FET cheat sheet

Newton–Raphson: x_{n+1} = x_n − f(x_n)/f′(x_n); quadratic convergence near a simple root, but it fails when f′ ≈ 0 or the initial guess is poor.

Trapezoidal rule: ∫f dx ≈ (h/2)[f₀ + 2f₁ + … + 2f_{n−1} + f_n], truncation error O(h²); Simpson's 1/3 rule needs an even number of intervals and has error O(h⁴).

Central difference: f′(x) ≈ [f(x+h) − f(x−h)]/(2h), error O(h²); forward and backward differences are only O(h) for the same step.

Gaussian elimination with partial pivoting costs about (2/3)n³ flops; a matrix is ill-conditioned when its condition number ‖A‖·‖A⁻¹‖ is very large.

Eigenvalues satisfy det(A − λI) = 0; the eigenvectors are the nonzero solutions of (A − λI)x = 0.

Bisection halves the bracket each iteration and converges linearly (one bit per step); it needs f(a)·f(b) < 0 to start.

Lagrange interpolation through n+1 distinct points gives a unique polynomial of degree ≤ n; high-degree fits can oscillate badly (Runge phenomenon).

Laplace transform pair: L{f′(t)} = sF(s) − f(0); the final value theorem lim_{t→∞} f(t) = lim_{s→0} sF(s) holds only if all poles of sF(s) lie in the left half-plane.

Bayes: P(A|B) = P(B|A)P(A)/P(B), with P(B) = ΣP(B|A_i)P(A_i) over mutually exclusive, exhaustive events A_i.

Binomial: mean np, variance np(1−p). Poisson: mean = variance = λ. Exponential: mean 1/λ, variance 1/λ², and memoryless.

Normal distribution: about 68.3% of values lie within ±1σ, 95.4% within ±2σ, and 99.7% within ±3σ of the mean.

Standard error of the mean = s/√n; the central limit theorem makes x̄ approximately normal for large n regardless of the population shape.

Confidence interval for a mean with known σ: x̄ ± z_{α/2}·σ/√n; use t_{α/2, n−1} when σ is estimated from a small sample.

Type I error = α (rejecting a true H₀); Type II error = β (failing to reject a false H₀); power = 1 − β.

Least-squares slope b₁ = S_xy/S_xx with S_xy = Σ(x_i − x̄)(y_i − ȳ); r² is the fraction of variance in y explained by the regression.

Sample variance uses divisor n − 1; for a Weibull shape parameter β < 1 the failure rate decreases, β = 1 is constant, and β > 1 increases.

Public safety, health, and welfare rank above employer or client interests; disclose conflicts of interest and accept only work within your demonstrated competence.

Never seal or sign plans outside your field of competence; document design decisions and require an independent qualified check of critical calculations.

Present worth: PW = F/(1+i)^n. Uniform series: P = A[(1+i)^n − 1]/[i(1+i)^n]. Sinking fund: A = F·i/[(1+i)^n − 1].

Capitalized cost for an infinite-life project: CC = P + A/i, where A is the recurring annual cost.

NPV > 0 at the MARR means the project is acceptable; IRR is the rate at which NPV = 0, and the project is acceptable when IRR ≥ MARR.

Effective annual rate: i_eff = (1 + r/m)^m − 1 for nominal r compounded m times per year; continuous compounding gives e^r − 1.

Straight-line depreciation = (first cost − salvage value)/life; MACRS uses declining-balance rates switching to straight line over a statutory recovery period.

Benefit–cost ratio ≥ 1 indicates a worthwhile public project; breakeven quantity = fixed cost/(unit price − unit variable cost).

Static equilibrium of a rigid body: ΣF_x = ΣF_y = ΣF_z = 0 and ΣM = 0 about any point; a 2D problem gives three independent scalar equations.

Dry friction: F ≤ μ_s·N before slipping, then F = μ_k·N while sliding; the friction angle φ satisfies tan φ = μ_s.

Parallel axis theorem: I = I_c + A·d², where d is the perpendicular distance between the centroidal axis and the parallel axis.

Trusses: method of joints handles two unknowns per joint, method of sections three; a two-member joint with no external load has zero-force members.

Rectilinear motion: v = ds/dt, a = dv/dt = v·dv/ds, so (v₂² − v₁²)/2 = ∫a ds when acceleration is known as a function of position.

Rigid-body rotation: v = ωr, tangential acceleration a_t = αr, normal acceleration a_n = ω²r = v²/r.

Work–energy: T₁ + V₁ + U_{1→2} = T₂ + V₂, where U includes work by non-conservative forces such as friction.

Impulse–momentum: ∫F dt = m(v₂ − v₁); for central impact, e = (v₂′ − v₁′)/(v₁ − v₂), with e = 1 elastic and e = 0 perfectly plastic.

Projectile range on level ground with no drag: R = v₀²sin(2θ)/g, maximum at θ = 45°.

Axial stress σ = F/A, strain ε = ΔL/L, and Hooke's law σ = Eε in the linear-elastic range; Poisson's ratio ν = −ε_lateral/ε_axial.

Bending: σ = Mc/I, with maximum stress at the outermost fiber; curvature is 1/ρ = M/(EI).

Transverse shear: τ = VQ/(It); it is zero at the free surfaces and maximum at the neutral axis of a rectangular section.

Torsion of a circular shaft: τ_max = Tc/J and angle of twist θ = TL/(JG), with J = πd⁴/32 for a solid shaft.

Euler buckling: P_cr = π²EI/(KL)²; K = 1 pinned–pinned, 0.5 fixed–fixed, 0.7 fixed–pinned, 2 fixed–free.

Thermal stress in a fully restrained member: σ = −EαΔT; free thermal strain ε = αΔT produces no stress.

Stress concentration: σ_max = K_t·σ_nom at holes, fillets, and notches; K_t depends on geometry and loading, not on the material.

Fatigue: S–N curves plot stress amplitude against cycles to failure; Miner's rule predicts failure when Σ(n_i/N_i) = 1, and many steels show an endurance limit.

Mohr's circle: σ₁,₂ = (σ_x + σ_y)/2 ± √[((σ_x − σ_y)/2)² + τ_xy²], and maximum in-plane shear τ_max = (σ₁ − σ₂)/2.

Continuity for steady flow: A₁V₁ = A₂V₂ for incompressible flow; ρ₁A₁V₁ = ρ₂A₂V₂ for compressible flow.

Bernoulli along a streamline (steady, incompressible, inviscid): p/γ + z + V²/(2g) = constant; add head-loss terms for real flow.

Reynolds number Re = ρVD/μ = VD/ν; pipe flow is laminar below about 2000, transitional to 4000, and turbulent above.

Darcy–Weisbach head loss h_f = f(L/D)(V²/2g); laminar pipe flow has f = 64/Re, and the Moody chart gives f for turbulent flow from Re and ε/D.

Hydrostatic pressure p = γh acts normal to surfaces; the force on a plane gate is F = γ·h_c·A, with the center of pressure below the centroid.

Orifice flow Q = C_d·A·√(2gH); the discharge coefficient accounts for vena contracta and losses and is about 0.6 for a sharp-edged orifice.

Pump affinity laws at constant efficiency: Q ∝ N, H ∝ N², P ∝ N³; the same proportionalities apply to changes in impeller diameter.

Cavitation occurs when local absolute pressure falls below the liquid vapor pressure; NPSH_available must exceed NPSH_required with margin.

Drag force F_D = ½ρV²C_D A; for Stokes flow around a sphere C_D = 24/Re.

First law, closed system: Q − W = ΔU; steady-flow energy equation per unit mass: q − w = Δh + Δke + Δpe.

Ideal gas: PV = mRT, cp − cv = R, k = cp/cv; for air near room temperature R ≈ 0.287 kJ/(kg·K) and cp ≈ 1.005 kJ/(kg·K).

Carnot efficiency η = 1 − T_L/T_H with absolute temperatures; no cycle operating between the same reservoirs can exceed it.

Entropy: ΔS = ∫δQ_rev/T; for an isolated system ΔS ≥ 0, with equality only for reversible processes.

Conduction through a plane wall: q = kAΔT/L, with thermal resistance R = L/(kA); series resistances add.

Convection q = hA(T_s − T_∞); radiation q = εσA(T_s⁴ − T_surr⁴) with σ = 5.67×10⁻⁸ W/(m²·K⁴).

Overall heat transfer: q = UAΔT with 1/(UA) = ΣR; for a heat exchanger use ΔT_lmtd = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂).

Isentropic ideal-gas relations: T₂/T₁ = (P₂/P₁)^((k−1)/k) and P₂/P₁ = (v₁/v₂)^k.

Heat-exchanger effectiveness ε = q/q_max with q_max = C_min(T_h,in − T_c,in); the NTU method avoids iteration when outlet temperatures are unknown.

Ohm's law V = IR; power P = VI = I²R = V²/R; energy in kWh = power in kW × time in hours.

Resistances in series add; in parallel 1/R_eq = Σ1/R_i; voltage divides in series and current divides in parallel.

Thevenin: V_th = open-circuit voltage, R_th = resistance seen with independent sources deactivated; the Norton equivalent is the current-source dual.

Impedances: Z_R = R, Z_L = jωL, Z_C = 1/(jωC); series resonance occurs at f₀ = 1/(2π√(LC)), where impedance is minimum and current maximum.

Balanced three-phase: P = √3·V_L·I_L·cos φ, Q = √3·V_L·I_L·sin φ, S = √3·V_L·I_L; in a wye connection line voltage is √3 times phase voltage.

Ideal transformer: V₁/V₂ = N₁/N₂ = I₂/I₁; real transformers add core loss, copper loss, and leakage reactance.

Power-factor correction: Q_c = P(tan φ₁ − tan φ₂) raises the power factor from cos φ₁ to cos φ₂.

Silicon diode forward drop ≈ 0.7 V, germanium ≈ 0.3 V; a half-wave rectifier ripples at the supply frequency and a full-wave rectifier at twice that.

Ideal op-amp: infinite open-loop gain and input impedance, zero output impedance; inverting gain = −R_f/R_in, non-inverting gain = 1 + R_f/R_in.

Closed-loop transfer function for negative feedback: T(s) = G(s)/[1 + G(s)H(s)]; stability requires all roots of 1 + GH = 0 in the left half-plane.

First-order step response reaches 63.2% of its final value at t = τ and is within about 2% after 4τ.

Second-order response is set by ω_n and damping ratio ζ; percent overshoot M_p = 100·exp(−πζ/√(1−ζ²)), and ζ < 1 gives an oscillatory response.

PID: u(t) = K_p·e + K_i∫e dt + K_d·de/dt; proportional action reduces rise time, integral removes steady-state error, derivative adds damping.

Steady-state error: a type 0 system has step error 1/(1 + K_p); a type 1 system has zero step error but finite ramp error 1/K_v.

Routh–Hurwitz counts right-half-plane roots from the first column of the Routh array; gain margin and phase margin quantify relative stability on a Bode plot.

Strain gauge sensitivity is the gauge factor GF = (ΔR/R)/ε, about 2 for metal foil gauges; a Wheatstone bridge converts the resistance change to a voltage.

Sampling must satisfy f_s > 2·f_max (Nyquist); otherwise aliasing folds higher frequencies into the measured band, so anti-alias filters are used.

A 4–20 mA current loop uses 4 mA as a live zero, so an open circuit (0 mA) is distinguishable from a valid minimum reading.

Factor of safety = failure strength/allowable stress, or allowable load/working load; it covers uncertainty in loads, material properties, and analysis.

Series reliability R = ΠR_i; parallel (redundant) R = 1 − Π(1 − R_i); MTBF = 1/λ and availability = MTBF/(MTBF + MTTR).

FMEA risk priority number RPN = severity × occurrence × detection, each scored 1–10; high-severity items warrant action even at moderate RPN.

Fault tree: AND gate probability = ΠP_i for independent inputs; OR gate probability = 1 − Π(1 − P_i).

Hierarchy of controls, most to least effective: elimination, substitution, engineering controls, administrative controls, personal protective equipment.

Tolerance stack-up: worst case = Σt_i; statistical (root-sum-square) = √(Σt_i²), assuming independent, centered tolerances.

Risk = probability × severity; ALARP means reducing risk until further reduction is grossly disproportionate to the benefit.

Design reviews, prototype testing, and verification against requirements are the main checks before release; traceability links each requirement to a test.

Safety instrumented functions are rated by SIL, derived from the required risk reduction and verified by PFD calculations.

Newton–Raphson: x_{n+1} = x_n − f(x_n)/f′(x_n); quadratic convergence near a simple root, but it fails when f′ ≈ 0 or the initial guess is poor.

Trapezoidal rule: ∫f dx ≈ (h/2)[f₀ + 2f₁ + … + 2f_{n−1} + f_n], truncation error O(h²); Simpson's 1/3 rule needs an even number of intervals and has error O(h⁴).

Central difference: f′(x) ≈ [f(x+h) − f(x−h)]/(2h), error O(h²); forward and backward differences are only O(h).

Gaussian elimination with partial pivoting costs about (2/3)n³ flops; a matrix is ill-conditioned when its condition number ‖A‖·‖A⁻¹‖ is very large.

Eigenvalues satisfy det(A − λI) = 0; the eigenvectors are the nonzero solutions of (A − λI)x = 0.

Bisection halves the bracket each iteration and converges linearly; it needs f(a)·f(b) < 0 to start.

Lagrange interpolation through n+1 distinct points gives a unique polynomial of degree ≤ n; high-degree fits can oscillate (Runge phenomenon).

Laplace: L{f′(t)} = sF(s) − f(0); the final value theorem lim_{t→∞} f(t) = lim_{s→0} sF(s) applies only if the poles of sF(s) lie in the left half-plane.

Bayes: P(A|B) = P(B|A)P(A)/P(B); P(B) = ΣP(B|A_i)P(A_i) for mutually exclusive, exhaustive A_i.

Binomial: mean np, variance np(1−p). Poisson: mean = variance = λ. Exponential: mean 1/λ, variance 1/λ², memoryless.

Normal: about 68.3% of values lie within ±1σ, 95.4% within ±2σ, 99.7% within ±3σ of the mean.

Standard error of the mean = s/√n; the central limit theorem makes x̄ approximately normal for large n regardless of the population distribution.

Confidence interval for a mean with known σ: x̄ ± z_{α/2}·σ/√n; use t_{α/2, n−1} when σ is estimated from a small sample.

Type I error = α (rejecting a true H₀); Type II error = β (failing to reject a false H₀); power = 1 − β.

Least-squares slope b₁ = S_xy/S_xx with S_xy = Σ(x_i − x̄)(y_i − ȳ); r² is the fraction of variance in y explained by the regression.

Sample variance uses divisor n − 1; for a Weibull shape parameter β < 1 the failure rate decreases, β = 1 is constant, β > 1 increases.

Public safety, health, and welfare rank above employer or client interests; disclose conflicts of interest and accept only work within your competence.

Never seal or sign plans outside your field of competence; document design decisions and require independent checking of critical calculations.

Present worth: PW = F/(1+i)^n. Uniform series: P = A[(1+i)^n − 1]/[i(1+i)^n]. Sinking fund: A = F·i/[(1+i)^n − 1].

Capitalized cost for an infinite-life project: CC = P + A/i, where A is the recurring annual cost.

NPV > 0 at the MARR means acceptable; IRR is the rate where NPV = 0, and the project is acceptable when IRR ≥ MARR.

Effective annual rate: i_eff = (1 + r/m)^m − 1 for nominal r compounded m times per year; continuous compounding gives e^r − 1.

Straight-line depreciation = (first cost − salvage)/life; MACRS uses declining-balance rates switching to straight line over a statutory recovery period.

Benefit–cost ratio ≥ 1 indicates a worthwhile public project; breakeven quantity = fixed cost/(unit price − unit variable cost).

Static equilibrium: ΣF_x = ΣF_y = ΣF_z = 0 and ΣM = 0 about any point; in 2D this gives three independent equations.

Dry friction: F ≤ μ_s·N before slipping, then F = μ_k·N while sliding; the friction angle φ satisfies tan φ = μ_s.

Parallel axis theorem: I = I_c + A·d², with d the perpendicular distance between the centroidal and parallel axes.

Trusses: method of joints for two unknowns per joint, method of sections for three; a two-member unloaded joint has zero-force members.

Rectilinear motion: v = ds/dt, a = dv/dt = v·dv/ds, so (v₂² − v₁²)/2 = ∫a ds when acceleration is known versus position.

Rigid-body rotation: v = ωr, tangential acceleration a_t = αr, normal acceleration a_n = ω²r = v²/r.

Work–energy: T₁ + V₁ + U_{1→2} = T₂ + V₂, where U includes work by non-conservative forces such as friction.

Impulse–momentum: ∫F dt = m(v₂ − v₁); for central impact, e = (v₂′ − v₁′)/(v₁ − v₂), with e = 1 elastic and e = 0 perfectly plastic.

Projectile range on level ground with no drag: R = v₀²sin(2θ)/g, maximum at θ = 45°.

Axial stress σ = F/A, strain ε = ΔL/L, Hooke's law σ = Eε in the linear-elastic range; Poisson's ratio ν = −ε_lateral/ε_axial.

Bending: σ = Mc/I, maximum at the outermost fiber; curvature is 1/ρ = M/(EI).

Transverse shear: τ = VQ/(It); zero at free surfaces and maximum at the neutral axis for a rectangular section.

Torsion of a circular shaft: τ_max = Tc/J and angle of twist θ = TL/(JG), with J = πd⁴/32 for a solid shaft.

Euler buckling: P_cr = π²EI/(KL)²; K = 1 pinned–pinned, 0.5 fixed–fixed, 0.7 fixed–pinned, 2 fixed–free.

Thermal stress in a fully restrained member: σ = −EαΔT; free thermal strain ε = αΔT produces no stress.

Stress concentration: σ_max = K_t·σ_nom at holes, fillets, and notches; K_t depends on geometry and loading, not material.

Fatigue: S–N curves plot stress amplitude versus cycles; Miner's rule predicts failure when Σ(n_i/N_i) = 1, and many steels show an endurance limit.

Mohr's circle: σ₁,₂ = (σ_x + σ_y)/2 ± √[((σ_x − σ_y)/2)² + τ_xy²], with maximum in-plane shear τ_max = (σ₁ − σ₂)/2.

Continuity for steady flow: A₁V₁ = A₂V₂ for incompressible flow; ρ₁A₁V₁ = ρ₂A₂V₂ for compressible flow.

Bernoulli along a streamline (steady, incompressible, inviscid): p/γ + z + V²/(2g) = constant; add head-loss terms for real flow.

Reynolds number Re = ρVD/μ = VD/ν; pipe flow is laminar below about 2000, transitional to 4000, and turbulent above.

Darcy–Weisbach head loss h_f = f(L/D)(V²/2g); laminar flow has f = 64/Re, and the Moody chart gives f for turbulent flow from Re and ε/D.

Hydrostatic pressure p = γh acts normal to surfaces; force on a plane gate is F = γ·h_c·A, with center of pressure below the centroid.

Orifice flow Q = C_d·A·√(2gH); the discharge coefficient accounts for vena contracta and losses and is about 0.6 for a sharp-edged orifice.

Pump affinity laws at constant efficiency: Q ∝ N, H ∝ N², P ∝ N³; the same relations apply to impeller diameter changes.

Cavitation occurs when local absolute pressure falls below the liquid vapor pressure; NPSH_available must exceed NPSH_required by a margin.

Drag force F_D = ½ρV²C_D A; for Stokes flow around a sphere C_D = 24/Re.

First law, closed system: Q − W = ΔU; steady-flow energy equation per unit mass: q − w = Δh + Δke + Δpe.

Ideal gas: PV = mRT, cp − cv = R, k = cp/cv; for air near room temperature R ≈ 0.287 kJ/(kg·K) and cp ≈ 1.005 kJ/(kg·K).

Carnot efficiency η = 1 − T_L/T_H with absolute temperatures; no cycle between the same reservoirs can exceed it.

Entropy: ΔS = ∫δQ_rev/T; for an isolated system ΔS ≥ 0, with equality only for reversible processes.

Conduction through a plane wall: q = kAΔT/L, thermal resistance R = L/(kA); series resistances add.

Convection q = hA(T_s − T_∞); radiation q = εσA(T_s⁴ − T_surr⁴) with σ = 5.67×10⁻⁸ W/(m²·K⁴).

Overall heat transfer: q = UAΔT with 1/(UA) = ΣR; for a heat exchanger use ΔT_lmtd = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂).

Isentropic ideal-gas relations: T₂/T₁ = (P₂/P₁)^((k−1)/k) and P₂/P₁ = (v₁/v₂)^k.

Heat-exchanger effectiveness ε = q/q_max with q_max = C_min(T_h,in − T_c,in); the NTU method avoids iteration when outlet temperatures are unknown.

Ohm's law V = IR; power P = VI = I²R = V²/R; energy in kWh = power in kW × time in hours.

Resistances in series add; in parallel 1/R_eq = Σ1/R_i; voltage divides in series and current divides in parallel.

Thevenin: V_th = open-circuit voltage, R_th = resistance seen with independent sources deactivated; Norton is the current-source dual.

Impedances: Z_R = R, Z_L = jωL, Z_C = 1/(jωC); series resonance at f₀ = 1/(2π√(LC)), where impedance is minimum and current maximum.

Balanced three-phase: P = √3·V_L·I_L·cos φ, Q = √3·V_L·I_L·sin φ, S = √3·V_L·I_L; in wye, line voltage is √3 times phase voltage.

Ideal transformer: V₁/V₂ = N₁/N₂ = I₂/I₁; real transformers add core loss, copper loss, and leakage reactance.

Power-factor correction: Q_c = P(tan φ₁ − tan φ₂) raises the power factor from cos φ₁ to cos φ₂.

Silicon diode forward drop ≈ 0.7 V, germanium ≈ 0.3 V; half-wave rectifiers ripple at supply frequency, full-wave at twice that.

Ideal op-amp: infinite open-loop gain and input impedance, zero output impedance; inverting gain = −R_f/R_in, non-inverting = 1 + R_f/R_in.

Closed-loop transfer function for negative feedback: T(s) = G(s)/[1 + G(s)H(s)]; stability requires all roots of 1 + GH = 0 in the left half-plane.

First-order step response reaches 63.2% of final value at t = τ and is within about 2% after 4τ.

Second-order response depends on ω_n and damping ratio ζ; percent overshoot M_p = 100·exp(−πζ/√(1−ζ²)), and ζ < 1 gives oscillation.

PID: u(t) = K_p·e + K_i∫e dt + K_d·de/dt; proportional action reduces rise time, integral removes steady-state error, derivative adds damping.

Steady-state error: type 0 system has step error 1/(1 + K_p); type 1 has zero step error but finite ramp error 1/K_v.

Routh–Hurwitz counts right-half-plane roots from the first column of the Routh array; gain margin and phase margin quantify relative stability on a Bode plot.

Strain gauge sensitivity is the gauge factor GF = (ΔR/R)/ε, about 2 for metal foil gauges; a Wheatstone bridge converts the resistance change to voltage.

Sampling must satisfy f_s > 2·f_max (Nyquist); otherwise aliasing folds higher frequencies into the measured band, so anti-alias filters are used.

A 4–20 mA current loop uses 4 mA as a live zero, so an open circuit (0 mA) is distinguishable from a valid minimum reading.

Factor of safety = failure strength/allowable stress, or allowable load/working load; it covers uncertainty in loads, material properties, and analysis.

Series reliability R = ΠR_i; parallel (redundant) R = 1 − Π(1 − R_i); MTBF = 1/λ and availability = MTBF/(MTBF + MTTR).

FMEA risk priority number RPN = severity × occurrence × detection, each scored 1–10; high-severity items warrant action even at moderate RPN.

Fault tree: AND gate probability = ΠP_i for independent inputs; OR gate probability = 1 − Π(1 − P_i).

Hierarchy of controls, most to least effective: elimination, substitution, engineering controls, administrative controls, personal protective equipment.

Tolerance stack-up: worst case = Σt_i; statistical root-sum-square = √(Σt_i²), assuming independent, centered tolerances.

Risk = probability × severity; ALARP means reducing risk until further reduction is grossly disproportionate to the benefit.

Design reviews, prototype testing, and verification against requirements are the main checks before release; traceability links each requirement to a test.

Safety instrumented functions are rated by SIL, derived from the required risk reduction and verified by PFD calculations.