ESE GS&EA cheat sheet

SI base units: length metre (m), mass kilogram (kg), time second (s), electric current ampere (A), thermodynamic temperature kelvin (K), amount of substance mole (mol), luminous intensity candela (cd).

Dimensional analysis: any valid physical equation must be dimensionally homogeneous; dimensionless groups (e.g., Reynolds number Re = ρvD/μ) are invariant under unit scaling and are used to check or derive relations.

Standard prefixes: nano 10⁻⁹, micro 10⁻⁶, milli 10⁻³, kilo 10³, mega 10⁶, giga 10⁹, tera 10¹²; 1 kWh = 3.6 × 10⁶ J; 1 bar = 10⁵ Pa; 1 atm = 101.325 kPa.

Error propagation: for z = f(x, y), fractional uncertainty adds in quadrature - (δz/z)² = (∂ln f/∂x · δx)² + (∂ln f/∂y · δy)²; for products/quotients relative errors add directly as an upper bound.

Significant figures: results carry the least number of significant figures of the inputs; in addition/subtraction, retain the least number of decimal places.

Engineering ethics: a professional obligation hierarchy runs public safety and health first, then client/employer interest, then profession, with statutory law and codes of conduct binding in all cases.

Environmental management: an Environmental Impact Assessment screens a project before clearance; ISO 14001 specifies a plan–do–check–act environmental management system; the precautionary principle permits action before full scientific certainty.

Project economics: payback period ignores time value of money; net present value (NPV) and internal rate of return (IRR) discount future cash flows, and a project is viable when NPV > 0 or IRR exceeds the discount rate.

Descriptive statistics: mean = Σx/n; sample variance s² = Σ(x − x̄)²/(n − 1); standard deviation is its square root; the coefficient of variation CV = s/x̄ compares dispersion across datasets.

Normal distribution: about 68% of values lie within ±1σ, 95% within ±2σ, and 99.7% within ±3σ of the mean; standard score z = (x − μ)/σ.

Linear regression: least-squares slope b = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)²; correlation coefficient r lies in [−1, 1] and r² is the fraction of variance explained.

Set and probability rules: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(A|B) = P(A ∩ B)/P(B); for independent events P(A ∩ B) = P(A)P(B).

Combinatorics: permutations nPr = n!/(n − r)!; combinations nCr = n!/[r!(n − r)!]; binomial term probability = nCr pʳ(1 − p)^(n − r).

Number theory and series: sum of first n natural numbers n(n + 1)/2; sum of squares n(n + 1)(2n + 1)/6; a number is divisible by 3 (or 9) if the sum of its digits is divisible by 3 (or 9).

Reasoning decision rules: in syllogisms, a valid conclusion must hold in every model satisfying the premises; in data-sufficiency items, decide whether each statement alone or only together yields a unique answer.

Stress σ = F/A; engineering strain ε = ΔL/L; Hooke's law σ = Eε holds within the proportional limit; Poisson's ratio ν = −lateral strain/longitudinal strain.

Thermal stress in a fully constrained bar: σ = EαΔT, where α is the coefficient of linear expansion; a free bar develops no thermal stress, only free strain ε = αΔT.

Beam relations: shear force V = dM/dx; for a simply supported beam with a central point load W and span L, maximum bending moment M = WL/4 at midspan.

Bernoulli equation for steady incompressible frictionless flow along a streamline: p/ρg + v²/2g + z = constant; the continuity equation is A₁v₁ = A₂v₂.

Heat transfer: Fourier conduction q = −kA dT/dx; Newton cooling q = hA(Ts − T∞); Stefan–Boltzmann radiation q = εσA(T⁴ − Tsurr⁴) with σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴.

Electric circuits: Ohm's law V = IR; series resistances add, parallel conductances add; power P = VI = I²R = V²/R; Kirchhoff's current and voltage laws express charge and energy conservation.

Measurement instruments: a vernier calliper least count = 1 main-scale division − 1 vernier-scale division; a micrometer least count = pitch/number of circular-scale divisions.

Material properties: Young's modulus measures stiffness, toughness is energy absorbed before fracture (area under stress–strain curve), hardness resists indentation, and fatigue failure occurs under cyclic stress below the ultimate strength.