Formula library

Mechanical · Thermodynamics: First Law

First law (closed system)

ΔU = Q − W

U internal energy, Q heat added, W work done by system

Mechanical · Thermodynamics: First Law

Boundary work (constant P)

W = P (V₂ − V₁)

P pressure, V volume

Mechanical · Stress & Strain

Normal stress

σ = F / A

F force, A cross-section area

Mechanical · Stress & Strain

Strain

ε = ΔL / L

ΔL change in length, L original length

Mechanical · Stress & Strain

Hooke's law

σ = E · ε

E Young's modulus

Mechanical · Fluid Mechanics: Bernoulli's Theorem

Bernoulli equation

P + ½ρv² + ρgh = constant

P pressure, ρ density, v velocity, h height

Mechanical · Fluid Mechanics: Bernoulli's Theorem

Continuity

A₁v₁ = A₂v₂

A area, v velocity

Civil · Beams: Shear Force & Bending Moment

Central point load (SS)

M_max = W L / 4

W load, L span

Civil · Beams: Shear Force & Bending Moment

UDL (SS)

M_max = w L² / 8

w load per length

Civil · Beams: Shear Force & Bending Moment

Cantilever end load

M_max = W L

at fixed end

Civil · Concrete Technology Basics

Water–cement ratio

w/c = mass of water / mass of cement

typical 0.4–0.6

Electrical · Ohm's Law & DC Circuits

Ohm's law

V = I R

V volts, I amps, R ohms

Electrical · Ohm's Law & DC Circuits

Power

P = V I = I²R = V²/R

P watts

Electrical · Ohm's Law & DC Circuits

Parallel

1/R = 1/R₁ + 1/R₂

Electrical · AC Fundamentals

RMS (sine)

V_rms = V_peak / √2

Electrical · AC Fundamentals

Real power

P = V I cos φ

φ phase angle

Electrical · AC Fundamentals

Frequency

f = 1 / T

T period

Electronics · PN Junction Diode

Diode current

I = I_s (e^(V/ηV_T) − 1)

V_T ≈ 26 mV at room temperature

Electronics · PN Junction Diode

Half-wave avg

V_dc = V_m / π

CS · Time Complexity (Big-O)

Binary search

T(n) = O(log₂ n)

CS · Time Complexity (Big-O)

Merge sort

T(n) = 2T(n/2) + n = O(n log n)

Physics · Newton's Laws of Motion

Second law

F = m a

F newtons, m kg, a m/s²

Physics · Newton's Laws of Motion

Momentum

p = m v

Physics · Newton's Laws of Motion

Weight

W = m g

g = 9.81 m/s²

Mathematics · Differentiation Basics

Power rule

d/dx (xⁿ) = n xⁿ⁻¹

Mathematics · Differentiation Basics

Chain rule

dy/dx = dy/du · du/dx

Mathematics · Differentiation Basics

Common

d/dx sin x = cos x ; d/dx eˣ = eˣ