PIEAS Test cheat sheet

Projectile range on level ground: R = v²sin(2θ)/g, maximum at θ = 45°; maximum height H = v²sin²θ/(2g); time of flight T = 2v sinθ/g.

Newton's second law in vector form: ΣF = ma; on an incline of angle θ, the downslope component is mg sinθ and the normal force is mg cosθ (friction f = μN opposes relative motion).

Work–energy theorem: W_net = ΔK = ½mv₂² − ½mv₁²; for a spring, elastic potential energy U = ½kx² and the restoring force is F = −kx.

Ideal gas law PV = nRT with R = 8.314 J mol⁻¹ K⁻¹; at STP one mole occupies 22.4 L; average translational kinetic energy per molecule = (3/2)k_BT.

Ohm's law V = IR; resistors in series add (R_eq = R₁ + R₂), in parallel 1/R_eq = 1/R₁ + 1/R₂; electrical power P = VI = I²R = V²/R.

Mole concept: number of moles n = mass/molar mass; one mole contains 6.022 × 10²³ particles; molarity M = moles of solute per litre of solution.

Acid–base neutralization: moles of H⁺ equal moles of OH⁻ at equivalence; for a strong acid/strong base titration pH at equivalence = 7 at 25 °C.

Quadratic formula x = [−b ± √(b² − 4ac)]/(2a); the discriminant b² − 4ac decides two real, one repeated, or two complex roots.

Derivative rules: d/dx(xⁿ) = nxⁿ⁻¹, d/dx(sin x) = cos x, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x; a function is increasing where f′(x) > 0.

Eigenvalue condition: det(A − λI) = 0; the trace equals the sum of eigenvalues and the determinant equals their product, so a singular matrix has λ = 0.

Gradient, divergence, and curl: ∇f gives steepest ascent; ∇·F = 0 indicates a solenoidal (incompressible) field; ∇×F = 0 indicates an irrotational (conservative) field.

First law of thermodynamics for a control mass: Q − W = ΔU; for an ideal gas ΔU = nC_vΔT and C_p − C_v = R.

Carnot efficiency of a heat engine: η = 1 − T_c/T_h with absolute temperatures; no engine operating between the same two reservoirs can exceed it.

Heat conduction (Fourier): q = −kA dT/dx; for a plane wall in steady state the thermal resistance is R_th = L/(kA), and series resistances add.

Reynolds number Re = ρVD/μ = VD/ν; pipe flow is generally laminar below Re ≈ 2300 and turbulent above ≈ 4000, with transition between.

Bernoulli equation along a streamline (steady, inviscid, incompressible): P + ½ρV² + ρgz = constant; it is an energy statement, not valid across losses.

Linear regression by least squares: slope b = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)²; the fitted line passes through the centroid (x̄, ȳ).

Standard error of the mean: SEM = s/√n; a 95% confidence interval for a large sample is approximately x̄ ± 1.96·SEM.

SI units: force in newtons (kg·m·s⁻²), energy in joules (N·m), power in watts (J·s⁻¹), pressure in pascals (N·m⁻²), frequency in hertz (s⁻¹).

Kinematics: v = u + at, s = ut + ½at², v² = u² + 2as - valid only for constant acceleration in a straight line.

Momentum conservation: total momentum of an isolated system is constant; in a perfectly inelastic collision the bodies stick and kinetic energy is not conserved.

Wave relation v = fλ; for a string fixed at both ends the fundamental frequency is f₁ = v/(2L), and harmonics are integer multiples of f₁.

Snell's law n₁sinθ₁ = n₂sinθ₂; total internal reflection occurs only when light travels from a denser to a rarer medium and the angle exceeds the critical angle sinθ_c = n₂/n₁.

Periodic trends: atomic radius decreases across a period and increases down a group; electronegativity increases across a period and decreases down a group.

Oxidation number rules: free elements are 0, oxygen is usually −2, hydrogen is +1 with nonmetals, and the sum equals the ion charge.

Stoichiometry decision rule: balance the equation first, then convert mass to moles, apply the mole ratio, and convert back to the requested quantity.

Ideal-gas stoichiometry at the same temperature and pressure uses volume ratios equal to mole ratios (Avogadro's law).

Matrix rank equals the number of nonzero rows in row-echelon form; a square system Ax = b has a unique solution only if det(A) ≠ 0.

Fourier series of a periodic function uses a₀/2 + Σ[aₙcos(nωt) + bₙsin(nωt)]; even functions have only cosine terms, odd functions only sine terms.

Laplace transforms: L{1} = 1/s, L{e^{at}} = 1/(s−a), L{sin ωt} = ω/(s²+ω²); the final-value theorem applies only if all poles of sF(s) lie in the left half-plane.

Second-order system: natural frequency ω_n = √(k/m) and damping ratio ζ = c/(2√(km)); ζ < 1 gives underdamped oscillation, ζ = 1 critical damping.

Beam bending: maximum bending stress σ = Mc/I with second moment of area I = bh³/12 for a rectangle; deflection scales with L³ for a point load.

Torsion of a solid circular shaft: shear stress τ = Tr/J with polar moment J = πd⁴/32, so stress grows with the cube of the diameter denominator.

Material selection: yield strength governs onset of permanent deformation, ultimate strength governs fracture, and the factor of safety is strength divided by working stress.

Dimensional analysis (Buckingham π): the number of dimensionless groups equals the number of variables minus the number of fundamental dimensions involved.

Hypothesis testing: a p-value below the chosen significance level α leads to rejection of the null hypothesis; failing to reject is not proof that the null is true.

Logarithm laws: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a; log_b x = ln x / ln b.

Arithmetic progression: nth term a + (n−1)d and sum n/2·[2a + (n−1)d]; geometric progression sum a(1 − rⁿ)/(1 − r) for r ≠ 1.

Trigonometric identities: sin²θ + cos²θ = 1, sin2θ = 2sinθcosθ, cos2θ = 1 − 2sin²θ; the sine rule a/sinA = b/sinB = c/sinC.

Permutations and combinations: nPr = n!/(n−r)! and nCr = n!/[r!(n−r)!]; nCr counts unordered selections.

Electrostatics: Coulomb force F = kq₁q₂/r² with k ≈ 9 × 10⁹ N·m²·C⁻²; the field inside a conductor in electrostatic equilibrium is zero.

Magnetic force on a moving charge F = qvB sinθ; a charge moving parallel to the field (θ = 0) experiences zero magnetic force.

Half-life relation: after n half-lives the remaining fraction is (1/2)ⁿ; activity A = λN with decay constant λ = ln2/t½.

Le Chatelier's principle: adding reactant or removing product shifts equilibrium forward; raising temperature favours the endothermic direction.

pH definitions: pH = −log[H⁺] and pH + pOH = 14 at 25 °C; a tenfold increase in [H⁺] lowers pH by exactly one unit.

Error propagation: for z = x ± y, absolute errors add; for z = xy or x/y, fractional (relative) errors add.

Numerical integration: the trapezoidal rule is exact for linear functions, while Simpson's 1/3 rule is exact for polynomials up to degree three.

Newton–Raphson iteration x_{n+1} = x_n − f(x_n)/f′(x_n) converges quadratically near a simple root but fails where f′(x) = 0.

Signal sampling: the Nyquist criterion requires a sampling rate above twice the highest signal frequency to avoid aliasing.

Stress–strain interpretation: the slope of the linear region is Young's modulus, the area under the curve up to fracture is toughness, and the yield point marks the elastic limit.

Fatigue design: repeated cyclic loading can cause failure below the static yield stress, so endurance limit or S–N data must be used rather than a single static check.

Heat exchanger effectiveness–NTU method: effectiveness rises with NTU and falls with heat-capacity-rate ratio; counterflow always exceeds parallel flow effectiveness.

Control systems: a negative feedback loop reduces sensitivity to plant variation but can cause instability; the closed-loop denominator 1 + G(s)H(s) = 0 gives the characteristic equation.

Experimental design: randomisation reduces systematic bias, replication estimates variability, and blocking removes the effect of a known nuisance factor.

g ≈ 9.8 m·s⁻² near Earth's surface; G = 6.67 × 10⁻¹¹ N·m²·kg⁻²; the escape velocity from Earth is about 11.2 km·s⁻¹.

Universal constants: speed of light c ≈ 3 × 10⁸ m·s⁻¹, Planck constant h ≈ 6.63 × 10⁻³⁴ J·s, elementary charge e ≈ 1.6 × 10⁻¹⁹ C.

Photoelectric effect: maximum kinetic energy K_max = hf − φ, where φ is the work function; no emission occurs below the threshold frequency.

Bohr model energy levels: E_n = −13.6 eV/n² for hydrogen; the transition energy equals the photon energy hf.

Thermal expansion: ΔL = αL₀ΔT for linear expansion; volume expansion coefficient is about three times the linear coefficient for isotropic solids.

Calorimetry: Q = mcΔT for a temperature change and Q = mL at a phase change, where temperature stays constant during melting or boiling.

Gas laws combined: P₁V₁/T₁ = P₂V₂/T₂ for a fixed amount of gas with temperatures in kelvin.

Periodic table position: group number gives valence electrons for main-group elements; period number gives the highest occupied principal shell.

Solubility rules: most nitrate, sodium, potassium, and ammonium salts are soluble; most carbonates and phosphates are insoluble except with alkali metals.

Vector calculus: the divergence theorem converts a closed-surface flux integral to a volume integral of ∇·F; Stokes' theorem converts a line integral to a surface integral of ∇×F.

Partial derivatives: for f(x, y) the total differential is df = (∂f/∂x)dx + (∂f/∂y)dy; extrema require both first partials zero and the Hessian test for type.

Linear independence: vectors are independent only if c₁v₁ + … + cₙvₙ = 0 forces all cᵢ = 0; otherwise they are dependent.

Complex numbers: Euler's formula e^{jθ} = cosθ + j sinθ; magnitude of a + jb is √(a² + b²) and phase is arctan(b/a).

Probability: for independent events P(A∩B) = P(A)P(B), while P(A∪B) = P(A) + P(B) − P(A∩B) in general.

Bayes' theorem: P(A|B) = P(B|A)P(A)/P(B), where the denominator is the total probability of B over all causes.

Normal distribution: about 68% of values lie within one standard deviation, 95% within two, and 99.7% within three.

Z-transform and discrete systems: a pole outside the unit circle in the z-plane indicates an unstable discrete-time system.

Optimisation with constraints uses Lagrange multipliers: ∇f = λ∇g at an extremum subject to g = 0.

Capacitance: C = Q/V; for a parallel-plate capacitor C = ε₀A/d; capacitors in parallel add, in series the reciprocals add.

Energy stored in a capacitor U = ½CV² = ½QV; in an inductor U = ½LI².

Kirchhoff's rules: the junction rule conserves charge (ΣI = 0) and the loop rule conserves energy (ΣV = 0 around a closed loop).

Faraday's law: induced emf = −N dΦ/dt; Lenz's law fixes the induced current direction so that it opposes the change in flux.

Transformer relation V_s/V_p = N_s/N_p = I_p/I_s for an ideal transformer, so stepping up voltage steps down current.

Radioactive decay law N = N₀e^{−λt}; activity is measured in becquerel (1 decay per second) and dose in gray or sievert.

Mass–energy equivalence E = mc²; mass defect in a nucleus appears as binding energy, and binding energy per nucleon peaks near iron.

de Broglie wavelength λ = h/p; for electrons accelerated through potential V, λ ≈ 1.23/√V nanometres.

Entropy statement of the second law: ΔS_total ≥ 0 for any process; for a reversible adiabatic process ΔS = 0.

Isentropic relations for an ideal gas: T₂/T₁ = (P₂/P₁)^{(γ−1)/γ} and P₂/P₁ = (v₁/v₂)^γ.

Nozzle behaviour: flow chokes when the throat reaches Mach 1, after which the mass flow rate stays fixed as downstream pressure drops further.

Continuity for steady incompressible flow: A₁V₁ = A₂V₂, so halving the area doubles the velocity.

Darcy–Weisbach head loss h_f = f(L/D)(V²/2g); friction factor f depends on Reynolds number and relative roughness.

Pump power required P = ρgQH/η, where Q is volumetric flow rate, H the total head, and η the pump efficiency.

Conduction through a composite wall: total resistance is the sum of layer resistances, and the heat rate is ΔT over that total resistance.

Radiation: Stefan–Boltzmann law q = εσAT⁴ with σ = 5.67 × 10⁻⁸ W·m⁻²·K⁻⁴; net exchange depends on the fourth power of absolute temperature.

Psychrometry: relative humidity is the ratio of actual vapour pressure to saturation vapour pressure at the same dry-bulb temperature.

Straight line: slope m = (y₂ − y₁)/(x₂ − x₁); parallel lines share the slope and perpendicular lines have slopes whose product is −1.

Circle equation (x − h)² + (y − k)² = r² has centre (h, k) and radius r; a general second-degree equation represents a circle only if the xy term is zero and x², y² coefficients are equal.

Binomial theorem: (a + b)ⁿ = Σ nCr a^{n−r} b^r; the general term is nCr a^{n−r} b^r.

Limits: lim_{x→0}(sin x)/x = 1 and lim_{x→0}(1 − cos x)/x = 0; L'Hôpital's rule applies only to 0/0 or ∞/∞ indeterminate forms.

Integration by parts: ∫u dv = uv − ∫v du; for polynomials times exponentials choose the polynomial as u.

Matrix multiplication is not commutative (AB ≠ BA in general); the inverse exists only when the determinant is nonzero.

Set relations: n(A∪B) = n(A) + n(B) − n(A∩B); for three sets the inclusion–exclusion correction subtracts pairwise intersections and adds the triple intersection.

Simple and compound interest: A = P(1 + rt) for simple interest and A = P(1 + r/n)^{nt} for compound interest compounded n times per year.

Stress and strain: normal stress σ = F/A and engineering strain ε = ΔL/L₀; shear stress τ = F/A_parallel with shear strain as angular deformation.

Poisson's ratio ν = −ε_lateral/ε_axial; for most metals it lies near 0.3 and for rubber near 0.5, which implies near-incompressibility.

Thermal stress in a fully constrained bar: σ = −EαΔT, so heating a clamped bar produces compression.

Mohr's circle gives principal stresses σ₁,₂ = (σx + σy)/2 ± √[((σx − σy)/2)² + τxy²], with maximum shear stress equal to the circle radius.

Operational amplifier inverting configuration gain = −R_f/R_in; the non-inverting gain is 1 + R_f/R₁, and the ideal op-amp has infinite input impedance.

Diode forward bias requires the anode above the cathode by roughly the barrier voltage (about 0.7 V for silicon, 0.3 V for germanium).

Bipolar transistor regions: cutoff (no base current), active (amplifier, base–emitter forward, base–collector reverse), and saturation (switch on).

Digital logic: NAND and NOR are universal gates; De Morgan's laws state (A·B)′ = A′ + B′ and (A + B)′ = A′·B′.

Sensor and measurement chains: sensitivity is output change per unit input, resolution is the smallest detectable change, and accuracy differs from precision.

Hydrocarbon naming: the suffix -ane denotes single bonds, -ene a double bond, and -yne a triple bond; the longest chain containing the multiple bond sets the parent name.

Alkane general formula CₙH₂ₙ₊₂; alkenes and cycloalkanes CₙH₂ₙ; alkynes CₙH₂ₙ₋₂.

Isomerism: structural isomers differ in connectivity, while stereoisomers share connectivity but differ in spatial arrangement (geometric and optical).

Functional group tests: bromine water decolourises with alkenes; Tollens' reagent gives a silver mirror with aldehydes but not ketones.

Carboxylic acids are stronger acids than alcohols because the carboxylate anion is resonance-stabilised; phenol is intermediate in acidity.

Ideal gas and solution colligative properties: boiling-point elevation ΔT_b = K_b·m and freezing-point depression ΔT_f = K_f·m, where m is molality.

Electrochemistry: at the cathode reduction occurs (gain of electrons) and at the anode oxidation; in a galvanic cell the more positive reduction potential electrode is the cathode.

Nernst relation: E = E° − (0.059/n)log Q at 25 °C; a positive cell potential means the reaction is spontaneous as written.

Transition metals show variable oxidation states and form coloured complexes because of d–d electronic transitions.

Correlation does not imply causation; a controlled experiment or a justified causal model is needed to support causal claims.

Sensitivity and specificity: sensitivity = true positives/(true positives + false negatives), specificity = true negatives/(true negatives + false positives).

Type I error is rejecting a true null hypothesis (probability α); Type II error is failing to reject a false null (probability β), and power is 1 − β.

Sample size for a mean scales roughly with (zσ/E)², so halving the desired margin of error E requires about four times the sample.

Regression interpretation: the coefficient of determination R² is the fraction of variance in the response explained by the model, and it does not establish causation.

Units and significant figures: a computed result carries no more significant figures than the least precise input, and units must be carried through every step.

Calibration: a linear calibration curve y = mx + b converts instrument response to concentration, and the limit of detection relates to the blank standard deviation.

Reliability versus validity: reliability is repeatability of a measurement, while validity is whether it measures the intended quantity.

Peer review and reproducibility: independent replication is the standard test of a reported result, and reporting uncertainty is required for meaningful comparison.

Sound intensity level in decibels: β = 10 log(I/I₀) with I₀ = 10⁻¹² W·m⁻²; a tenfold intensity rise adds 10 dB.

Doppler effect for sound: observed frequency rises when source and observer approach and falls when they recede; the shift depends on both speeds relative to the medium.

Thin lens and mirror equation 1/f = 1/v + 1/u with magnification m = −v/u; a converging lens has positive focal length.

Young's double slit: fringe spacing Δy = λL/d, so spacing grows with wavelength and screen distance and shrinks with slit separation.

Diffraction grating: maxima occur at d sinθ = nλ; the maximum observable order is the largest n with nλ/d ≤ 1.

Heat transfer modes: conduction needs a medium, convection involves bulk fluid motion, and radiation needs no medium and travels at the speed of light.

Specific heat of water is about 4200 J kg⁻¹ K⁻¹, roughly ten times that of iron, so water resists temperature change far more.

Kirchhoff's radiation law: a good absorber is a good emitter at the same wavelength; a black body has emissivity equal to one.

Convolution in time corresponds to multiplication in frequency; the system output is the convolution of input with impulse response.

Fourier transform pair: a rectangular pulse in time gives a sinc function in frequency, and the narrower the pulse the wider the spectrum.

Transfer function H(s) = Y(s)/X(s); a pole in the right half of the s-plane makes the continuous-time system unstable.

Bode plot slopes: a single pole gives −20 dB/decade, a double pole −40 dB/decade, and a zero adds +20 dB/decade.

PID control: proportional action reduces rise time, integral action removes steady-state error, and derivative action adds damping and reduces overshoot.

Finite difference approximation: forward difference (f(x+h) − f(x))/h has first-order error, while central difference (f(x+h) − f(x−h))/(2h) is second order.

Euler's method for y′ = f(x, y) uses y_{n+1} = y_n + h f(x_n, y_n); its global error is first order in step size h.

Root finding: the bisection method always converges for a sign change on an interval but is slow, with error halving each iteration.

Condition number of a matrix measures sensitivity of the solution to input perturbations; a large condition number signals an ill-conditioned system.

Multiple-choice strategy with certainty: eliminate options that violate a known law or unit; a dimensional mismatch alone can rule out an answer without solving.

Order-of-magnitude check: estimate with rounded values (g ≈ 10, π ≈ 3) first; if the estimate is far from every option, recheck the setup rather than the arithmetic.

Unit consistency rule: convert all quantities to SI before substituting into a formula; mixing grams with kilograms is the most common source of factor-of-1000 errors.

Sign convention rule: choose a positive direction once, state it, and apply it to displacement, velocity, force, and work throughout the problem.

Limiting-case test: substitute θ = 0° or 90°, or m → 0, into a derived expression; a physically impossible limit signals an algebra error.

Ratio technique: when a quantity is scaled (doubling radius, halving length), form a ratio of the two states so constants cancel and only the exponent matters.

Graph reading rule: slope of a distance–time graph is velocity, slope of a velocity–time graph is acceleration, and area under a velocity–time graph is displacement.

Compound amount factor: F = P(1 + i)ⁿ; present worth factor is the reciprocal, P = F(1 + i)^{−n}.

Annuity present worth: P = A[(1 + i)ⁿ − 1]/[i(1 + i)ⁿ]; as n grows the factor approaches A/i.

Net present value decision rule: accept an independent project when NPV > 0 at the chosen discount rate; between mutually exclusive projects choose the highest positive NPV.

Internal rate of return is the discount rate where NPV equals zero; IRR comparison is unreliable for non-conventional cash flows with multiple sign changes.

Depreciation by straight line: annual amount = (cost − salvage)/useful life, giving a constant charge each year.

Break-even quantity = fixed cost/(unit price − unit variable cost); contribution margin per unit is the denominator.

Life-cycle costing includes capital, operation, maintenance, and disposal costs, so the lowest first cost is not necessarily the lowest total cost.

Uncertainty in estimates: sensitivity analysis identifies which input most affects the outcome, and a tornado diagram ranks variables by that influence.

Energy in food and fuels is measured in joules or calories; 1 calorie (the food Calorie, kcal) equals 4184 J.

Osmosis moves water across a semipermeable membrane from lower to higher solute concentration; osmotic pressure π = iMRT for dilute solutions.

Enzyme kinetics: the Michaelis constant K_m is the substrate concentration at half maximum velocity, and a lower K_m means higher apparent affinity.

Photosynthesis summary: 6CO₂ + 6H₂O → C₆H₁₂O₆ + 6O₂ in the presence of light and chlorophyll; oxygen comes from water.

Respiration summary: aerobic oxidation of glucose releases about 38 ATP in textbook accounting, with CO₂ and water as products.

DNA base pairing: adenine pairs with thymine (two hydrogen bonds) and guanine with cytosine (three), so G–C-rich DNA has a higher melting temperature.

Buffers resist pH change; the Henderson–Hasselbalch relation pH = pK_a + log([A⁻]/[HA]) shows maximum buffering near pH = pK_a.

Hardy–Weinberg equilibrium allele frequencies p + q = 1 and genotype frequencies p² + 2pq + q² = 1 apply only without selection, mutation, migration, or drift.

Schrödinger equation for a particle in a one-dimensional infinite well of width L gives Eₙ = n²h²/(8mL²), so energy scales with the inverse square of width.

Quantum harmonic oscillator levels are Eₙ = (n + ½)ħω, equally spaced, with zero-point energy ½ħω that cannot be removed.

Heisenberg uncertainty: ΔxΔp ≥ ħ/2; a confined state necessarily has momentum spread.

Semiconductor doping: n-type uses donor impurities (group V in silicon) with electrons as majority carriers, p-type uses acceptors (group III) with holes.

p–n junction: the built-in potential is about 0.7 V for silicon, and the depletion width narrows under forward bias and widens under reverse bias.

MOSFET regions: cutoff below threshold, linear (triode) when V_DS < V_GS − V_th, and saturation above it, where drain current is nearly independent of V_DS.

Laser action requires population inversion so stimulated emission exceeds absorption; the emitted photons are coherent in phase and wavelength.

Superconductivity: below the critical temperature resistance vanishes and magnetic flux is expelled (Meissner effect); a Type II material admits flux in vortices.

Equilibrium constant K_c uses molar concentrations and K_p uses partial pressures; K_p = K_c(RT)^{Δn}, where Δn is the change in moles of gas.

Reaction quotient Q compared with K: Q < K drives the reaction forward, Q > K drives it backward, and Q = K means equilibrium.

Gibbs energy criterion: ΔG = ΔH − TΔS; a reaction is spontaneous when ΔG < 0, and the sign of ΔH and ΔS together decide the temperature dependence.

Rate law and order: for rate = k[A]^m[B]^n the overall order is m + n, and the units of k depend on that overall order.

First-order kinetics: half-life t½ = ln2/k ≈ 0.693/k is independent of initial concentration.

Solubility product: for a salt dissolving to give ions, K_sp is the product of ion concentrations raised to stoichiometric powers; a smaller K_sp means lower solubility for the same formula type.

Molarity versus molality: molarity depends on solution volume and changes with temperature, while molality uses solvent mass and does not.

Percentage yield = (actual yield/theoretical yield) × 100; limiting reagent determines the theoretical yield.

Axial deformation of a prismatic bar: δ = PL/(AE), so stiffness grows with area and modulus and falls with length.

Euler buckling load for a pinned–pinned column P_cr = π²EI/L²; fixed ends multiply the effective load by four, and a free end divides it by four.

Combined loading design uses a failure criterion: von Mises stress for ductile metals, maximum principal stress for brittle materials.

Fatigue crack growth follows the Paris law da/dN = C(ΔK)^m, so the growth rate accelerates as the stress-intensity range increases.

Welded and bolted joints: the allowable load is governed by the weaker of base metal, weld throat area, or bolt shear area, with a factor of safety applied.

Casting versus machining: casting suits complex internal shapes with low waste, while machining gives tighter tolerance and better surface finish.

Tolerance stack-up: worst-case analysis sums absolute tolerances, whereas statistical (root-sum-square) analysis assumes independent random deviations.

Non-destructive testing: ultrasonic testing detects internal flaws, dye penetrant finds surface-breaking defects, and radiography reveals density variations.

Subject–verb agreement: a singular subject takes a singular verb even when a plural phrase intervenes; collective nouns are usually singular in formal usage.

Dangling modifier rule: an introductory participial phrase must describe the subject of the main clause, otherwise the sentence is ill-formed.

Logical reasoning structure: a valid argument's conclusion follows necessarily from its premises; a sound argument additionally has true premises.

Common fallacies: ad hominem attacks the person rather than the argument, while circular reasoning assumes the conclusion among its premises.

Analogy items test the relationship type (cause–effect, part–whole, degree, function); state the relationship in a sentence before choosing.

Data interpretation: read axis scales and units before comparing; percentage change is (new − old)/old × 100, not new/old × 100.

Sequence problems: check successive differences first, then ratios, then alternating patterns between odd and even terms.

Algorithmic complexity: binary search is O(log n) on a sorted array, while linear search is O(n); comparison sorting has an O(n log n) lower bound in the general case.

Data structures: a stack is last-in-first-out, a queue is first-in-first-out, and a balanced binary search tree gives O(log n) search, insert, and delete.

Floating-point arithmetic is not exact; comparing two computed reals requires a tolerance rather than an equality test.

Array indexing convention matters: a one-based language and a zero-based language differ by one in every loop bound, a frequent off-by-one source.

Recursion requires a base case; without it the call stack overflows, and deep recursion can be replaced by an explicit stack.

Database normalisation reduces redundancy: a table in third normal form has no transitive dependency of a non-key attribute on the key.

Numerical stability: subtracting nearly equal numbers causes catastrophic cancellation, so formulas are sometimes rearranged algebraically to avoid it.

Units in physical simulation: a consistent unit system throughout prevents silent scaling errors, and stiffness or mass matrices must use the same basis.

Speed of sound in air at room temperature ≈ 343 m·s⁻¹; in water ≈ 1500 m·s⁻¹; in steel ≈ 5000 m·s⁻¹.

Density of water = 1000 kg·m⁻³, so 1 litre of water has a mass of about 1 kg and a 1 m³ volume weighs about 9.8 kN.

Atmospheric pressure ≈ 1.01 × 10⁵ Pa = 760 mmHg = 101.3 kPa, equivalent to about 10.3 m of water column.

Avogadro's number 6.022 × 10²³ mol⁻¹; Faraday constant F = 96485 C·mol⁻¹; gas constant R = 8.314 J mol⁻¹ K⁻¹.

Specific heat capacity of water = 4186 J kg⁻¹ K⁻¹; latent heat of fusion of ice ≈ 3.34 × 10⁵ J·kg⁻¹ and of vaporisation of water ≈ 2.26 × 10⁶ J·kg⁻¹.

Refractive index of water ≈ 1.33 and of glass ≈ 1.5; the critical angle for a water–air interface is about 48.6°.

Earth's surface gravity 9.81 m·s⁻²; Earth radius ≈ 6.37 × 10⁶ m; mass of Earth ≈ 5.97 × 10²⁴ kg.

Standard atomic masses: H = 1, C = 12, N = 14, O = 16, Na = 23, Mg = 24, Al = 27, S = 32, Cl = 35.5, K = 39, Ca = 40, Fe = 56, Cu = 63.5.

Green's theorem in the plane: ∮(P dx + Q dy) = ∬(∂Q/∂x − ∂P/∂y)dA for a positively oriented simple closed curve.

Line integral of a conservative field is path-independent and equals the potential difference φ(B) − φ(A).

Partial differential equations: the heat equation u_t = αu_xx is parabolic, the wave equation u_tt = c²u_xx is hyperbolic, and Laplace's equation is elliptic.

Separation of variables solves linear PDEs with homogeneous boundary conditions; the allowed eigenvalues come from the boundary conditions, not the initial condition.

Fourier transform of a Gaussian is a Gaussian, and the product of the time and frequency widths is bounded below (uncertainty relation).

Numerical linear algebra: LU decomposition solves Ax = b in O(n³) for the factorisation and O(n²) per subsequent right-hand side.

Matrix norms and conditioning: the 2-norm condition number is σ_max/σ_min, the ratio of largest to smallest singular value.

Monte Carlo integration error decreases as 1/√N regardless of dimension, which makes it attractive for high-dimensional integrals.

Stability of numerical schemes: the CFL condition for explicit advection requires the Courant number cΔt/Δx ≤ 1.

Electric field of a point charge E = kq/r²; potential V = kq/r; potential energy of two charges U = kq₁q₂/r.

Uniform field between charged plates E = V/d; the force on a charge in that field is F = qE and the acceleration is qE/m.

Current definition I = Q/t; resistance R = ρL/A, so a longer or thinner wire has more resistance and a hotter metal has higher resistivity.

Equivalent resistance: n equal resistors in parallel give R/n, while n in series give nR.

Magnetic field of a long straight wire B = μ₀I/(2πr); inside a solenoid B = μ₀nI, where n is turns per metre.

Force between parallel current-carrying wires: equal currents attract, opposite currents repel, with force per length μ₀I₁I₂/(2πd).

Motional emf for a rod moving in a field emf = Bℓv, and the induced current direction follows Lenz's law.

RC charging: charge q(t) = Q_max(1 − e^{−t/RC}) and time constant τ = RC; the capacitor is about 63% charged after one τ.

Wheatstone bridge balance condition R₁/R₂ = R₃/R₄; at balance the galvanometer current is zero, which allows precise unknown resistance measurement.

Strain gauge relation ΔR/R = GF·ε, where GF is the gauge factor; a quarter bridge gives roughly one-quarter the sensitivity of a full bridge.

Thermocouple output is a small Seebeck voltage that requires cold-junction compensation; the measured emf depends on the temperature difference, not absolute temperature alone.

Sampling and quantisation: an n-bit ADC has 2ⁿ levels and resolution equal to full-scale range divided by 2ⁿ.

Aliasing avoidance requires an anti-aliasing low-pass filter before the sampler, not after it.

Sensor linearity and hysteresis: hysteresis is the difference between readings on increasing and decreasing input sweeps at the same input.

PID tuning by Ziegler–Nichols ultimate-gain method uses the ultimate gain K_u and period P_u from a sustained oscillation test.

Feedback loop stability: gain margin and phase margin quantify robustness; a negative phase margin indicates an unstable closed loop.

Noise reduction: averaging N independent measurements reduces random noise by √N, so quadrupling samples halves the noise standard deviation.

First law: ΔU = Q − W, where W is work done by the system; for an isothermal ideal-gas process ΔU = 0 and W = nRT ln(V₂/V₁).

Adiabatic ideal-gas process: PV^γ = constant, with γ = C_p/C_v equal to 1.67 for monatomic and about 1.4 for diatomic gases.

Efficiency of a heat engine η = W/Q_h = 1 − Q_c/Q_h; a coefficient of performance for a refrigerator is Q_c/W.

Root-mean-square speed of gas molecules v_rms = √(3RT/M), so lighter gases move faster at the same temperature.

Mean free path grows as pressure falls; at fixed temperature it is inversely proportional to number density.

Entropy change for a reversible isothermal process ΔS = Q_rev/T; for heating a solid ΔS = mc ln(T₂/T₁).

Triple point of water is 273.16 K and 611.7 Pa; the Celsius and Kelvin scales differ by the offset 273.15.

Internal energy of an ideal gas depends only on temperature; for a monatomic gas U = (3/2)nRT.

Solar photovoltaic output depends on irradiance and cell temperature; efficiency falls as cell temperature rises above the rated 25 °C.

Wind power available in the wind scales with the cube of wind speed and with the square of rotor diameter: P = ½ρAV³.

Betz limit: no wind turbine can extract more than 16/27 (about 59.3%) of the kinetic energy in the wind stream.

Combustion stoichiometry: methane CH₄ + 2O₂ → CO₂ + 2H₂O; the stoichiometric air requirement follows from the oxygen demand of the fuel.

Air–fuel ratio and excess air: excess air above stoichiometric reduces flame temperature and raises exhaust heat losses.

Water quality: biochemical oxygen demand measures biodegradable organic load, and chemical oxygen demand measures total oxidisable matter including non-biodegradable.

Pollutant dispersion: Gaussian plume concentration falls with wind speed and with the square of downwind distance for a ground-level receptor.

Energy storage round-trip efficiency multiplies charge and discharge efficiencies, so two 90% stages give about 81% overall.

Life-cycle assessment counts emissions per unit of delivered energy, so capacity factor strongly affects the carbon intensity of a renewable plant.