Formulas
Math
Area of a Rectangle
A = l × w
A Area
l Length
w Width
Area of a Triangle
A = (1/2) × b × h
A Area
b Base
h Height
Area of a Circle
A = πr²
A Area
π Pi Constant
r Radius
Circumference of a Circle
C = 2πr
C Circumference
π Pi Constant
r Radius
Volume of a Rectangular Prism
V = l × w × h
V Volume
l Length
w Width
h Height
Volume of a Cylinder
V = πr²h
V Volume
π Pi Constant
r Radius
h Height
Volume of a Sphere
V = (4/3)πr³
V Volume
π Pi Constant
r Radius
Pythagorean Theorem
a² + b² = c²
a Length of One Leg
b Length of Other Leg
c Length of Hypotenuse
Quadratic Formula
x = [-b ± √(b² – 4ac)] / (2a)
x Solutions to the Equation
a Coefficient of x²
b Coefficient of x
c Constant Term
Distance Formula
d = √[(x₂ – x₁)² + (y₂ – y₁)²]
d Distance Between Two Points
x₁ X Coordinate of First Point
y₁ Y Coordinate of First Point
x₂ X Coordinate of Second Point
y₂ Y Coordinate of Second Point
Slope of a Line
m = (y₂ – y₁) / (x₂ – x₁)
m Slope
x₁ X Coordinate of First Point
y₁ Y Coordinate of First Point
x₂ X Coordinate of Second Point
y₂ Y Coordinate of Second Point
Midpoint Formula
M = [(x₁ + x₂)/2, (y₁ + y₂)/2]
M Midpoint
x₁ X Coordinate of First Point
y₁ Y Coordinate of First Point
x₂ X Coordinate of Second Point
y₂ Y Coordinate of Second Point
Simple Interest
I = Prt
I Interest
P Principal Amount
r Interest Rate
t Time
Compound Interest
A = P(1 + r/n)^(nt)
A Final Amount
P Principal Amount
r Interest Rate
n Number of Times Interest Applied Per Time Period
t Time
Percent Change
% Change = [(New – Original) / Original] × 100%
% Change Percentage Change
New New Value
Original Original Value
Mean (Average)
Mean = (Sum of all values) / (Number of values)
Mean Average Value
Sum of all values Total of All Data Points
Number of values Count of Data Points
Median
The middle value in an ordered list
Median Middle Value in a Sorted Dataset
Mode
The value that appears most frequently
Mode Most Frequently Occurring Value
Perimeter of a Square
P = 4s
P Perimeter
s Side Length
Perimeter of a Rectangle
P = 2l + 2w
P Perimeter
l Length
w Width
Area of a Trapezoid
A = (1/2)h(b₁ + b₂)
A Area
h Height
b₁ First Base Length
b₂ Second Base Length
Area of a Parallelogram
A = b × h
A Area
b Base Length
h Height
Surface Area of a Cube
SA = 6s²
SA Surface Area
s Side Length
Surface Area of a Sphere
SA = 4πr²
SA Surface Area
π Pi Constant
r Radius
Volume of a Cone
V = (1/3)πr²h
V Volume
π Pi Constant
r Radius
h Height
Volume of a Pyramid
V = (1/3)Bh
V Volume
B Base Area
h Height
Sum of Interior Angles of a Polygon
Sum = (n – 2) × 180°
Sum Sum of Interior Angles
n Number of Sides
Arc Length of a Circle
L = (θ/360°) × 2πr
L Arc Length
θ Central Angle
π Pi Constant
r Radius
Sector Area of a Circle
A = (θ/360°) × πr²
A Sector Area
θ Central Angle
π Pi Constant
r Radius
Law of Sines
sin(A)/a = sin(B)/b = sin(C)/c
A Angle A
a Side Opposite Angle A
B Angle B
b Side Opposite Angle B
C Angle C
c Side Opposite Angle C
Law of Cosines
c² = a² + b² – 2ab cos(C)
a Side a
b Side b
c Side c
C Angle Opposite Side c
Exponential Growth and Decay
A = A₀(1 + r)^t
A Final Amount
A₀ Initial Amount
r Growth Rate
t Time
Standard Deviation
σ = √[ Σ(xᵢ – μ)² / N ]
σ Standard Deviation
xᵢ Individual Data Point
μ Mean
N Number of Data Points
Probability of an Event
P(A) = Number of favorable outcomes / Total number of outcomes
P(A) Probability of Event A
Number of favorable outcomes Count of Desired Outcomes
Total number of outcomes Count of All Possible Outcomes
Quadratic Equation Roots
The solutions to ax² + bx + c = 0
a Coefficient of x²
b Coefficient of x
c Constant Term
Logarithm Definition
log_b(a) = c if and only if b^c = a
log_b(a) Logarithm of a to Base b
a Argument
b Base
c Exponent
Distance, Rate, and Time
d = rt
d Distance
r Rate
t Time
Slope-Intercept Form of a Line
y = mx + b
y Y Coordinate
m Slope
x X Coordinate
b Y Intercept
Point-Slope Form of a Line
y – y₁ = m(x – x₁)
y Y Coordinate
y₁ Known Y Coordinate
m Slope
x X Coordinate
x₁ Known X Coordinate
The Pythagorean Identity (Trigonometry)
sin²(θ) + cos²(θ) = 1
θ Angle
sin Sine Function
cos Cosine Function
Area of an Ellipse
A = πab
A Area
π Pi Constant
a Semi Major Axis
b Semi Minor Axis
Equation of a Circle
(x – h)² + (y – k)² = r²
x X Coordinate
y Y Coordinate
h X Coordinate of Center
k Y Coordinate of Center
r Radius
Equation of an Ellipse
(x-h)²/a² + (y-k)²/b² = 1
x X Coordinate
y Y Coordinate
h X Coordinate of Center
k Y Coordinate of Center
a Semi Major Axis
b Semi Minor Axis
Equation of a Hyperbola
(x-h)²/a² – (y-k)²/b² = 1
x X Coordinate
y Y Coordinate
h X Coordinate of Center
k Y Coordinate of Center
a Distance to Vertex
b Distance to Covertex
Difference of Squares
a² – b² = (a + b)(a – b)
a First Term
b Second Term
Perfect Square Trinomial
a² + 2ab + b² = (a + b)²
a First Term
b Second Term
Sum of Cubes
a³ + b³ = (a + b)(a² – ab + b²)
a First Term
b Second Term
Difference of Cubes
a³ – b³ = (a – b)(a² + ab + b²)
a First Term
b Second Term
Binomial Theorem
(a + b)^n = Σ [n! / (k!(n-k)!)] a^(n-k) b^k
a First Term
b Second Term
n Power
k Term Index
Arithmetic Sequence nth Term
a_n = a₁ + (n – 1)d
a_n nth Term
a₁ First Term
n Term Number
d Common Difference
Geometric Sequence nth Term
a_n = a₁ * r^(n-1)
a_n nth Term
a₁ First Term
r Common Ratio
n Term Number
Arithmetic Series Sum
S_n = n/2 (a₁ + a_n)
S_n Sum of First n Terms
n Number of Terms
a₁ First Term
a_n nth Term
Geometric Series Sum
S_n = a₁(1 – r^n)/(1 – r)
S_n Sum of First n Terms
a₁ First Term
r Common Ratio
n Number of Terms
Infinite Geometric Series Sum
S = a₁ / (1 – r)
S Sum of Infinite Series
a₁ First Term
r Common Ratio
Permutations (No Repetition)
nPr = n! / (n – r)!
nPr Number of Permutations
n Total Items
r Items Selected
Combinations
nCr = n! / [r!(n – r)!]
nCr Number of Combinations
n Total Items
r Items Selected
Fundamental Counting Principle
m * n ways
m Number of Ways for First Event
n Number of Ways for Second Event
Conditional Probability
P(A|B) = P(A ∩ B) / P(B)
P(A|B) Probability of A Given B
P(A ∩ B) Probability of Both A and B
P(B) Probability of B
Bayes’ Theorem
P(A|B) = [P(B|A) * P(A)] / P(B)
P(A|B) Probability of A Given B
P(B|A) Probability of B Given A
P(A) Probability of A
P(B) Probability of B
Quadratic Discriminant
Δ = b² – 4ac
Δ Discriminant
a Coefficient of x²
b Coefficient of x
c Constant Term
Vertex Form of a Parabola
y = a(x – h)² + k
y Y Coordinate
a Coefficient Determining Width and Direction
x X Coordinate
h X Coordinate of Vertex
k Y Coordinate of Vertex
Standard Form of a Parabola
y = ax² + bx + c
y Y Coordinate
a Coefficient of x²
b Coefficient of x
c Constant Term
Distance from Point to Line
d = |Ax₀ + By₀ + C| / √(A² + B²)
d Distance
A Coefficient of x in Line Equation
B Coefficient of y in Line Equation
C Constant in Line Equation
x₀ X Coordinate of Point
y₀ Y Coordinate of Point
Dot Product (Vectors)
a · b = |a||b|cosθ
a First Vector
b Second Vector
|a| Magnitude of First Vector
|b| Magnitude of Second Vector
θ Angle Between Vectors
Cross Product (Vectors)
|a × b| = |a||b|sinθ
a First Vector
b Second Vector
|a| Magnitude of First Vector
|b| Magnitude of Second Vector
θ Angle Between Vectors
Magnitude of a Vector
|v| = √(v₁² + v₂² + v₃²)
|v| Magnitude of Vector
v₁ X Component
v₂ Y Component
v₃ Z Component
Scalar Projection
comp_b a = (a · b) / |b|
comp_b a Scalar Projection of a onto b
a First Vector
b Second Vector
|b| Magnitude of Second Vector
Equation of a Plane
A(x – x₀) + B(y – y₀) + C(z – z₀) = 0
A Coefficient of x
B Coefficient of y
C Coefficient of z
x₀ X Coordinate of Point on Plane
y₀ Y Coordinate of Point on Plane
z₀ Z Coordinate of Point on Plane
Sine of Sum of Angles
sin(α + β) = sinα cosβ + cosα sinβ
α First Angle
β Second Angle
sin Sine Function
cos Cosine Function
Cosine of Sum of Angles
cos(α + β) = cosα cosβ – sinα sinβ
α First Angle
β Second Angle
sin Sine Function
cos Cosine Function
Tangent of Sum of Angles
tan(α + β) = (tanα + tanβ) / (1 – tanα tanβ)
α First Angle
β Second Angle
tan Tangent Function
Double-Angle Formula (Sine)
sin(2θ) = 2 sinθ cosθ
θ Angle
sin Sine Function
cos Cosine Function
Double-Angle Formula (Cosine)
cos(2θ) = cos²θ – sin²θ
θ Angle
sin Sine Function
cos Cosine Function
Half-Angle Formula (Sine)
sin(θ/2) = ±√[(1 – cosθ)/2]
θ Angle
sin Sine Function
cos Cosine Function
Half-Angle Formula (Cosine)
cos(θ/2) = ±√[(1 + cosθ)/2]
θ Angle
cos Cosine Function
Product-to-Sum Formula
sinα sinβ = 1/2[cos(α-β) – cos(α+β)]
α First Angle
β Second Angle
sin Sine Function
cos Cosine Function
Euler’s Formula
e^(iθ) = cosθ + i sinθ
e Base of Natural Logarithm
i Imaginary Unit
θ Angle
cos Cosine Function
sin Sine Function
De Moivre’s Theorem
(cosθ + i sinθ)^n = cos(nθ) + i sin(nθ)
θ Angle
n Power
i Imaginary Unit
cos Cosine Function
sin Sine Function
Derivative of a Function
f'(x) = lim_(h->0) [f(x+h) – f(x)] / h
f'(x) Derivative of f at x
h Small Change in x
f(x) Function Value at x
Power Rule for Derivatives
d/dx[x^n] = nx^(n-1)
d/dx Derivative with Respect to x
x Variable
n Power
Product Rule for Derivatives
d/dx[uv] = u’v + uv’
d/dx Derivative with Respect to x
u First Function
v Second Function
u’ Derivative of First Function
v’ Derivative of Second Function
Quotient Rule for Derivatives
d/dx[u/v] = (u’v – uv’) / v²
d/dx Derivative with Respect to x
u Numerator Function
v Denominator Function
u’ Derivative of Numerator
v’ Derivative of Denominator
Chain Rule for Derivatives
d/dx[f(g(x))] = f'(g(x)) * g'(x)
d/dx Derivative with Respect to x
f Outer Function
g Inner Function
f’ Derivative of Outer Function
g’ Derivative of Inner Function
Derivative of sin(x)
d/dx[sin(x)] = cos(x)
d/dx Derivative with Respect to x
sin Sine Function
cos Cosine Function
Derivative of cos(x)
d/dx[cos(x)] = -sin(x)
d/dx Derivative with Respect to x
cos Cosine Function
sin Sine Function
Derivative of e^x
d/dx[e^x] = e^x
d/dx Derivative with Respect to x
e Base of Natural Logarithm
Derivative of ln(x)
d/dx[ln(x)] = 1/x
d/dx Derivative with Respect to x
ln Natural Logarithm
Indefinite Integral (Antiderivative)
∫ f(x) dx = F(x) + C
∫ Integral Symbol
f(x) Function to Integrate
dx Differential of x
F(x) Antiderivative
C Constant of Integration
Fundamental Theorem of Calculus
∫_a^b f(x) dx = F(b) – F(a)
∫ Integral Symbol
a Lower Limit
b Upper Limit
f(x) Function to Integrate
dx Differential of x
F Antiderivative
Integration by Parts
∫ u dv = uv – ∫ v du
∫ Integral Symbol
u First Function
dv Differential of Second Function
v Second Function
du Differential of First Function
Pythagorean Theorem in 3D
d = √(x² + y² + z²)
d Distance
x X Coordinate Difference
y Y Coordinate Difference
z Z Coordinate Difference
Surface Area of a Cylinder
SA = 2πrh + 2πr²
SA Surface Area
π Pi Constant
r Radius
h Height
Surface Area of a Cone
SA = πrl + πr²
SA Surface Area
π Pi Constant
r Radius
l Slant Height
Volume of a Torus
V = 2π²Rr²
V Volume
π Pi Constant
R Major Radius
r Minor Radius
Heron’s Formula (Area of Triangle)
A = √[s(s-a)(s-b)(s-c)]
A Area
s Semi Perimeter
a Side a
b Side b
c Side c
Law of Tangents
(a – b)/(a + b) = tan[(A-B)/2] / tan[(A+B)/2]
a Side a
b Side b
A Angle A
B Angle B
tan Tangent Function
Covariance
Cov(X,Y) = Σ[(xᵢ – μₓ)(yᵢ – μᵧ)] / N
Cov(X,Y) Covariance Between X and Y
xᵢ Individual X Value
yᵢ Individual Y Value
μₓ Mean of X
μᵧ Mean of Y
N Number of Data Points
Correlation Coefficient
r = Cov(X,Y) / (σₓ σᵧ)
r Correlation Coefficient
Cov(X,Y) Covariance Between X and Y
σₓ Standard Deviation of X
σᵧ Standard Deviation of Y
Binomial Probability
P(k) = nCk * p^k * (1-p)^(n-k)
P(k) Probability of k Successes
n Number of Trials
k Number of Successes
p Probability of Success
Poisson Probability
P(k) = (λ^k * e^(-λ)) / k!
P(k) Probability of k Events
λ Average Rate of Occurrence
e Base of Natural Logarithm
k Number of Events
Normal Distribution
f(x) = (1 / (σ√(2π))) * e^(-(x-μ)²/(2σ²))
f(x) Probability Density Function
x Random Variable
μ Mean
σ Standard Deviation
π Pi Constant
e Base of Natural Logarithm
Physics
Mechanics
Velocity
v = Δs / Δt
v Velocity
Δs Change in Displacement or Position
Δt Change in Time
Acceleration
a = Δv / Δt
a Acceleration
Δv Change in Velocity
Δt Change in Time
Equations of Motion
v = v₀ + at
v Final Velocity
v₀ Initial Velocity
a Acceleration
t Time
Equations of Motion
s = s₀ + v₀t + ½at²
s Final Position
s₀ Initial Position
v₀ Initial Velocity
a Acceleration
t Time
Equations of Motion
v² = v₀² + 2a(s − s₀)
v Final Velocity
v₀ Initial Velocity
a Acceleration
s Final Position
s₀ Initial Position
Newton’s Second Law
∑F = ma
∑F Net Force the Vector Sum of All Forces
m Mass
a Acceleration
Weight
W = mg
W Weight Force Due to Gravity
m Mass
g Acceleration Due to Gravity
Dry Static Friction
f_s ≤ μ_s N
f_s Force of Static Friction
μ_s Coefficient of Static Friction
N Normal Force the Perpendicular Force Exerted by a Surface
Dry Kinetic Friction
f_k = μ_k N
f_k Force of Kinetic Friction
μ_k Coefficient of Kinetic Friction
N Normal Force
Centripetal Acceleration
a_c = v² / r
a_c Centripetal Acceleration
v Tangential Speed Magnitude of Velocity
r Radius of the Circular Path
Impulse
J = FΔt
J Impulse
F Average Force
Δt Time Interval Over Which the Force Acts
Impulse-Momentum Theorem
FΔt = mΔv
F Average Force
Δt Time Interval
m Mass
Δv Change in Velocity
Work
W = FΔs cos θ
W Work
F Magnitude of the Force
Δs Magnitude of the Displacement
θ Angle Between the Force and Displacement Vectors
Work-Energy Theorem
W = ΔK
W Net Work Done on an Object
ΔK Change in Kinetic Energy K Final K Initial
Kinetic Energy
K = ½mv²
K Kinetic Energy
m Mass
v Speed
Gravitational Potential Energy
U_g = mgh
U_g Gravitational Potential Energy
m Mass
g Acceleration Due to Gravity
h Height Above a Reference Point
General Potential Energy Change
ΔU = -∫ F ⋅ ds
ΔU Change in Potential Energy
F Conservative Force
ds Infinitesimal Displacement Vector
∫ … ⋅ ds Line Integral Work Done by the Force Along a Path
Power
P = ΔW / Δt
P Power
ΔW Change in Work Amount of Work Done
Δt Change in Time Time Interval
Power-Velocity Relation
P = Fv cos θ
P Instantaneous Power
F Magnitude of the Force
v Magnitude of the Velocity
θ Angle Between the Force and Velocity Vectors
Angular Velocity
ω = Δθ / Δt
ω Angular Velocity
Δθ Change in Angular Displacement
Δt Change in Time
Linear to Angular Velocity
v = ωr
v Tangential Linear Speed
ω Angular Velocity
r Radius
Angular Acceleration
α = Δω / Δt
α Angular Acceleration
Δω Change in Angular Velocity
Δt Change in Time
Equations of Rotational Motion
ω = ω₀ + αt
ω Final Angular Velocity
ω₀ Initial Angular Velocity
α Angular Acceleration
t Time
Equations of Rotational Motion
θ = θ₀ + ω₀t + ½αt²
θ Final Angular Position
θ₀ Initial Angular Position
ω₀ Initial Angular Velocity
α Angular Acceleration
t Time
Torque
τ = rF sin θ
τ Torque
r Distance from the Pivot Point to the Point Where Force is Applied Lever Arm
F Magnitude of the Force
θ Angle Between the Force Vector and the Lever Arm Vector
Newton’s Second Law for Rotation
∑τ = Iα
∑τ Net Torque
I Moment of Inertia
α Angular Acceleration
Moment of Inertia Point Mass
I = ∑mr²
I Moment of Inertia
m Mass of a Point Particle
r Perpendicular Distance from the Particle to the Axis of Rotation
∑ Sum Over All Particles
Rotational Work
W = τΔθ
W Work Done by a Torque
τ Torque
Δθ Angular Displacement
Rotational Power
P = τω
P Power Delivered by a Torque
τ Torque
ω Angular Velocity
Angular Momentum Point Mass
L = mrv sin θ
L Angular Momentum
m Mass
r Distance from the Point Mass to the Pivot Point
v Tangential Velocity
θ Angle Between the Position and Velocity Vectors
Angular Momentum Rigid Body
L = Iω
L Angular Momentum
I Moment of Inertia
ω Angular Velocity
Angular Impulse
H = τΔt
H Angular Impulse
τ Average Torque
Δt Time Interval
Angular Impulse-Momentum Theorem
τΔt = ΔL
τ Average Torque
Δt Time Interval
ΔL Change in Angular Momentum
Newton’s Law of Universal Gravitation
F_g = G m₁ m₂ / r²
F_g Force of Gravitational Attraction
G Gravitational Constant
m₁ m₂ Masses of the Two Objects
r Distance Between the Centers of the Two Masses
Gravitational Potential Energy General
U_g = -G m₁ m₂ / r
U_g Gravitational Potential Energy Defined as Zero at Infinite Separation
G Gravitational Constant
m₁ m₂ Masses of the Two Objects
r Distance Between Their Centers
Escape Speed
v_esc = √(2Gm / r)
v_esc Escape Speed Minimum Speed to Escape a Gravitational Field
G Gravitational Constant
m Mass of the Celestial Body e g Planet Star
r Radius of the Celestial Body
Hooke’s Law
F = -kΔx
F Restoring Force Exerted by the Spring
k Spring Constant Stiffness
Δx Displacement from the Spring’s Equilibrium Position
Spring Potential Energy
U_s = ½k(Δx)²
U_s Elastic Potential Energy Stored in the Spring
k Spring Constant
Δx Displacement from Equilibrium
Period of Simple Harmonic Oscillator
T = 2π√(m / k)
T Period Time for One Complete Cycle
m Mass of the Oscillating Object
k Spring Constant
Density
ρ = m / V
ρ Density
m Mass
V Volume
Buoyancy Force
B = ρ_fluid g V_displaced
B Buoyant Force
ρ_fluid Density of the Fluid
g Acceleration Due to Gravity
V_displaced Volume of Fluid Displaced by the Object
Mass Flow Rate
q_m = Δm / Δt
q_m Mass Flow Rate
Δm Mass
Δt Time Interval
Volume Flow Rate
q_V = ΔV / Δt
q_V Volume Flow Rate
ΔV Volume
Δt Time Interval
Equation of Continuity
ρ₁A₁v₁ = ρ₂A₂v₂
ρ Density of the Fluid
A Cross Sectional Area
v Fluid Velocity
Subscripts 1 2 Refer to Two Different Points in the Flow
Bernoulli’s Equation
P₁ + ρgy₁ + ½ρv₁² = P₂ + ρgy₂ + ½ρv₂²
P Pressure
ρ Density of the Fluid
g Acceleration Due to Gravity
y Height Above a Reference Level
v Fluid Speed
Subscripts 1 2 Refer to Two Different Points Along a Streamline
Drag Force
F_d = ½ ρ C_d A v²
F_d Drag Force
ρ Density of the Fluid
C_d Drag Coefficient Depends on the Object’s Shape
A Cross Sectional Area Area Perpendicular to Flow
v Speed of the Object Relative to the Fluid
Thermal Physics
Linear Thermal Expansion
ΔL = α L₀ ΔT
ΔL Change in Length
α Coefficient of Linear Expansion
L₀ Original Length
ΔT Change in Temperature
Volumetric Thermal Expansion
ΔV = 3α V₀ ΔT
ΔV Change in Volume
α Coefficient of Linear Expansion
V₀ Original Volume
ΔT Change in Temperature
Average Molecular Kinetic Energy
⟨K⟩ = (3/2)kT
⟨K⟩ Average Translational Kinetic Energy of a Molecule
k Boltzmann Constant
T Absolute Temperature in Kelvin
Root Mean Square Speed
v_rms = √(3kT / m)
v_rms Root Mean Square Speed of Molecules
k Boltzmann Constant
T Absolute Temperature
m Mass of a Single Molecule
Heat Flow Rate
P = ΔQ / Δt
P Power Rate of Heat Transfer
ΔQ Heat Transferred
Δt Time Interval
Stefan-Boltzmann Law
P = εσA(T⁴ – T₀⁴)
P Net Radiated Power
ε Emissivity of the Object’s Surface
σ Stefan Boltzmann Constant
A Surface Area
T Temperature of the Object
T₀ Temperature of the Surroundings
Wien’s Displacement Law
λ_max = b / T
λ_max Wavelength of Peak Emission Blackbody Radiation
b Wien’s Displacement Constant
T Absolute Temperature
Change in Internal Energy Ideal Monatomic Gas
ΔU = (3/2)nRΔT
ΔU Change in Internal Energy
n Number of Moles
R Ideal Gas Constant
ΔT Change in Temperature
Entropy Change
ΔS = ΔQ_rev / T
ΔS Change in Entropy
ΔQ_rev Heat Transferred in a Reversible Process
T Absolute Temperature
Efficiency of a Heat Engine
η = 1 – Q_c / Q_h
η Efficiency
Q_c Heat Exhausted to the Cold Reservoir
Q_h Heat Absorbed from the Hot Reservoir
Maximum Carnot Efficiency
η_carnot = 1 – T_c / T_h
η_carnot Maximum Possible Carnot Efficiency
T_c Absolute Temperature of the Cold Reservoir
T_h Absolute Temperature of the Hot Reservoir
Coefficient of Performance Refrigerator
COP_real = Q_c / (Q_h – Q_c)
COP Coefficient of Performance
Q_c Heat Removed from the Cold Reservoir Inside
Q_h Heat Delivered to the Hot Reservoir Outside
Ideal Gas Law
PV = nRT
P Pressure
V Volume
n Number of Moles of Gas
R Ideal Gas Constant
T Absolute Temperature
First Law of Thermodynamics
ΔU = Q – W
ΔU Change in Internal Energy of the System
Q Heat Added to the System
W Work Done by the System on its Surroundings
Work Done in Isothermal Expansion
W = nRT ln(V_f / V_i)
W Work Done by the Gas
n Number of Moles
R Ideal Gas Constant
T Constant Absolute Temperature
V_f Final Volume
V_i Initial Volume
Heat Transfer at Constant Pressure
Q = n C_p ΔT
Q Heat Transferred
n Number of Moles
C_p Molar Specific Heat at Constant Pressure
ΔT Change in Temperature
Waves & Optics
Wave Velocity
v = fλ
v Wave Velocity Speed
f Frequency
λ Wavelength
Wave Function Sinusoidal
y(x,t) = A sin(2π(x/λ – ft) + φ)
y Wave Displacement at Position x and Time t
x Position
t Time
A Amplitude Maximum Displacement
λ Wavelength
f Frequency
φ Phase Constant Initial Phase
Beat Frequency
f_beat = |f₁ – f₂|
f_beat Beat Frequency the Frequency of the Amplitude Variation
f₁ f₂ Frequencies of the Two Interfering Waves
Intensity of a Wave
I = P / A
I Intensity Power per Unit Area
P Power
A Area
Sound Intensity Level
β = 10 log(I / I₀)
β Sound Intensity Level in Decibels dB
I Sound Intensity
I₀ Reference Intensity Threshold of Hearing Typically 10⁻¹² W m²
Doppler Effect Approaching Source
f_o = f_s [v / (v – v_s)]
f_o Observed Frequency
f_s Source Frequency
v Speed of Sound in the Medium
v_s Speed of the Source Toward the Observer
Snell’s Law of Refraction
n₁ sinθ₁ = n₂ sinθ₂
n Index of Refraction of a Medium
θ Angle Measured from the Normal Perpendicular
Subscripts 1 2 Refer to the First and Second Media
Critical Angle for Total Internal Reflection
sin θ_c = n₂ / n₁
θ_c Critical Angle
n₁ Index of Refraction of the Initial Denser Medium
n₂ Index of Refraction of the Second Less Dense Medium
Lens Mirror Equation
1/f = 1/d_o + 1/d_i
f Focal Length
d_o Object Distance
d_i Image Distance
Magnification
m = h_i / h_o = -d_i / d_o
m Magnification
h_i Image Height
h_o Object Height
d_i Image Distance
d_o Object Distance
Thin Lens Formula
1/f = 1/i + 1/o
f Focal Length
i Image Distance
o Object Distance
Speed of Sound in a Medium
v = √(B / ρ)
v Speed of Sound
B Bulk Modulus of the Medium
ρ Density of the Medium
Wavelength
λ = v / f
λ Wavelength
v Wave Speed
f Frequency
Acoustic Impedance
Z = ρ c
Z Acoustic Impedance
ρ Density of the Medium
c Speed of Sound in the Medium
Electricity & Magnetism
Coulomb’s Law
F = k q₁ q₂ / r²
F Electrostatic Force Between Two Point Charges
k Coulomb’s Constant 1 4πε₀
q₁ q₂ Electric Charges
r Distance Between the Charges
Electric Field
E = F / q
E Electric Field
F Force Experienced by a Test Charge
q Magnitude of the Test Charge
Electric Field and Potential
E = -ΔV / Δx
E Electric Field Component in the X Direction
ΔV Change in Electric Potential
Δx Change in Position Displacement
Electric Potential Point Charge
V = k q / r
V Electric Potential Voltage at a Distance r from the Charge
k Coulomb’s Constant
q Source Charge Creating the Potential
r Distance from the Source Charge
Capacitance Parallel Plate
C = κε₀ A / d
C Capacitance
κ Dielectric Constant
ε₀ Permittivity of Free Space
A Area of One Plate
d Distance Between the Plates
Energy Stored in a Capacitor
U = ½ Q V = ½ C V²
U Energy Stored
Q Charge on One Plate
V Voltage Potential Difference Between Plates
C Capacitance
Electric Current
I = Δq / Δt
I Electric Current
Δq Net Charge Flowing Past a Point
Δt Time Interval
Current Density
J = I / A
J Current Density
I Electric Current
A Cross Sectional Area
Ohm’s Law
V = I R
V Voltage Potential Difference
I Current
R Resistance
Resistivity
ρ = R A / L
ρ Resistivity of the Material
R Resistance of a Sample of the Material
A Cross Sectional Area of the Sample
L Length of the Sample
Electric Power
P = V I = I² R
P Power Dissipated
V Voltage Drop Across a Component
I Current Through the Component
R Resistance of the Component
Magnetic Force on a Moving Charge
F = q v B sin θ
F Magnetic Force
q Charge of the Particle
v Speed of the Particle
B Magnetic Field Strength
θ Angle Between the Velocity and Magnetic Field Vectors
Magnetic Force on a Current Carrying Wire
F = I L B sin θ
F Magnetic Force on the Wire
I Current in the Wire
L Length of the Wire Inside the Magnetic Field
B Magnetic Field Strength
θ Angle Between the Current and Magnetic Field Vectors
Biot Savart Law
dB = (μ₀ / 4π) (I ds × r̂) / r²
dB Infinitesimal Magnetic Field Produced by a Current Element
μ₀ Permeability of Free Space
I Current
ds Infinitesimal Length Vector of the Wire Direction of Current
r̂ Unit Vector Pointing from the Source Element to the Point Where the Field is Measured
r Distance from the Source Element to the Point
Magnetic Field Inside a Solenoid
B = μ₀ n I
B Magnetic Field Inside the Solenoid
μ₀ Permeability of Free Space
n Number of Turns of Wire per Unit Length n N L
I Current in the Wire
Magnetic Field of a Straight Wire
B = (μ₀ I) / (2π r)
B Magnetic Field Strength
μ₀ Permeability of Free Space
I Current in the Wire
r Perpendicular Distance from the Wire
Force Between Parallel Wires
F / L = (μ₀ I₁ I₂) / (2π d)
F / L Force per Unit Length Between the Wires
μ₀ Permeability of Free Space
I₁ I₂ Currents in the Two Wires
d Distance Separating the Wires
Magnetic Flux
Φ_B = B A cos θ
Φ_B Magnetic Flux
B Magnetic Field Strength
A Area
θ Angle Between the Magnetic Field Vector and the Area Vector Normal to the Surface
Faraday’s Law of Induction
ℰ = -dΦ_B / dt
ℰ Induced Electromotive Force EMF or Voltage
dΦ_B / dt Rate of Change of Magnetic Flux
Self Inductance EMF
ℰ = -L dI/dt
ℰ Self Induced EMF
L Self Inductance
dI/dt Rate of Change of Current
Impedance of RLC Circuit
Z = √[R² + (X_L – X_C)²]
Z Impedance AC Analogue of Resistance
R Resistance
X_L Inductive Reactance X L ωL
X_C Capacitive Reactance X C 1 ωC
Gauss’s Law for Electricity
∮ E ⋅ dA = Q_enclosed / ε₀
∮ … ⋅ dA Closed Surface Integral Calculates Flux Through a Gaussian Surface
E Electric Field
Q_enclosed Total Charge Enclosed Inside the Surface
ε₀ Permittivity of Free Space
Gauss’s Law for Magnetism
∮ B ⋅ dA = 0
∮ … ⋅ dA Closed Surface Integral Calculates Flux Through a Gaussian Surface
B Magnetic Field
0 Signifies that There are No Magnetic Monopoles All Magnetic Field Lines are Closed Loops
Ampere’s Law with Maxwell’s Correction
∮ B ⋅ ds = μ₀ε₀ (dΦ_E / dt) + μ₀ I
∮ B ⋅ ds Line Integral of the Magnetic Field Around a Closed Loop
μ₀ Permeability of Free Space
ε₀ Permittivity of Free Space
dΦ_E / dt Rate of Change of Electric Flux Through the Loop
I Current Enclosed by the Loop
Modern Physics
Mass Energy Equivalence
E = m c²
E Rest Energy
m Rest Mass
c Speed of Light in a Vacuum
Energy of a Photon
E = h f
E Energy of a Single Photon
h Planck’s Constant
f Frequency of the Photon
Photoelectric Effect Max Kinetic Energy
K_max = h f – φ
K_max Maximum Kinetic Energy of the Ejected Photoelectron
h Planck’s Constant
f Frequency of the Incident Photon
φ Work Function Minimum Energy Needed to Eject an Electron from the Material
De Broglie Wavelength
λ = h / p
λ de Broglie Wavelength
h Planck’s Constant
p Momentum of the Particle
Heisenberg Uncertainty Principle Position Momentum
Δx Δp ≥ ħ / 2
Δx Uncertainty in Position
Δp Uncertainty in Momentum
ħ Reduced Planck’s Constant h 2π
Time Dependent Schrödinger Equation
iħ ∂Ψ/∂t = Ĥ Ψ
i Imaginary Unit √ 1
ħ Reduced Planck’s Constant
∂Ψ/∂t Partial Derivative of the Wavefunction with Respect to Time
Ĥ Hamiltonian Operator an Operator Representing the Total Energy of the System
Ψ Wavefunction of the Quantum System
Relativistic Time Dilation
Δt = Δt₀ / √(1 – v²/c²)
Δt Time Interval Measured in a Frame of Reference Where the Clock is Moving
Δt₀ Proper Time Interval Measured in the Clock’s Rest Frame
v Relative Speed Between the Two Reference Frames
c Speed of Light
Relativistic Length Contraction
L = L₀ √(1 – v²/c²)
L Length Measured in a Frame Where the Object is Moving
L₀ Proper Length Length in the Object’s Rest Frame
v Relative Speed of the Object
c Speed of Light
Relativistic Momentum
p = γ m v
p Relativistic Momentum
γ Lorentz Factor γ 1 √ 1 v² c²
m Rest Mass
v Velocity
Relativistic Energy
E = γ m c²
E Total Relativistic Energy Rest Energy Kinetic Energy
γ Lorentz Factor
m Rest Mass
c Speed of Light
Rydberg Formula
1/λ = R (1/n₁² – 1/n₂²)
λ Wavelength of the Emitted Absorbed Photon
R Rydberg Constant
n₁ n₂ Principal Quantum Numbers n₂ gt n₁
Hubble’s Law
v = H₀ d
v Recessional Velocity of a Galaxy How Fast it is Moving Away from Us
H₀ Hubble Constant Current Rate of Expansion of the Universe
d Proper Distance to the Galaxy
Engineering
Mechanical Engineering: Mechanics & Materials
Stress (Engineering Stress)
σ = F / A₀
σ Stress
F Applied Force
A₀ Original Cross Sectional Area
Strain (Engineering Strain)
ε = ΔL / L₀
ε Strain
ΔL Change in Length
L₀ Original Length
Young’s Modulus (Modulus of Elasticity)
E = σ / ε
E Young’s Modulus
σ Stress
ε Strain
Shear Modulus (Modulus of Rigidity)
G = τ / γ
G Shear Modulus
τ Shear Stress
γ Shear Strain
Bulk Modulus
K = -V (ΔP / ΔV)
K Bulk Modulus
V Original Volume
ΔP Pressure Change
ΔV Volume Change
Poisson’s Ratio
ν = -ε_lateral / ε_axial
ν Poisson’s Ratio
ε_lateral Lateral Strain
ε_axial Axial Strain
Strain Energy (Axial Loading)
U = (σ² / 2E) * V
U Strain Energy
σ Stress
E Young’s Modulus
V Volume
Torsion Formula (Shear Stress in a Shaft)
τ = T * r / J
τ Shear Stress
T Applied Torque
r Radial Distance from the Center
J Polar Moment of Inertia
Polar Moment of Inertia (Solid Shaft)
J = π * d⁴ / 32
J Polar Moment of Inertia
d Diameter of the Shaft
Area Moment of Inertia (Rectangular Beam)
I = b * h³ / 12
I Area Moment of Inertia
b Base Width
h Height of the Beam
Bending Stress (Euler-Bernoulli Beam Theory)
σ = M * y / I
σ Bending Stress
M Bending Moment
y Distance from the Neutral Axis
I Area Moment of Inertia
Deflection of a Simply Supported Beam (Central Point Load)
δ = (P * L³) / (48 * E * I)
δ Deflection
P Point Load
L Length of the Beam
E Young’s Modulus
I Area Moment of Inertia
Critical Buckling Load (Euler’s Formula)
P_cr = π² * E * I / (K * L)²
P_cr Critical Buckling Load
E Young’s Modulus
I Area Moment of Inertia
K Column Effective Length Factor
L Actual Length
Mohr’s Circle for Stress (Radius)
R = √[((σ_x – σ_y)/2)² + τ_xy²]
R Radius of Mohr’s Circle
σ_x Normal Stress in X Direction
σ_y Normal Stress in Y Direction
τ_xy Shear Stress
Von Mises Stress (Ductile Materials)
σ’ = √[σ_x² – σ_xσ_y + σ_y² + 3τ_xy²]
σ’ Von Mises Stress
σ_x Normal Stress in X Direction
σ_y Normal Stress in Y Direction
τ_xy Shear Stress
Mechanical Engineering: Thermodynamics
First Law of Thermodynamics (Closed System)
ΔU = Q – W
ΔU Change in Internal Energy
Q Heat Added to the System
W Work Done by the System
Enthalpy
h = u + P * v
h Specific Enthalpy
u Specific Internal Energy
P Pressure
v Specific Volume
Heat Transfer by Conduction (Fourier’s Law)
Q = -k * A * (dT/dx)
Q Heat Transfer Rate
k Thermal Conductivity
A Cross Sectional Area
dT/dx Temperature Gradient
Heat Transfer by Convection (Newton’s Law of Cooling)
Q = h * A * ΔT
Q Heat Transfer Rate
h Convective Heat Transfer Coefficient
A Surface Area
ΔT Temperature Difference
Heat Exchanger Effectiveness (NTU Method)
ε = (1 – exp[-NTU(1-C_r)]) / (1 – C_r * exp[-NTU(1-C_r)])
ε Effectiveness
NTU Number of Transfer Units
C_r Heat Capacity Rate Ratio C_min C_max
Log Mean Temperature Difference (LMTD)
ΔT_lm = (ΔT₁ – ΔT₂) / ln(ΔT₁ / ΔT₂)
ΔT_lm Log Mean Temperature Difference
ΔT₁ Temperature Difference at End One
ΔT₂ Temperature Difference at End Two
Isentropic Efficiency (Turbine)
η_t = (h₁ – h₂) / (h₁ – h₂s)
η_t Isentropic Efficiency
h₁ Inlet Enthalpy
h₂ Actual Outlet Enthalpy
h₂s Isentropic Outlet Enthalpy
Coefficient of Performance (Refrigerator)
COP_R = Q_L / W_in
COP_R Coefficient of Performance
Q_L Heat Removed from the Cold Reservoir
W_in Work Input
Carnot Efficiency (Heat Engine)
η_th = 1 – T_C / T_H
η_th Theoretical Carnot Efficiency
T_C Cold Reservoir Temperature
T_H Hot Reservoir Temperature
Mass Flow Rate
ṁ = ρ * A * V
ṁ Mass Flow Rate
ρ Density
A Cross Sectional Area
V Flow Velocity
Mechanical Engineering: Fluid Mechanics
Reynolds Number
Re = (ρ * V * L) / μ
Re Reynolds Number
ρ Density
V Velocity
L Characteristic Length
μ Dynamic Viscosity
Darcy-Weisbach Equation (Head Loss)
h_f = f * (L/D) * (V² / (2g))
h_f Head Loss Due to Friction
f Darcy Friction Factor
L Pipe Length
D Pipe Diameter
V Velocity
g Gravity
Drag Force
F_D = (1/2) * ρ * V² * A * C_D
F_D Drag Force
ρ Density
V Velocity
A Reference Area
C_D Drag Coefficient
Lift Force
F_L = (1/2) * ρ * V² * A * C_L
F_L Lift Force
ρ Density
V Velocity
A Reference Area e g Wing Area
C_L Lift Coefficient
Bernoulli’s Equation (Incompressible Flow)
P₁ + (1/2)ρV₁² + ρgh₁ = P₂ + (1/2)ρV₂² + ρgh₂
P Pressure
ρ Density
V Velocity
g Gravity
h Elevation Height
Dynamic Pressure
q = (1/2) * ρ * V²
q Dynamic Pressure
ρ Density
V Velocity
Viscous Shear Stress (Newton’s Law of Viscosity)
τ = μ * (du/dy)
τ Shear Stress
μ Dynamic Viscosity
du/dy Velocity Gradient Perpendicular to the Flow
Hagen–Poiseuille Equation (Laminar Flow in a Pipe)
Q = (π * ΔP * r⁴) / (8 * μ * L)
Q Volumetric Flow Rate
ΔP Pressure Drop
r Pipe Radius
μ Dynamic Viscosity
L Pipe Length
Pump Hydraulic Power
W_pump = ṁ * g * H
W_pump Hydraulic Power
ṁ Mass Flow Rate
g Gravity
H Pump Head
Mach Number
M = V / a
M Mach Number
V Velocity of the Object
a Speed of Sound in the Medium
Electrical Engineering: Circuits & Power
Ohm’s Law
V = I * R
V Voltage
I Current
R Resistance
Electrical Power (DC)
P = V * I
P Power
V Voltage
I Current
Joule’s Law (Resistive Heating)
P = I² * R
P Power Dissipated as Heat
I Current
R Resistance
Capacitance
C = Q / V
C Capacitance
Q Charge Stored
V Voltage
Inductance
V = L * (di/dt)
V Voltage Across the Inductor
L Inductance
di/dt Rate of Change of Current
Impedance (RL Series Circuit)
Z = √(R² + (ωL)²)
Z Impedance
R Resistance
ω Angular Frequency
L Inductance
Resonant Frequency (LC Circuit)
f_r = 1 / (2π √(L C))
f_r Resonant Frequency
L Inductance
C Capacitance
Real Power (AC Circuit)
P = V_rms * I_rms * cos(θ)
P Real Power
V_rms RMS Voltage
I_rms RMS Current
θ Phase Angle Between Voltage and Current
Reactive Power
Q = V_rms * I_rms * sin(θ)
Q Reactive Power
V_rms RMS Voltage
I_rms RMS Current
θ Phase Angle Between Voltage and Current
Apparent Power
S = V_rms * I_rms
S Apparent Power
V_rms RMS Voltage
I_rms RMS Current
Electrical Engineering: Electromagnetics & Machines
Force on a Current Carrying Conductor
F = B * I * L * sin(θ)
F Force
B Magnetic Flux Density
I Current
L Conductor Length
θ Angle Between B and I
Magnetic Flux
Φ_B = B * A * cos(θ)
Φ_B Magnetic Flux
B Magnetic Flux Density
A Area
θ Angle Between B and the Normal to A
Transformer Equation (Turns Ratio)
V_s / V_p = N_s / N_p
V_s Secondary Voltage
V_p Primary Voltage
N_s Number of Secondary Turns
N_p Number of Primary Turns
Back EMF in a DC Motor
E_b = V – I_a * R_a
E_b Back Electromotive Force
V Terminal Voltage
I_a Armature Current
R_a Armature Resistance
Synchronous Speed (AC Motor)
N_s = (120 * f) / P
N_s Synchronous Speed in RPM
f Line Frequency
P Number of Magnetic Poles
Motor Slip
s = (N_s – N_r) / N_s
s Slip
N_s Synchronous Speed
N_r Rotor Speed
Power Factor
PF = cos(θ) = P / S
PF Power Factor
θ Phase Angle
P Real Power
S Apparent Power
Energy Consumption (kWh)
E = P * t / 1000
E Energy in Kilowatt Hours
P Power in Watts
t Time in Hours
Charge and Current
I = dQ / dt
I Current
dQ Change in Charge
dt Change in Time
Skin Depth
δ = √(2 / (ω * μ * σ))
δ Skin Depth
ω Angular Frequency
μ Permeability
σ Conductivity
Civil & Structural Engineering
Allowable Stress (Factor of Safety)
σ_allow = σ_yield / FoS
σ_allow Allowable Stress
σ_yield Yield Stress
FoS Factor of Safety
Bearing Capacity (Terzaghi’s Equation)
q_u = c*N_c + q*N_q + 0.5*γ*B*N_γ
q_u Ultimate Bearing Capacity
c Cohesion
q Surcharge
γ Soil Unit Weight
B Footing Width
N_c N_q N_γ Bearing Capacity Factors
Vertical Stress Under a Point Load (Boussinesq)
Δσ_z = (3P * z³) / (2π * R⁵)
Δσ_z Vertical Stress Increase
P Point Load
z Depth
R Radial Distance from the Load Point
Flow Rate (Manning’s Equation)
V = (1/n) * R_h^(2/3) * S^(1/2)
V Flow Velocity
n Manning’s Roughness Coefficient
R_h Hydraulic Radius
S Channel Slope
Hydraulic Radius
R_h = A / P_w
R_h Hydraulic Radius
A Cross Sectional Area of Flow
P_w Wetted Perimeter
Euler’s Critical Buckling Stress
σ_cr = π² * E / (L_e / r)²
σ_cr Critical Buckling Stress
E Young’s Modulus
L_e Effective Column Length
r Radius of Gyration
Section Modulus
S = I / y_max
S Section Modulus
I Area Moment of Inertia
y_max Distance to the Outermost Fiber
Deflection of a Cantilever Beam (End Load)
δ = (P * L³) / (3 * E * I)
δ Deflection at the End
P Point Load
L Length of the Beam
E Young’s Modulus
I Area Moment of Inertia
Chemical Engineering
Reynolds Number (Pipe Flow)
Re = (ρ * u * d) / μ
Re Reynolds Number
ρ Fluid Density
u Fluid Velocity
d Pipe Diameter
μ Dynamic Viscosity
Friction Factor (Colebrook Equation)
1/√f = -2 log₁₀[(ε/D)/3.7 + 2.51/(Re √f)]
f Darcy Friction Factor
ε Pipe Roughness
D Pipe Diameter
Re Reynolds Number
Ideal Gas Law
P * V = n * R * T
P Absolute Pressure
V Volume
n Number of Moles
R Ideal Gas Constant
T Absolute Temperature
Raoult’s Law (Vapor-Liquid Equilibrium)
P_i = x_i * P_i^*
P_i Partial Pressure of Component i
x_i Mole Fraction in the Liquid
P_i^* Pure Component Vapor Pressure
Henry’s Law
C_i = P_i * H_i
C_i Concentration of Dissolved Gas
P_i Partial Pressure
H_i Henry’s Law Constant
Fick’s First Law (Diffusion)
J = -D * (dc/dx)
J Diffusion Flux
D Diffusion Coefficient
dc/dx Concentration Gradient
Arrhenius Equation (Reaction Rate)
k = A * exp(-E_a / (R T))
k Reaction Rate Constant
A Pre Exponential Factor
E_a Activation Energy
R Gas Constant
T Temperature
Reynolds Transport Theorem
dB_sys/dt = ∂/∂t ∫_cv ρ b dV + ∫_cs ρ b (v · n) dA
B_sys Extensive System Property
b Corresponding Intensive Property
ρ Density
v Velocity
n Normal Vector
Control Systems & Robotics
Transfer Function (General Form)
G(s) = Y(s) / U(s)
G(s) Transfer Function
Y(s) Laplace Transform of the Output
U(s) Laplace Transform of the Input
Damping Ratio (Second-Order System)
ζ = c / (2 √(m k))
ζ Damping Ratio
c Damping Coefficient
m Mass
k Spring Constant
Natural Frequency
ω_n = √(k / m)
ω_n Natural Frequency
k Spring Constant
m Mass
Gain Margin
GM = 1 / |G(jω_pc)|
GM Gain Margin
|G(jω_pc)| Magnitude of the Open Loop Transfer Function at the Phase Crossover Frequency
Phase Margin
PM = 180° + ∠G(jω_gc)
PM Phase Margin
∠G(jω_gc) Phase Angle of the Open Loop Transfer Function at the Gain Crossover Frequency
Engineering Economics
Present Worth Analysis
PW = Σ [F_t / (1 + i)^t]
PW Present Worth
F_t Net Cash Flow in Period t
i Discount Rate
t Time Period
Future Worth Analysis
FW = PW * (1 + i)^n
FW Future Worth
PW Present Worth
i Interest Rate
n Number of Periods
Annual Worth Analysis
AW = PW * [i(1+i)^n / ((1+i)^n – 1)]
AW Annual Worth
PW Present Worth
i Interest Rate
n Number of Periods
Internal Rate of Return (IRR)
0 = Σ [CF_t / (1 + IRR)^t]
CF_t Cash Flow in Time Period t
IRR Internal Rate of Return
Benefit-Cost Ratio
BCR = Σ (Benefits) / Σ (Costs)
BCR Benefit Cost Ratio
General & Interdisciplinary
Root Mean Square (RMS) Value
X_rms = √(1/T ∫_0^T x(t)² dt)
X_rms Root Mean Square Value of the Signal x(t) Over a Period T
Signal-to-Noise Ratio (SNR)
SNR = P_signal / P_noise
SNR Signal to Noise Ratio
P_signal Power of the Signal
P_noise Power of the Noise
Weibull Distribution (Failure Rate)
f(t) = (β/η) (t/η)^{β-1} exp[-(t/η)^β]
f(t) Probability Density Function
t Time
β Shape Parameter
η Scale Parameter
Binomial Coefficient
C(n, k) = n! / (k! (n-k)!)
C(n, k) Number of Combinations
n Total Number of Items
k Number of Items to Choose
Bayes’ Theorem
P(A|B) = [P(B|A) * P(A)] / P(B)
P(A|B) Conditional Probability of A Given B
P(B|A) Probability of B Given A
P(A) Probability of A
P(B) Probability of B
Information Entropy (Shannon)
H = -Σ [p_i * log₂(p_i)]
H Information Entropy
p_i Probability of the i th Possible Value of the Source Symbol
Euler’s Formula
e^(iθ) = cos(θ) + i sin(θ)
e Base of the Natural Logarithm
i Imaginary Unit
θ Angle in Radians
cos Cosine Function
sin Sine Function
Euler’s Identity
e^(iπ) + 1 = 0
e Base of the Natural Logarithm
i Imaginary Unit
π Pi
1 Integer One
0 Integer Zero