The Ultimate GATE ME Cheatsheet

# Matrix Fundamentals
  Rank: max linearly independent rows/columns
  Rank(A) = Rank(Aᵀ) = Rank(AAᵀ)

  det(A) = 0 → Singular, no inverse
  A⁻¹ = adj(A)/det(A)

# Eigenvalues & Eigenvectors
  det(A − λI) = 0   (Characteristic equation)
  (A − λI)v = 0     (Eigenvector for each λ)
  |A| = product of eigenvalues
  tr(A) = sum of eigenvalues

# Solving Ax = b
  rank(A) = rank([A|b]) = n → Unique solution
  rank(A) = rank([A|b]) < n → Infinite solutions
  rank(A) < rank([A|b]) → No solution

# Cayley-Hamilton: A satisfies its own char eqn
  A² − (tr A)A + (det A)I = 0  (for 2×2)

# Key Derivatives
  d/dx (xⁿ) = nx^(n−1),  d/dx (eˣ) = eˣ,  d/dx (ln x) = 1/x
  d/dx (sin x) = cos x,  d/dx (cos x) = −sin x,  d/dx (tan x) = sec²x
  Product: (uv)' = u'v + uv'
  Quotient: (u/v)' = (u'v − uv')/v²
  Chain: (f∘g)' = f'(g(x))·g'(x)

# Key Integrals
  ∫ xⁿ dx = x^(n+1)/(n+1),  ∫ eˣ dx = eˣ
  ∫ sin x dx = −cos x,  ∫ cos x dx = sin x
  ∫ dx/x = ln|x|
  ∫₀^∞ e^(−ax) dx = 1/a
  ∫₋∞^∞ e^(−x²) dx = √π
  ∫₀^(2π) sin²x dx = π,  ∫₀^(2π) cos²x dx = π

# Partial Derivatives
  ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)  — Gradient
  ∇·F = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z  — Divergence
  ∇×F  — Curl (determinant of partial derivatives)
  ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²  — Laplacian

# Taylor Series
  f(x) = f(a) + f'(a)(x−a) + f''(a)(x−a)²/2! + ...

# Maxima/Minima
  ∂f/∂x = 0, ∂f/∂y = 0 → critical point
  D = fₓₓfᵧᵧ − f²ₓᵧ
  D > 0, fₓₓ > 0 → Min | D > 0, fₓₓ < 0 → Max | D < 0 → Saddle

# First Order ODE: dy/dx + Py = Q
  Integrating Factor: IF = e^(∫P dx)
  Solution: y × IF = ∫(Q × IF)dx + C

# Second Order ODE: ay'' + by' + cy = 0
  Characteristic: ar² + br + c = 0
  D = b² − 4ac:
    D > 0: y = C₁e^(r₁x) + C₂e^(r₂x)  (distinct real)
    D = 0: y = (C₁ + C₂x)e^(rx)        (repeated real)
    D < 0: y = e^(αx)(C₁cosβx + C₂sinβx)  (complex α±jβ)

# Special Forms
  Euler-Cauchy: ax²y'' + bxy' + cy = 0
    Try y = xᵐ, solve for m

# Laplace Transform
  L{tⁿ} = n!/s^(n+1)
  L{e^(at)} = 1/(s−a)
  L{sin(at)} = a/(s²+a²)
  L{cos(at)} = s/(s²+a²)
  L{f'(t)} = sF(s) − f(0)
  L{f''(t)} = s²F(s) − sf(0) − f'(0)

# Probability
  P(A∪B) = P(A) + P(B) − P(A∩B)
  P(A|B) = P(A∩B)/P(B)
  Bayes: P(Aᵢ|B) = P(B|Aᵢ)P(Aᵢ) / ΣP(B|Aⱼ)P(Aⱼ)

# Random Variables
  E[X] = μ,  Var(X) = E[X²]−μ² = σ²
  E[aX+b] = aμ+b,  Var(aX+b) = a²σ²

# Distributions
  Binomial(n,p):  E = np,  Var = np(1−p)
  Poisson(λ):     E = λ,   Var = λ
  Normal(μ,σ²):   Z = (X−μ)/σ ~ N(0,1)
  Exponential(λ): E = 1/λ, Var = 1/λ²

  Standard Normal: P(X ≤ x) = Φ(x)

# Regression
  y = a + bx
  b = Σ(xᵢ−x̄)(yᵢ−ȳ) / Σ(xᵢ−x̄)² = r·(Sy/Sx)
  a = ȳ − bx̄
  r = correlation coefficient = covariance/(Sx·Sy)

# Central Limit Theorem
  (X̄ − μ)/(σ/√n) ~ N(0,1) for large n

# Root Finding
  Bisection: x = (a+b)/2,  Error ≤ (b−a)/2ⁿ after n iterations
  Newton-Raphson: x_{n+1} = x_n − f(x_n)/f'(x_n)
    Convergence: order 2 (quadratic)
  Secant: x_{n+1} = x_n − f(x_n)(x_n−x_{n-1})/(f(x_n)−f(x_{n-1}))
  Regula-Falsi: Like bisection but uses linear interpolation

# Numerical Integration (Trapezoidal, Simpson's)
  Trapezoidal: I ≈ (h/2)[y₀ + 2(y₁+...+y_{n-1}) + yₙ]
  Simpson's 1/3: I ≈ (h/3)[y₀ + 4y₁ + 2y₂ + 4y₃ + ... + yₙ]  (n = even)
  Error: Trapezoidal O(h²), Simpson O(h⁴)

# ODE — Euler's Method
  y_{n+1} = y_n + h·f(x_n, y_n)
  Error: O(h) — first order
  Modified Euler (Heun): y* = y_n + hf(x_n,y_n), y_{n+1} = y_n + (h/2)[f(x_n,y_n)+f(x_{n+1},y*)]

# Curve Fitting
  Least squares: minimize Σ(yᵢ−(a+bxᵢ))²
  System of equations solved for a, b

# Fundamental Relations
  σ = F/A        (Stress = Force / Area)
  ε = ΔL/L       (Strain = Change in length / Original length)
  E = σ/ε        (Young's Modulus)
  δ = FL/AE      (Elongation = Force × Length / (Area × Modulus))

  Poisson's Ratio: μ = −ε_lateral/ε_axial  (0.25–0.35 for metals)

# Stresses
  Normal stress: σ = P/A
  Shear stress:  τ = P/A  (parallel to cross-section)
  Bearing stress: σ_b = P/(d×t)
  Thermal stress: σ_th = E·α·ΔT  (if constrained)

# Shear Strain & Modulus
  G = τ/γ        (Shear Modulus = Shear stress / Shear strain)
  E = 2G(1+μ)    (Relation between E, G, μ)

# Bulk Modulus
  K = −V·dP/dV
  E = 3K(1−2μ)

# Compound Bar
  P = P₁ + P₂
  δ₁ = δ₂ (same deformation for parallel bars)
  σ₁A₁ + σ₂A₂ = P

# Three-Dimensional State
  σ_x, σ_y, σ_z = normal stresses
  τ_xy, τ_yz, τ_zx = shear stresses
  Principal stresses from stress tensor
  ε_x = (1/E)[σ_x − μ(σ_y + σ_z)]
  γ_xy = τ_xy/G

# Principal Stresses (2D)
  σ₁,₂ = (σₓ + σᵧ)/2 ± √[((σₓ−σᵧ)/2)² + τ²ₓᵧ]

  τ_max = (σ₁ − σ₂)/2 = √[((σₓ−σᵧ)/2)² + τ²ₓᵧ]

  Angle of principal plane:
  tan(2θ_p) = 2τₓᵧ / (σₓ − σᵧ)

  Angle of max shear:
  tan(2θ_s) = −(σₓ − σᵧ) / (2τₓᵧ)

# Mohr's Circle
  Center: C = ((σₓ+σᵧ)/2, 0)
  Radius: R = √[((σₓ−σᵧ)/2)² + τ²ₓᵧ]
  σ₁ = C + R,  σ₂ = C − R
  τ_max = R

  Points on circle:
  (σₓ, τₓᵧ) and (σᵧ, −τₓᵧ)

# Theories of Failure
  1. Maximum Principal Stress (Rankine): σ₁ ≤ σ_yield
  2. Maximum Shear Stress (Tresca): τ_max = (σ₁−σ₃)/2 ≤ σ_y/2
  3. Maximum Distortion Energy (von Mises):
     σ_eq = √[½((σ₁−σ₂)² + (σ₂−σ₃)² + (σ₃−σ₁)²)] ≤ σ_y
  4. Maximum Strain Energy (Haigh-Beltrami)
  5. Maximum Principal Strain (St. Venant)

  von Mises is most widely used for ductile materials

# Bending Equation
  M/I = σ/y = E/R
  M = bending moment, I = moment of inertia
  σ = bending stress, y = distance from neutral axis
  R = radius of curvature

  σ_max = M·y_max/I = M/Z   (Z = section modulus = I/y_max)

# Shear Force & Bending Moment
  SF: V = dM/dx
  BM: dV/dx = −w (distributed load w per unit length)
  d²M/dx² = −w

  Point load → SF changes by load magnitude
  UDL → SF varies linearly, BM varies parabolically

# Common Beam Results (Simply Supported, length L)
  Central point load P:
    M_max = PL/4 (at center)
    R_A = R_B = P/2

  UDL w throughout:
    M_max = wL²/8 (at center)
    R_A = R_B = wL/2

# Cantilever, length L, tip load P:
  M_max = PL (at fixed end)
  Deflection δ = PL³/(3EI)

  UDL w on cantilever:
  M_max = wL²/2 (at fixed end)
  Deflection δ = wL⁴/(8EI)

# Deflection Methods
  Double Integration: EI d²y/dx² = M(x)
  Macaulay's Method: Use singularity functions
  Superposition: Add deflections from individual loads

# Torsion (Circular Shaft)
  T/J = τ/r = Gθ/L
  T = torque, J = polar moment of inertia
  τ = shear stress at radius r
  θ = angle of twist, L = length

  J = πd⁴/32 (solid shaft)
  J = π(D⁴−d⁴)/32 (hollow shaft)
  Power: P = T × ω = 2πNT/60

  τ_max = T·r_max/J = 16T/(πd³) (solid)

# Columns & Euler's Buckling
  P_cr = π²EI/(Le)²   (Euler's critical load)

  End conditions → Effective length Le:
    Both pinned:      Le = L
    Fixed-free:       Le = 2L
    Fixed-fixed:      Le = L/2
    Fixed-pinned:     Le = 0.7L

  Slenderness ratio: λ = Le/r_min (r = radius of gyration)
  Rankine: 1/P_r = 1/P_e + 1/P_c

# Springs
  Helical Spring:
    Deflection: δ = 8PD³n/(Gd⁴)
    Stiffness: k = Gd⁴/(8D³n)
    Shear stress: τ = 8PD/(πd³) × K (Wahl factor)
    K = (4C−1)/(4C−4) + 0.615/C  where C = D/d (spring index)

  Spring in series: 1/k_eq = 1/k₁ + 1/k₂
  Spring in parallel: k_eq = k₁ + k₂

# Thin Cylinders & Spheres
  Thin cylinder (r/t > 20):
    Hoop stress σ_h = Pr/t
    Longitudinal stress σ_l = Pr/(2t)
    Volumetric strain: ΔV/V = (Pr/tE)(5/2 − μ)

  Thin sphere:
    σ = Pr/(2t)
    ΔV/V = 3Pr/(2tE)(1−μ)

# Bolted Joints
  Bolt strength: P = σ_t × A_tc  (A_tc = tensile stress area)

  Preload: F_i = K_i × Proof load
  External load sharing: F_b = F_e × k_b/(k_b+k_m)
  F_m = F_i − F_e × k_b/(k_b+k_m)  (must remain > 0)

  Shear: τ = P/(n × A)  where n = number of bolts
  Bearing: σ_b = P/(n × d × t)

# Welded Joints
  Fillet weld (throat area):
    Throat = 0.707 × leg size (s)
    Strength = σ × (throat × length)

  Butt weld:
    Tensile strength = σ_all × throat area
    Throat = thickness of thinner plate

# Keys
  Square key: width w = d/4, height h = d/4
  Shear area = w × L
  Crushing area = h/2 × L

  Shaft-key coupling:
    Torque by shear: T = τ × w × L × (d/2)
  Torque by crushing: T = σ_c × (h/2) × L × (d/2)

# Spur Gear Terminology
  Module: m = d/z (pitch diameter / teeth)
  Addendum: a = m  (tooth height above pitch circle)
  Dedendum: b = 1.25m  (tooth below pitch circle)
  Clearance: c = 0.25m
  Tooth height: h = a + b = 2.25m

  Circular pitch: p = πd/z = πm
  Base pitch: p_b = p × cos(φ)

# Gear Forces
  Tangential: F_t = 2T/d
  Radial: F_r = F_t × tan(φ)
  Total: F = F_t / cos(φ)  where φ = pressure angle

# Lewis Equation (Bending)
  σ = F_t / (b × m × Y)
  Y = Lewis form factor (depends on # teeth)
  F_t = σ_all × b × m × Y

# Gear Ratio
  Speed ratio = N₁/N₂ = z₂/z₁ = d₂/d₁

# Helical Gears
  F_t = 2T/d
  F_a = F_t × tan(ψ)  (axial)
  F_r = F_t × tan(φ)/cos(ψ)

  Normal module: m_n = m × cos(ψ)

# Journal Bearings
  Sommerfeld Number: S = (r/c)² × μN/P
  r = radius, c = clearance, μ = viscosity, N = speed, P = load/area

  L/D ratio: typically 1 (for optimal performance)

# Rolling Contact Bearings
  Life: L₁₀ = (C/P)^p × 10⁶ revolutions
  C = dynamic capacity, P = equivalent load
  p = 3 (ball bearings), p = 10/3 (roller bearings)

  For reliability R: L = L₁₀ × (log(1/R)/log(1/0.9))^(1/p)

  Equivalent load: P = XF_r + YF_a
  X, Y depend on F_a/(VF_r) ratio (V = rotation factor)

# Fatigue Design
  Endurance limit: σ_e = 0.5 × σ_ut (for steel)
  Modified: σ_e' = σ_e × k_a × k_b × k_c × k_d × k_e

  Soderberg line: σ_a/σ_e + σ_m/σ_y = 1/n
  Goodman line: σ_a/σ_e + σ_m/σ_ut = 1/n
  Gerber parabola: (nσ_a/σ_e)² + nσ_m/σ_ut = 1
  Modified Goodman: σ_a/σ_e + σ_m/σ_ut = 1/n (most used)

  Stress concentration: K_t (theoretical), K_f (fatigue)
  Notch sensitivity: q = (K_f − 1)/(K_t − 1)

# Zeroth Law
  If A is in thermal equilibrium with B, and B with C → A in eq with C
  Basis of temperature measurement

# First Law (Energy Conservation)
  δQ = δW + dU  (closed system)
  dU = m·cv·dT  (ideal gas, constant volume)
  δQ = n·Cᵥ·dT  or  δW = P·dV

  Steady Flow Energy Equation (SFEE):
    Q̇ − Ẇ = ṁ[(h₂ − h₁) + (V₂²−V₁²)/2 + g(z₂−z₁)]

# Second Law
  Kelvin-Planck: No 100% efficient heat engine
  Clausius: Heat cannot flow from cold to hot without work

  Entropy: dS ≥ δQ/T
  For reversible: ΔS = ∫δQ/T = Q/T (isothermal)

  Carnot: η = 1 − T_cold/T_hot (maximum possible)
  COP (refrigerator) = T_cold/(T_hot − T_cold)
  COP (heat pump) = T_hot/(T_hot − T_cold)

# Ideal Gas Relations
  PV = nRT,  PV = mRT
  R_universal = 8.314 J/(mol·K)
  R_specific = R/M
  cp − cv = R,  γ = cp/cv
  For air: cp = 1.005 kJ/kgK, cv = 0.718 kJ/kgK, γ = 1.4

# Processes (Ideal Gas)
  Isothermal:    PV = const,    W = nRT ln(V₂/V₁)
  Adiabatic:     PV^γ = const,  T₁V₁^(γ−1) = T₂V₂^(γ−1)
  Polytropic:    PVⁿ = const
  Isobaric:      P = const,     W = P(V₂−V₁)
  Isochoric:     V = const,     W = 0

# Otto Cycle (Petrol Engine)
  Processes: 1-2 Isentropic comp, 2-3 Const vol heat add
             3-4 Isentropic exp, 4-1 Const vol heat reject
  η = 1 − 1/r^(γ−1)
  r = compression ratio = V₁/V₂

# Diesel Cycle
  1-2 Isentropic comp, 2-3 Const pressure heat add
  3-4 Isentropic exp, 4-1 Const vol heat reject
  η = 1 − (1/r^(γ−1)) × [(ρ^γ − 1)/(γ(ρ−1))]
  ρ = cutoff ratio = V₃/V₂

# Dual Cycle
  Combination of Otto + Diesel
  η = 1 − (1/r^(γ−1)) × [(r_p·ρ^γ − 1)/((r_p−1)+γ·r_p(ρ−1))]
  r_p = pressure ratio = P₃/P₂

# Rankine Cycle (Steam Power)
  1-2 Pump (isentropic), 2-3 Boiler (const P)
  3-4 Turbine (isentropic), 4-4' Condenser (const P)

  η = (h₁ − h₂ − h₄ + h₃) / (h₁ − h₃)
  W_pump = h₃ − h₂ = v_f₂(P₂ − P₁) ≈ v_f(P_high − P_low)

  With reheat & regeneration improves efficiency

# Brayton Cycle (Gas Turbine)
  1-2 Isentropic comp, 2-3 Const P heat add
  3-4 Isentropic exp (turbine), 4-1 Const P heat reject
  η = 1 − 1/r_p^((γ−1)/γ)
  r_p = pressure ratio = P₂/P₁

# Refrigeration Cycles
  VCC (Vapor Compression):
    COP = (h₁ − h₄)/(h₂ − h₁)
  h₂ from isentropic compression of superheated vapor

# Conduction (Fourier's Law)
  q = −kA·dT/dx   (W)
  Q = kA(T₁−T₂)/L   (flat wall, steady state)
  Composite wall: Q = ΔT_total / Σ(Lᵢ/kᵢAᵢ)

  Thermal resistance: R_th = L/(kA)  [K/W]
  Series: R_total = R₁ + R₂ + ... (add resistances)
  Parallel: 1/R_total = 1/R₁ + 1/R₂ + ...

  Cylindrical: Q = 2πkL(T₁−T₂)/ln(r₂/r₁)
  Spherical: Q = 4πk(T₁−T₂)/(1/r₁ − 1/r₂)

  Critical insulation radius (cylinder): r_cr = k/h

# Convection (Newton's Law)
  q = hA(T_s − T_∞)
  h = convective heat transfer coefficient [W/m²K]

  Dimensionless Numbers:
    Nu = hL/k  (Nusselt)
    Re = ρVL/μ (Reynolds)
    Pr = μcp/k (Prandtl)
    Gr = ρ²gβΔTL³/μ² (Grashof)

  Forced convection correlations:
    Laminar (Re < 5×10⁵): Nu = 0.664·Re^(1/2)·Pr^(1/3)
    Turbulent (Re > 5×10⁵): Nu = 0.037·Re^(4/5)·Pr^(1/3)

  Natural convection: Nu = C·(Gr·Pr)^n

# Radiation (Stefan-Boltzmann)
  q = εσA(T_s⁴ − T_surr⁴)
  σ = 5.67×10⁻⁸ W/(m²·K⁴)
  ε = emissivity (0–1)

  View factors: F₁₂ = radiation from surface 1 reaching surface 2
    Reciprocity: A₁F₁₂ = A₂F₂₁
    Summation: ΣF₁ᵢ = 1
    Radiation shield: R_shield = R₁ + R₂ (parallel for series of 2)

# Heat Exchanger
  LMTD: ΔT_m = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂)
  Counter-flow is more effective than parallel-flow
  NTU (Number of Transfer Units): NTU = UA/C_min
  ε (Effectiveness) = Q_actual/Q_max

# Casting
  Solidification time (Chvorinov's Rule):
    t = C × (V/A)²
    V = volume, A = surface area, C = mold constant

  Riser design: t_riser > t_casting
  Risers should solidify last → placed near heavy sections

  Pattern allowances:
    Shrinkage (1–2%), Draft (1–3°), Machining, Shake, Distortion

  Types: Sand, Die, Investment, Centrifugal, Shell molding

# Welding
  Heat input: H = V × I / (v × 1000)  [kJ/mm]
  V = voltage, I = current, v = welding speed

  Melt efficiency: μ_melt = H_melt / H_input

  Types: Arc (SMAW, GTAW/TIG, GMAW/MIG), Gas, Resistance, Laser

  Weld defects: Porosity, Slag inclusion, Lack of fusion,
    Cracking (hot/cold), Undercut, Spatter

# Metal Forming
  True strain: ε = ln(A₀/A_f) = ln(L_f/L₀)
  Engineering strain: e = (L_f − L₀)/L₀

  Forging:
    Force: F = σ_f × A_f × (1 + μD/(3h))
    Open-die vs Closed-die

  Rolling:
    Bite condition: μ ≥ tan(α) where α = entry angle
    Draft: Δh = h₁ − h₂ = 2R(1 − cosα)
    Power: P = 2πNR × Torque

  Extrusion:
    Force: F = A₀ × σ_avg × ln(A₀/A_f)  (ideal)
    Extrusion ratio: R = A₀/A_f

  Sheet Metal:
    Deep drawing: LDR = blank dia / punch dia (max ≈ 2 for steel)
    Blanking force: F = πdt × τ_u
    Punching force: F = πdt × τ_u

# Turning
  Cutting speed: V = πDN/1000 (m/min)
  Material Removal Rate: MRR = V × f × d (mm³/min)
  f = feed rate (mm/rev), d = depth of cut (mm)

  Machine time: T_m = L/(f×N)
  L = length of cut + approach + overtravel

  Power: P = (Fc × V) / (60 × 10³)  [kW]
  Fc = cutting force

# Taylor's Tool Life
  VT^n = C
  V = cutting speed, T = tool life, n, C = constants

  Economic (Taylor): VT^n = C for minimum cost
  Modified: V₁T₁^n₁ = V₂T₂^n₂ (same tool/work)

# Drilling
  MRR = (π/4)D² × f × N
  Thrust force: F_thrust ≈ 2 × F_c

# Milling
  MRR = w × d × f × N (w = width, d = depth)
  Up milling: cutter rotation opposes feed
  Down milling: cutter rotation same as feed (better finish)

# Thread Cutting
  Pitch = 1/N (N = TPI)
  Lead = pitch × number of starts

# Tool Geometry (ASA)
  Rake angle (α), Clearance angle (γ), Cutting edge angle (φ)
  Inclination angle (λ), Nose radius (r)

  Merchant's circle:
  F = Fc/cos(β−α)  where β = friction angle
  Chip thickness ratio: r = t₁/t₂ = sin(φ)/sin(φ+β−α)
  Shear angle: φ = π/4 + α/2 − β/2  (Merchant's relation)

  Shear strain: γ = cos α / sin φ cos(φ−α)

# Limits, Fits & Tolerances
  Fundamental deviation: H (hole basis), h (shaft basis)
  IT grades: IT01 to IT18 (IT01 = finest, IT18 = coarsest)
  Allowance = Min(hole) − Max(shaft) = negative for interference fit

  Fits:
    Clearance: hole > shaft
    Interference: shaft > hole
    Transition: may be either

# Interchangeability & Selective Assembly
  Tolerance = Upper limit − Lower limit

# Ergonomics & Work Study
  Method study: Record, Examine, Develop, Install, Maintain
  Time study: Observed time × Performance rating × Allowance

  Normal time = Observed time × Rating factor
  Standard time = Normal time × (1 + Allowance %)

# Forecasting
  Moving average: F(t+1) = (1/n) Σ D(t−i+1)
  Exponential smoothing: F(t+1) = αD(t) + (1−α)F(t)

# Inventory
  EOQ = √(2DS/H)
  D = demand, S = order cost, H = holding cost
  Total cost = (D/Q)S + (Q/2)H

  Reorder point: ROP = d × L  (demand × lead time)
  Safety stock: SS = z × σ_d × √L

# Linear Programming (Simplex Method)
  Standard form: Maximize Z = c^T x
    Subject to: Ax ≤ b, x ≥ 0

  Add slack variables for ≤ constraints
  Add surplus & artificial for ≥ constraints
  Pivot on most negative entry in Z row (maximization)

  Degeneracy: zero in RHS → multiple optimal solutions
  Unbounded: no pivot restriction in min ratio test
  Infeasible: artificial variable in final basis

# Transportation Problem
  Initial solution: North-West Corner, VOGEL (best), LCM
  Optimality test: MODI method (u_i + v_j = c_ij)
  Allocate to most negative cell (δᵢⱼ = cᵢⱼ − uᵢ − vⱼ)

  Degeneracy: #allocations ≠ m+n−1 → add ε

# Assignment Problem
  Special case of transportation (all supplies = 1)
  Hungarian method: row reduction, column reduction, covering zeros
  Min lines = n → optimal, else adjust and repeat

# Network (CPM/PERT)
  CPM: deterministic times, single estimate
  PERT: probabilistic times, 3 estimates
    t_e = (t_o + 4t_m + t_p)/6
    σ² = ((t_p − t_o)/6)²

  ES, EF, LS, LF, Float = LS − ES = LF − EF
  Critical path: longest path, zero float

# Queuing Theory (M/M/1)
  λ = arrival rate, μ = service rate
  ρ = λ/μ (utilization, must be < 1)
  L_s = λ/(μ−λ),  L_q = ρ²/(1−ρ) = λ²/[μ(μ−λ)]
  W_s = 1/(μ−λ),   W_q = λ/[μ(μ−λ)]

# Properties
  Density: ρ = m/V (kg/m³)
  Specific weight: γ = ρg (N/m³)
  Specific gravity: SG = ρ/ρ_water
  Viscosity: μ (dynamic, Pa·s), ν = μ/ρ (kinematic, m²/s)
  Surface tension: σ (N/m)
  Compressibility: β = −(1/V)(dV/dP)

  Water at 20°C: ρ = 998 kg/m³, μ = 1.002×10⁻³ Pa·s

# Fluid Statics
  P = P₀ + ρgh   (Pascal's law)
  Gauge pressure: P_gauge = P − P_atm
  P_atm = 101.325 kPa = 1 atm = 760 mmHg

  Manometer: P = ρgh (simple)
  Differential: ΔP = (ρ₂−ρ₁)gh (U-tube)

  Force on submerged surface:
    F = ρg × h̄ × A
    h̄ = centroid depth
    Center of pressure: h* = h̄ + I_G/(Ah̄)
    I_G = second moment of area about centroidal axis

# Buoyancy
  F_b = ρ_fluid × g × V_displaced (Archimedes)
  Metacentric height (GM):
    GM > 0 → stable, GM < 0 → unstable
    GM = BM − BG = I/V_sub − BG

# Continuity Equation
  A₁V₁ = A₂V₂   (incompressible, steady)
  ṁ = ρAV = const (mass conservation)

# Bernoulli's Equation (along streamline)
  P/ρg + V²/2g + z = const  (head form)
  P + ½ρV² + ρgz = const    (energy form)

  V₁²/(2g) + P₁/(ρg) + z₁ = V₂²/(2g) + P₂/(ρg) + z₂ + losses

# Reynolds Number
  Re = ρVD/μ = VD/ν
  Pipe flow:
    Re < 2300 → Laminar
    Re > 4000 → Turbulent
    2300 < Re < 4000 → Transition

# Laminar Flow in Pipes
  V_max = 2V_avg (parabolic profile)
  V(r) = (ΔP/4μL)(R²−r²)
  f = 64/Re (Darcy friction factor)
  h_f = fLV²/(2gD) = 32μLV/(ρgD²) (Hagen-Poiseuille)

# Turbulent Flow
  f = 0.316/Re^(1/4)  (Blasius, smooth pipe, Re < 10⁵)
  Colebrook-White: 1/√f = −2log₁₀[(ε/D)/3.7 + 2.51/(Re√f)]

# Minor Losses
  h_L = K V²/(2g)
  K = loss coefficient
  Sudden expansion: K = (1−A₁/A₂)²
  Sudden contraction: K ≈ 0.5 (typical)
  Exit to reservoir: K = 1.0
  Entrance from reservoir: K = 0.5

# Pumps & Turbines
  Pump power: P = ρgQH/η_pump
  Specific speed: N_s = N√Q/H^(3/4)
  Cavitation: NPSH_available > NPSH_required