wave speed · form 2

T = μv²

PhysicsWaves

String wave speed · form 3

μ = T/v²

PhysicsElectricity

Coulomb's law

F = k|q₁q₂|/r²

PhysicsElectricity

Coulomb's law · form 2

k = Fr²/|q₁q₂|

PhysicsElectricity

Coulomb's law · form 3

r = √(k|q₁q₂|/F)

PhysicsElectricity

Electric field

E = F/q

PhysicsElectricity

Electric field · form 2

F = qE

PhysicsElectricity

Electric field · form 3

q = F/E

PhysicsElectricity

Electric potential

V = U/q

PhysicsElectricity

Electric potential · form 2

U = qV

PhysicsElectricity

Electric potential · form 3

q = U/V

PhysicsElectricity

Capacitance

C = Q/V

PhysicsElectricity

Capacitance · form 2

Q = CV

PhysicsElectricity

Capacitance · form 3

V = Q/C

PhysicsElectricity

Capacitor energy

U = ½CV²

PhysicsElectricity

Capacitor energy · form 2

U = Q²/(2C)

PhysicsElectricity

Capacitor energy · form 3

U = ½QV

PhysicsElectricity

Ohm's law

V = IR

PhysicsElectricity

Ohm's law · form 2

I = V/R

PhysicsElectricity

Ohm's law · form 3

R = V/I

PhysicsElectricity

Resistance of wire

R = ρL/A

PhysicsElectricity

Resistance of wire · form 2

ρ = RA/L

PhysicsElectricity

Resistance of wire · form 3

A = ρL/R

PhysicsElectricity

Electrical power

P = VI

PhysicsElectricity

Electrical power · form 2

P = I²R

PhysicsElectricity

Electrical power · form 3

P = V²/R

PhysicsMagnetism

Magnetic force on charge

F = qvB sinθ

PhysicsMagnetism

Magnetic force on charge · form 2

B = F/(qv sinθ)

PhysicsMagnetism

Magnetic force on charge · form 3

v = F/(qB sinθ)

PhysicsMagnetism

Magnetic force on wire

F = BIL sinθ

PhysicsMagnetism

Magnetic force on wire · form 2

B = F/(IL sinθ)

PhysicsMagnetism

Magnetic force on wire · form 3

I = F/(BL sinθ)

PhysicsMagnetism

Flux

Φ = BA cosθ

PhysicsMagnetism

Flux · form 2

B = Φ/(A cosθ)

PhysicsMagnetism

Flux · form 3

A = Φ/(B cosθ)

PhysicsMagnetism

Faraday induction

ε = −NΔΦ/Δt

PhysicsMagnetism

Faraday induction · form 2

|ε| = N|ΔΦ|/Δt

PhysicsMagnetism

Faraday induction · form 3

|ΔΦ| = |ε|Δt/N

PhysicsAC circuits

RMS values

Vrms = Vpeak/√2

PhysicsAC circuits

RMS values · form 2

Vpeak = √2 Vrms

PhysicsAC circuits

RMS values · form 3

Irms = Ipeak/√2

PhysicsOptics

Refractive index

n = c/v

PhysicsOptics

Refractive index · form 2

v = c/n

PhysicsOptics

Refractive index · form 3

c = nv

PhysicsOptics

Snell's law

n₁ sin i = n₂ sin r

PhysicsOptics

Snell's law · form 2

sin r = n₁ sin i/n₂

PhysicsOptics

Snell's law · form 3

n₂ = n₁ sin i/sin r

PhysicsOptics

Thin lens equation

1/f = 1/v − 1/u

PhysicsOptics

Thin lens equation · form 2

1/v = 1/f + 1/u

PhysicsOptics

Thin lens equation · form 3

1/u = 1/v − 1/f

PhysicsOptics

Lens power

P = 1/f (f in metres)

PhysicsOptics

Lens power · form 2

f = 1/P (metres)

PhysicsOptics

Lens power · form 3

Ptotal = P₁ + P₂

PhysicsOptics

Interference fringe width

β = λD/d

PhysicsOptics

Interference fringe width · form 2

λ = βd/D

PhysicsOptics

Interference fringe width · form 3

d = λD/β

PhysicsModern physics

Photon energy

E = hν

PhysicsModern physics

Photon energy · form 2

E = hc/λ

PhysicsModern physics

Photon energy · form 3

ν = c/λ

PhysicsModern physics

Photoelectric equation

Kmax = hν − φ

PhysicsModern physics

Photoelectric equation · form 2

φ = hν − Kmax

PhysicsModern physics

Photoelectric equation · form 3

eVstop = Kmax

PhysicsModern physics

de Broglie wavelength

λ = h/p

PhysicsModern physics

de Broglie wavelength · form 2

p = h/λ

PhysicsModern physics

de Broglie wavelength · form 3

λ = h/(mv)

PhysicsModern physics

Radioactive decay

N = N₀e^(−λt)

PhysicsModern physics

Radioactive decay · form 2

A = λN

PhysicsModern physics

Radioactive decay · form 3

t½ = ln 2/λ

PhysicsModern physics

Mass-energy equivalence

E = mc²

PhysicsModern physics

Mass-energy equivalence · form 2

m = E/c²

PhysicsModern physics

Mass-energy equivalence · form 3

ΔE = Δmc²

ChemistryMole concept

Moles from mass

n = m/M

ChemistryMole concept

Moles from mass · form 2

m = nM

ChemistryMole concept

Moles from mass · form 3

M = m/n

ChemistryMole concept

Particle count

N = nNA

ChemistryMole concept

Particle count · form 2

n = N/NA

ChemistryMole concept

Particle count · form 3

NA = N/n

ChemistryMole concept

Molarity

C = n/V(L)

ChemistryMole concept

Molarity · form 2

n = CV

ChemistryMole concept

Molarity · form 3

V(L) = n/C

ChemistryMole concept

Molality

b = nsolute/kg solvent

ChemistryMole concept

Molality · form 2

nsolute = b × kg solvent

ChemistryMole concept

Molality · form 3

kg solvent = nsolute/b

ChemistryMole concept

Mole fraction

xA = nA/(nA+nB)

ChemistryMole concept

Mole fraction · form 2

xA + xB = 1

ChemistryMole concept

Mole fraction · form 3

nA/nB = xA/xB

ChemistryAtomic structure

Photon frequency

ν = c/λ

ChemistryAtomic structure

Photon frequency · form 2

λ = c/ν

ChemistryAtomic structure

Photon frequency · form 3

E = hν

ChemistryAtomic structure

Bohr energy

En = −13.6Z²/n² eV

ChemistryAtomic structure

Bohr energy · form 2

ΔE = Ef − Ei

ChemistryAtomic structure

Bohr energy · form 3

|ΔE| = hν

ChemistryAtomic structure

Bohr radius

rn = a₀n²/Z

ChemistryAtomic structure

Bohr radius · form 2

r₁ = a₀/Z

ChemistryAtomic structure

Bohr radius · form 3

rn/r₁ = n²

ChemistryAtomic structure

Orbital capacity

Maximum electrons in shell n = 2n²

ChemistryAtomic structure

Orbital capacity · form 2

Orbitals in shell n = n²

ChemistryAtomic structure

Orbital capacity · form 3

Maximum electrons in subshell l = 2(2l+1)

ChemistryAtomic structure

Matter wavelength

λ = h/p

ChemistryAtomic structure

Matter wavelength · form 2

λ = h/(mv)

ChemistryAtomic structure

Matter wavelength · form 3

p = h/λ

ChemistryBonding

Formal charge

FC = valence − lone electrons − bonds

ChemistryBonding

Formal charge · form 2

sum of formal charges = net charge

ChemistryBonding

Formal charge · form 3

bond order = (bonding e⁻ − antibonding e⁻)/2

ChemistryBonding

Dipole moment

μ = qd

ChemistryBonding

Dipole moment · form 2

q = μ/d

ChemistryBonding

Dipole moment · form 3

d = μ/q

ChemistryGases

Ideal gas

PV = nRT

ChemistryGases

Ideal gas · form 2

P = nRT/V

ChemistryGases

Ideal gas · form 3

V = nRT/P

ChemistryGases

Combined gas law

P₁V₁/T₁ = P₂V₂/T₂

ChemistryGases

Combined gas law · form 2

P₂ = P₁V₁T₂/(T₁V₂)

ChemistryGases

Combined gas law · form 3

V₂ = P₁V₁T₂/(T₁P₂)

ChemistryGases

Dalton partial pressures

Ptotal = ΣPi

ChemistryGases

Dalton partial pressures · form 2

Pi = xiPtotal

ChemistryGases

Dalton partial pressures · form 3

xi = Pi/Ptotal

ChemistryGases

Gas density

ρ = PM/(RT)

ChemistryGases

Gas density · form 2

M = ρRT/P

ChemistryGases

Gas density · form 3

P = ρRT/M

ChemistryThermodynamics

Enthalpy relation

ΔH = ΔU + ΔngRT

ChemistryThermodynamics

Enthalpy relation · form 2

ΔU = ΔH − ΔngRT

ChemistryThermodynamics

Enthalpy relation · form 3

Δng = (ΔH − ΔU)/(RT)

ChemistryThermodynamics

Gibbs free energy

ΔG = ΔH − TΔS

ChemistryThermodynamics

Gibbs free energy · form 2

ΔH = ΔG + TΔS

ChemistryThermodynamics

Gibbs free energy · form 3

ΔS = (ΔH − ΔG)/T

ChemistryThermodynamics

Gibbs and equilibrium

ΔG° = −RT ln K

ChemistryThermodynamics

Gibbs and equilibrium · form 2

ln K = −ΔG°/(RT)

ChemistryThermodynamics

Gibbs and equilibrium · form 3

K = e^(−ΔG°/RT)

ChemistryThermodynamics

Calorimetry

q = mcΔT

ChemistryThermodynamics

Calorimetry · form 2

c = q/(mΔT)

ChemistryThermodynamics

Calorimetry · form 3

ΔT = q/(mc)

ChemistrySolutions

Raoult's law

PA = xA PA°

ChemistrySolutions

Raoult's law · form 2

xA = PA/PA°

ChemistrySolutions

Raoult's law · form 3

Ptotal = ΣxiPi°

ChemistrySolutions

Boiling point elevation

ΔTb = iKb b

ChemistrySolutions

Boiling point elevation · form 2

b = ΔTb/(iKb)

ChemistrySolutions

Boiling point elevation · form 3

i = ΔTb/(Kb b)

ChemistrySolutions

Freezing point depression

ΔTf = iKf b

ChemistrySolutions

Freezing point depression · form 2

b = ΔTf/(iKf)

ChemistrySolutions

Freezing point depression · form 3

i = ΔTf/(Kf b)

ChemistrySolutions

Osmotic pressure

π = iCRT

ChemistrySolutions

Osmotic pressure · form 2

C = π/(iRT)

ChemistrySolutions

Osmotic pressure · form 3

i = π/(CRT)

ChemistryEquilibrium

Reaction quotient

Q = product activities/reactant activities

ChemistryEquilibrium

Reaction quotient · form 2

Q < K: forward favored

ChemistryEquilibrium

Reaction quotient · form 3

Q > K: reverse favored

ChemistryEquilibrium

pH and pOH

pH = −log[H⁺]

ChemistryEquilibrium

pH and pOH · form 2

pOH = −log[OH⁻]

ChemistryEquilibrium

pH and pOH · form 3

pH + pOH = 14 (at 25°C)

ChemistryEquilibrium

Acid dissociation

Ka = [H⁺][A⁻]/[HA]

ChemistryEquilibrium

Acid dissociation · form 2

pKa = −log Ka

ChemistryEquilibrium

Acid dissociation · form 3

KaKb = Kw (conjugate pair)

ChemistryEquilibrium

Buffer equation

pH = pKa + log([A⁻]/[HA])

ChemistryEquilibrium

Buffer equation · form 2

[A⁻]/[HA] = 10^(pH−pKa)

ChemistryEquilibrium

Buffer equation · form 3

at [A⁻] = [HA], pH = pKa

ChemistryElectrochemistry

Standard cell potential

E°cell = E°cathode − E°anode

ChemistryElectrochemistry

Standard cell potential · form 2

ΔG° = −nFE°cell

ChemistryElectrochemistry

Standard cell potential · form 3

E°cell = (RT/nF) ln K

ChemistryElectrochemistry

Nernst equation

E = E° − (RT/nF)ln Q

ChemistryElectrochemistry

Nernst equation · form 2

E = E° − (0.0591/n)log Q at 298 K

ChemistryElectrochemistry

Nernst equation · form 3

at equilibrium E = 0

ChemistryElectrochemistry

Faraday electrolysis

Q = It

ChemistryElectrochemistry

Faraday electrolysis · form 2

m = MIt/(nF)

ChemistryElectrochemistry

Faraday electrolysis · form 3

m/M = Q/(nF)

ChemistryKinetics

First order rate

ln([A]₀/[A]) = kt

ChemistryKinetics

First order rate · form 2

[A] = [A]₀e^(−kt)

ChemistryKinetics

First order rate · form 3

t½ = 0.693/k

ChemistryKinetics

Zero order rate

[A] = [A]₀ − kt

ChemistryKinetics

Zero order rate · form 2

t½ = [A]₀/(2k)

ChemistryKinetics

Zero order rate · form 3

k = ([A]₀ − [A])/t

ChemistryKinetics

Arrhenius law

k = Ae^(−Ea/RT)

ChemistryKinetics

Arrhenius law · form 2

ln k = ln A − Ea/(RT)

ChemistryKinetics

Arrhenius law · form 3

ln(k₂/k₁) = (Ea/R)(1/T₁ − 1/T₂)

ChemistryKinetics

Rate law

rate = k[A]^m[B]^n

ChemistryKinetics

Rate law · form 2

overall order = m+n

ChemistryKinetics

Rate law · form 3

k = rate/([A]^m[B]^n)

ChemistryOrganic

Degree of unsaturation

DBE = (2C + 2 + N − H − X)/2

ChemistryOrganic

Degree of unsaturation · form 2

one ring contributes 1 DBE

ChemistryOrganic

Degree of unsaturation · form 3

one triple bond contributes 2 DBE

ChemistryOrganic

Chromatography ratio

Rf = distance of spot/distance of solvent

ChemistryOrganic

Chromatography ratio · form 2

distance of spot = Rf × solvent distance

ChemistryOrganic

Chromatography ratio · form 3

0 ≤ Rf ≤ 1 (typical planar chromatography)

ChemistryOrganic

Percentage yield

% yield = actual/theoretical × 100

ChemistryOrganic

Percentage yield · form 2

actual = theoretical × yield fraction

ChemistryOrganic

Percentage yield · form 3

theoretical = actual/yield fraction

ChemistryOrganic

Enantiomeric excess

ee = |%R − %S|

ChemistryOrganic

Enantiomeric excess · form 2

% major = (100 + ee)/2

ChemistryOrganic

Enantiomeric excess · form 3

% minor = (100 − ee)/2

ChemistryInorganic

Oxidation number sum

Σ oxidation states = ionic charge

ChemistryInorganic

Oxidation number sum · form 2

elemental substance has oxidation state 0

ChemistryInorganic

Oxidation number sum · form 3

monatomic ion oxidation state = ion charge

ChemistryInorganic

Coordination number

CN = number of donor atoms bonded to metal

ChemistryInorganic

Coordination number · form 2

metal oxidation state + ligand charges = complex charge

ChemistryInorganic

Coordination number · form 3

for octahedral complexes CN = 6

MathsAlgebra

Quadratic roots

x = (−b ± √(b²−4ac))/(2a)

MathsAlgebra

Quadratic roots · form 2

sum of roots = −b/a

MathsAlgebra

Quadratic roots · form 3

product of roots = c/a

MathsAlgebra

Discriminant

D = b² − 4ac

MathsAlgebra

Discriminant · form 2

D > 0: two distinct real roots

MathsAlgebra

Discriminant · form 3

D = 0: one repeated real root

MathsAlgebra

Difference of squares

a² − b² = (a−b)(a+b)

MathsAlgebra

Difference of squares · form 2

a² = (a+b)(a−b) + b²

MathsAlgebra

Difference of squares · form 3

b² = a² − (a−b)(a+b)

MathsAlgebra

Square of sum

(a+b)² = a² + 2ab + b²

MathsAlgebra

Square of sum · form 2

2ab = (a+b)² − a² − b²

MathsAlgebra

Square of sum · form 3

(a−b)² = a² − 2ab + b²

MathsAlgebra

Cube of sum

(a+b)³ = a³+3a²b+3ab²+b³

MathsAlgebra

Cube of sum · form 2

a³+b³ = (a+b)(a²−ab+b²)

MathsAlgebra

Cube of sum · form 3

a³−b³ = (a−b)(a²+ab+b²)

MathsAlgebra

Complex modulus

z = a+bi

MathsAlgebra

Complex modulus · form 2

|z| = √(a²+b²)

MathsAlgebra

Complex modulus · form 3

z z̄ = |z|²

MathsAlgebra

Complex polar form

z = r(cosθ+i sinθ)

MathsAlgebra

Complex polar form · form 2

r = |z|

MathsAlgebra

Complex polar form · form 3

arg(z₁z₂) = arg z₁ + arg z₂ (mod 2π)

MathsAlgebra

Arithmetic progression term

an = a + (n−1)d

MathsAlgebra

Arithmetic progression term · form 2

d = (an−a)/(n−1)

MathsAlgebra

Arithmetic progression term · form 3

a = an−(n−1)d

MathsAlgebra

Arithmetic progression sum

Sn = n(a+an)/2

MathsAlgebra

Arithmetic progression sum · form 2

Sn = n(2a+(n−1)d)/2

MathsAlgebra

Arithmetic progression sum · form 3

an = 2Sn/n − a

MathsAlgebra

Geometric progression term

an = ar^(n−1)

MathsAlgebra

Geometric progression term · form 2

r = an+1/an

MathsAlgebra

Geometric progression term · form 3

a = an/r^(n−1)

MathsAlgebra

Geometric progression sum

Sn = a(1−r^n)/(1−r)

MathsAlgebra

Geometric progression sum · form 2

S∞ = a/(1−r) for |r|<1

MathsAlgebra

Geometric progression sum · form 3

a = S∞(1−r)

MathsAlgebra

Binomial coefficient

nCr = n!/[r!(n−r)!]

MathsAlgebra

Binomial coefficient · form 2

nCr = nC(n−r)

MathsAlgebra

Binomial coefficient · form 3

nPr = nCr · r!

MathsAlgebra

Binomial theorem

(x+y)^n = Σ(nCr)x^(n−r)y^r

MathsAlgebra

Binomial theorem · form 2

sum of nCr from r=0 to n = 2^n

MathsAlgebra

Binomial theorem · form 3

alternating sum of nCr = 0 (n>0)

MathsAlgebra

Determinant 2×2

det[[a,b],[c,d]] = ad−bc

MathsAlgebra

Determinant 2×2 · form 2

inverse exists iff ad−bc ≠ 0

MathsAlgebra

Determinant 2×2 · form 3

det(AB) = det(A)det(B)

MathsFunctions

Function composition

(f∘g)(x) = f(g(x))

MathsFunctions

Function composition · form 2

(f∘g)⁻¹ = g⁻¹∘f⁻¹

MathsFunctions

Function composition · form 3

f(f⁻¹(x)) = x (on valid domain)

MathsTrigonometry

Pythagorean identity

sin²x + cos²x = 1

MathsTrigonometry

Pythagorean identity · form 2

1 + tan²x = sec²x

MathsTrigonometry

Pythagorean identity · form 3

1 + cot²x = csc²x

MathsTrigonometry

Angle addition sine

sin(A+B) = sin A cos B + cos A sin B

MathsTrigonometry

Angle addition sine · form 2

sin(A−B) = sin A cos B − cos A sin B

MathsTrigonometry

Angle addition sine · form 3

sin 2A = 2 sin A cos A

MathsTrigonometry

Angle addition cosine

cos(A+B) = cos A cos B − sin A sin B

MathsTrigonometry

Angle addition cosine · form 2

cos(A−B) = cos A cos B + sin A sin B

MathsTrigonometry

Angle addition cosine · form 3

cos 2A = cos²A − sin²A

MathsTrigonometry

Double-angle tangent

tan 2A = 2tan A/(1−tan²A)

MathsTrigonometry

Double-angle tangent · form 2

tan(A+B) = (tan A+tan B)/(1−tan A tan B)

MathsTrigonometry

Double-angle tangent · form 3

tan(A−B) = (tan A−tan B)/(1+tan A tan B)

MathsTrigonometry

Half angle

sin²(A/2) = (1−cos A)/2

MathsTrigonometry

Half angle · form 2

cos²(A/2) = (1+cos A)/2

MathsTrigonometry

Half angle · form 3

tan(A/2) = sin A/(1+cos A)

MathsCoordinate geometry

Distance in plane

d = √((x₂−x₁)²+(y₂−y₁)²)

MathsCoordinate geometry

Distance in plane · form 2

midpoint = ((x₁+x₂)/2,(y₁+y₂)/2)

MathsCoordinate geometry

Distance in plane · form 3

slope = (y₂−y₁)/(x₂−x₁)

MathsCoordinate geometry

Straight line

y−y₁ = m(x−x₁)

MathsCoordinate geometry

Straight line · form 2

y = mx+c

MathsCoordinate geometry

Straight line · form 3

x/a + y/b = 1

MathsCoordinate geometry

Point-line distance

d = |Ax₀+By₀+C|/√(A²+B²)

MathsCoordinate geometry

Point-line distance · form 2

parallel lines: A₁B₂−A₂B₁ = 0

MathsCoordinate geometry

Point-line distance · form 3

perpendicular lines: A₁A₂+B₁B₂ = 0

MathsCoordinate geometry

Circle equation

(x−h)²+(y−k)² = r²

MathsCoordinate geometry

Circle equation · form 2

centre = (h,k)

MathsCoordinate geometry

Circle equation · form 3

radius = √(h²+k²−c) for x²+y²−2hx−2ky+c=0

MathsCoordinate geometry

Parabola

y² = 4ax

MathsCoordinate geometry

Parabola · form 2

focus = (a,0) for y²=4ax

MathsCoordinate geometry

Parabola · form 3

directrix x = −a for y²=4ax

MathsCoordinate geometry

Ellipse

x²/a² + y²/b² = 1

MathsCoordinate geometry

Ellipse · form 2

c² = a²−b² (a>b)

MathsCoordinate geometry

Ellipse · form 3

e = c/a

MathsCoordinate geometry

Hyperbola

x²/a² − y²/b² = 1

MathsCoordinate geometry

Hyperbola · form 2

c² = a²+b²

MathsCoordinate geometry

Hyperbola · form 3

e = c/a

MathsCoordinate geometry

3D distance

d = √((Δx)²+(Δy)²+(Δz)²)

MathsCoordinate geometry

3D distance · form 2

midpoint = ((x₁+x₂)/2,(y₁+y₂)/2,(z₁+z₂)/2)

MathsCoordinate geometry

3D distance · form 3

direction cosines satisfy l²+m²+n²=1

MathsVectors

Dot product

a·b = |a||b|cosθ

MathsVectors

Dot product · form 2

a·b = a₁b₁+a₂b₂+a₃b₃

MathsVectors

Dot product · form 3

cosθ = (a·b)/(|a||b|)

MathsVectors

Cross product

|a×b| = |a||b|sinθ

MathsVectors

Cross product · form 2

a×b is perpendicular to a and b

MathsVectors

Cross product · form 3

parallelogram area = |a×b|

MathsCalculus

Derivative power rule

d(x^n)/dx = nx^(n−1)

MathsCalculus

Derivative power rule · form 2

d(c x^n)/dx = cnx^(n−1)

MathsCalculus

Derivative power rule · form 3

d(√x)/dx = 1/(2√x)

MathsCalculus

Derivative trigonometric

d(sin x)/dx = cos x

MathsCalculus

Derivative trigonometric · form 2

d(cos x)/dx = −sin x

MathsCalculus

Derivative trigonometric · form 3

d(tan x)/dx = sec²x

MathsCalculus

Derivative logarithmic

d(ln x)/dx = 1/x

MathsCalculus

Derivative logarithmic · form 2

d(e^x)/dx = e^x

MathsCalculus

Derivative logarithmic · form 3

d(a^x)/dx = a^x ln a

MathsCalculus

Product and quotient rules

(fg)' = f'g + fg'

MathsCalculus

Product and quotient rules · form 2

(f/g)' = (gf'−fg')/g²

MathsCalculus

Product and quotient rules · form 3

[f(g(x))]' = f'(g(x))g'(x)

MathsCalculus

Standard integrals

∫x^n dx = x^(n+1)/(n+1)+C (n≠−1)

MathsCalculus

Standard integrals · form 2

∫(1/x)dx = ln|x|+C

MathsCalculus

Standard integrals · form 3

∫e^x dx = e^x+C

MathsCalculus

Trig integrals

∫sin x dx = −cos x+C

MathsCalculus

Trig integrals · form 2

∫cos x dx = sin x+C

MathsCalculus

Trig integrals · form 3

∫sec²x dx = tan x+C

MathsCalculus

Integration by parts

∫u dv = uv − ∫v du

MathsCalculus

Integration by parts · form 2

∫ₐᵇ f(x)dx = F(b)−F(a)

MathsCalculus

Integration by parts · form 3

∫ₐᵇ f(x)dx = −∫ᵇₐ f(x)dx

MathsCalculus

Differential equation

dy/dx = ky ⇒ y = Ce^(kx)

MathsCalculus

Differential equation · form 2

dy/dx = f(x) ⇒ y = ∫f(x)dx+C

MathsCalculus

Differential equation · form 3

dy/y = k dx ⇒ ln|y| = kx+C

MathsProbability

Classical probability

P(A) = favorable/total equally likely

MathsProbability

Classical probability · form 2

P(Aᶜ) = 1−P(A)

MathsProbability

Classical probability · form 3

0 ≤ P(A) ≤ 1

MathsProbability

Addition rule

P(A∪B) = P(A)+P(B)−P(A∩B)

MathsProbability

Addition rule · form 2

if disjoint: P(A∪B) = P(A)+P(B)

MathsProbability

Addition rule · form 3

P(A∩B) = P(A)+P(B)−P(A∪B)

MathsProbability

Conditional probability

P(A|B) = P(A∩B)/P(B)

MathsProbability

Conditional probability · form 2

P(A∩B) = P(A|B)P(B)

MathsProbability

Conditional probability · form 3

independent: P(A∩B) = P(A)P(B)

MathsProbability

Bayes theorem

P(A|B) = P(B|A)P(A)/P(B)

MathsProbability

Bayes theorem · form 2

P(B) = Σ P(B|Ai)P(Ai)

MathsProbability

Bayes theorem · form 3

odds(A) = P(A)/(1−P(A))

MathsStatistics

Mean and variance

x̄ = Σxi/n

MathsStatistics

Mean and variance · form 2

variance = Σ(xi−x̄)²/n

MathsStatistics

Mean and variance · form 3

standard deviation = √variance

Formula bank

Average speed

Average speed · form 2

Average speed · form 3

Average velocity

Average velocity · form 2

Average velocity · form 3

Acceleration

Acceleration · form 2

Acceleration · form 3

Uniform acceleration displacement

Uniform acceleration displacement · form 2

Uniform acceleration displacement · form 3

Projectile horizontal range

Projectile horizontal range · form 2

Projectile horizontal range · form 3

Projectile maximum height

Projectile maximum height · form 2

Projectile maximum height · form 3

Projectile flight time

Projectile flight time · form 2

Projectile flight time · form 3

Newton's second law

Newton's second law · form 2

Newton's second law · form 3

Linear momentum

Linear momentum · form 2

Linear momentum · form 3

Impulse

Impulse · form 2

Impulse · form 3

Limiting friction

Limiting friction · form 2

Limiting friction · form 3

Centripetal acceleration

Centripetal acceleration · form 2

Centripetal acceleration · form 3

Centripetal force

Centripetal force · form 2

Centripetal force · form 3

Work

Work · form 2

Work · form 3

Kinetic energy

Kinetic energy · form 2

Kinetic energy · form 3

Gravitational potential energy

Gravitational potential energy · form 2

Gravitational potential energy · form 3

Power

Power · form 2

Power · form 3

Torque

Torque · form 2

Torque · form 3

Angular momentum

Angular momentum · form 2

Angular momentum · form 3

Rotational kinetic energy

Rotational kinetic energy · form 2

Rotational kinetic energy · form 3

Rolling without slipping

Rolling without slipping · form 2

Rolling without slipping · form 3

Newton's gravitation

Newton's gravitation · form 2

Newton's gravitation · form 3

Gravitational acceleration

Gravitational acceleration · form 2

Gravitational acceleration · form 3

Escape speed

Escape speed · form 2

Escape speed · form 3

Orbital speed

Orbital speed · form 2

Orbital speed · form 3

Hydrostatic pressure

Hydrostatic pressure · form 2

Hydrostatic pressure · form 3

Buoyancy