- Static equilibrium conditions
- A body is in static equilibrium when the sum of all forces equals zero (ΣF = 0) and the sum of all moments equals zero (ΣM = 0).
- Derivative of sin(x)
- cos(x). The derivative measures the instantaneous rate of change of the function.
- First law of thermodynamics
- Energy is conserved: the change in internal energy ΔU = Q − W, where Q is heat added to the system and W is work done by the system.
- Ohm's law
- V = I × R. Voltage equals current times resistance.
- Continuity equation (incompressible)
- A₁V₁ = A₂V₂. Volumetric flow rate Q is constant, so velocity rises where the cross-sectional area shrinks.
- What is a scalar vs a vector?
- A scalar has magnitude only (e.g., temperature, mass). A vector has both magnitude and direction (e.g., force, velocity).
- Dot product of two vectors
- A·B = |A||B|cos θ. A scalar result; zero when the vectors are perpendicular.
- Cross product of two vectors
- A×B = |A||B|sin θ, giving a vector perpendicular to both A and B; its magnitude is the area of the parallelogram they form.
- Derivative of cos(x)
- −sin(x).
- Derivative of eˣ
- eˣ. The exponential function is its own derivative.
- Derivative of ln(x)
- 1/x.
- Integral of 1/x dx
- ln|x| + C.
- Chain rule
- d/dx f(g(x)) = f′(g(x)) · g′(x). Differentiate the outer function, then multiply by the derivative of the inner function.
- Product rule
- d/dx (uv) = u′v + uv′.
- Quotient rule
- d/dx (u/v) = (u′v − uv′) / v².
- What does a definite integral represent geometrically?
- The signed area between the curve and the x-axis over the interval of integration.
- Fundamental theorem of calculus
- Integration and differentiation are inverse operations: ∫ₐᵇ f(x) dx = F(b) − F(a), where F′ = f.
- Order of a differential equation
- The order is the highest derivative that appears. y″ + 3y′ + 2y = 0 is second-order.
- Solution to dP/dt = kP
- P = P₀ eᵏᵗ — exponential growth (k > 0) or decay (k < 0), where P₀ is the initial value.
- Linear vs nonlinear ODE
- Linear: the dependent variable and its derivatives appear only to the first power and are not multiplied together. Otherwise it is nonlinear.
- Homogeneous vs nonhomogeneous ODE
- Homogeneous: the forcing term (right-hand side) is zero. Nonhomogeneous: it has a nonzero forcing function.
- Determinant of a 2×2 matrix [[a,b],[c,d]]
- ad − bc.
- When is a system of linear equations solvable by a unique solution?
- When the coefficient matrix is square and its determinant is nonzero (the matrix is nonsingular / invertible).
- Eigenvalue definition
- A scalar λ for which Av = λv has a nonzero vector v; found from det(A − λI) = 0.
- Law of cosines
- c² = a² + b² − 2ab·cos C. Generalizes the Pythagorean theorem to any triangle.
- sin²θ + cos²θ = ?
- 1. The fundamental Pythagorean trigonometric identity.
- Euler's formula
- e raised to the power iθ equals cos θ + i·sin θ. It links the exponential and trigonometric functions for complex numbers.
- Magnitude of complex number a + bi
- √(a² + b²).
- Slope of a line through (x₁,y₁) and (x₂,y₂)
- (y₂ − y₁) / (x₂ − x₁) — rise over run.
- Taylor series purpose
- Approximates a function near a point as a power series using the function's derivatives at that point.
- Gradient of a scalar field
- A vector ∇f pointing in the direction of steepest increase, with magnitude equal to the maximum rate of change.
- Divergence vs curl
- Divergence (∇·F) measures a field's net outflow (a scalar). Curl (∇×F) measures its rotation (a vector).
- Logarithm identity: log(ab)
- log a + log b. Multiplication becomes addition in log space.
- Roots of ax² + bx + c = 0
- x = (−b ± √(b² − 4ac)) / (2a) — the quadratic formula.
- Discriminant of a quadratic
- b² − 4ac. Positive = two real roots; zero = one repeated root; negative = two complex roots.
- Mean vs median
- Mean is the arithmetic average (sum ÷ count). Median is the middle value when data are ordered; it resists outliers.
- Standard deviation
- The variance's positive root (√variance); a measure of how spread out data are around the mean.
- Variance
- The average of the squared deviations from the mean. Standard deviation squared.
- Probability of two independent events both occurring
- Multiply: P(A and B) = P(A) × P(B).
- Probability of A or B (mutually exclusive)
- Add: P(A or B) = P(A) + P(B).
- General addition rule for P(A or B)
- P(A) + P(B) − P(A and B). Subtract the overlap so it is not double-counted.
- Conditional probability P(A|B)
- P(A and B) / P(B) — the probability of A given that B has occurred.
- Independent events condition
- Two events are independent if P(A|B) = P(A); knowing B happened does not change the probability of A.
- Normal distribution shape
- A symmetric, bell-shaped curve fully described by its mean and standard deviation.
- Empirical (68-95-99.7) rule
- For a normal distribution, about 68%, 95%, and 99.7% of data fall within 1, 2, and 3 standard deviations of the mean.
- Z-score
- (x − mean) / standard deviation. The number of standard deviations a value lies from the mean.
- Binomial distribution use
- Models the number of successes in n independent trials, each with the same success probability p.
- Expected value of a discrete random variable
- The probability-weighted sum of its possible values: E(X) = Σ xᵢ P(xᵢ).
- Permutation vs combination
- Permutations count ordered arrangements (order matters); combinations count selections (order does not matter).
- Number of combinations of n items taken r at a time
- C(n,r) = n! / [r!(n − r)!].
- Coefficient of correlation (r) range
- −1 to +1. It measures the strength and direction of a linear relationship between two variables.
- Least-squares regression goal
- Find the line that minimizes the sum of the squared vertical distances between the data points and the line.
- Type I vs Type II error
- Type I rejects a true null hypothesis (false positive); Type II fails to reject a false null (false negative).
- Confidence interval meaning
- A range that, for a given confidence level (e.g., 95%), is expected to contain the true population parameter.
- Mode
- The most frequently occurring value in a data set.
- Sample space
- The set of all possible outcomes of a random experiment.
- Time value of money principle
- A dollar today is worth more than a dollar in the future because money can earn a return over time.
- Future value (single payment)
- F = P(1 + i)ⁿ, where P is present value, i is the interest rate per period, and n is the number of periods.
- Present value (single payment)
- P = F / (1 + i)ⁿ — discount a future amount back to today.
- Simple vs compound interest
- Simple interest is earned only on the principal. Compound interest is earned on principal plus accumulated interest.
- Nominal vs effective interest rate
- Nominal is the stated annual rate. Effective accounts for compounding within the year: i_eff = (1 + r/m) raised to the m-th power, minus 1.
- Net present value (NPV) decision rule
- Accept a project if its NPV (sum of discounted cash flows minus initial cost) is positive.
- Internal rate of return (IRR)
- The discount rate that makes a project's NPV equal to zero; compare it to the minimum acceptable rate (MARR).
- Annuity
- A series of equal payments made at regular intervals.
- Capitalized cost
- The present value of an asset assumed to last forever: P = A / i, where A is the annual cost and i the interest rate.
- Straight-line depreciation
- Equal annual depreciation = (initial cost − salvage value) / useful life.
- Sunk cost
- A past cost that has already been incurred and cannot be recovered; it should be ignored in current decisions.
- MARR
- Minimum Attractive (Acceptable) Rate of Return — the lowest return a project must earn to be worthwhile.
- Benefit-cost ratio decision rule
- A project is economically justified when the benefit-cost ratio is greater than or equal to 1.
- Inflation effect on interest
- Real rate ≈ nominal rate − inflation rate. Inflation erodes the purchasing power of future cash flows.
- Payback period
- The time required for a project's cumulative cash inflows to recover its initial investment; ignores the time value of money.
- A/P (capital recovery) factor purpose
- Converts a present amount into an equivalent uniform series of annual payments over n periods.
- Engineer's paramount obligation (NSPE Code)
- To hold paramount the safety, health, and welfare of the public.
- When the Code and a less-strict law conflict
- The engineer must meet the higher standard — protecting the public can require action beyond the legal minimum.
- Practicing only in areas of competence
- Engineers must perform services only in their areas of competence and not sign or seal work outside their expertise.
- Conflict of interest duty
- Disclose all known or potential conflicts of interest to clients or employers promptly and in writing.
- Confidentiality vs public safety
- Confidentiality yields to the duty to protect the public; a danger to public safety can require disclosure to the proper authority.
- Acting as a faithful agent
- Engineers must act for each employer or client as faithful agents or trustees, avoiding deception.
- Truthful public statements
- Engineers must be objective and truthful in professional reports, statements, and testimony, including all relevant information.
- Accepting gifts or bribes
- Engineers must not solicit or accept gratuities, directly or indirectly, to influence their professional judgment.
- Credit for engineering work
- Engineers must give credit for work to those to whom credit is due and recognize the proprietary interests of others.
- Whistleblowing duty
- If a judgment is overruled where public safety is endangered, the engineer must notify the proper authorities and may withdraw.
- Continuing competence
- Engineers should continue professional development throughout their careers and keep current in their field.
- Plan stamping / sealing rule
- Engineers may seal only work prepared by them or under their responsible charge; sealing others' work is unethical and illegal.
- Sustainability and environment
- Engineers are encouraged to adhere to principles of sustainable development to protect the environment for future generations.
- Purpose of professional licensure (PE)
- To protect the public by ensuring only qualified individuals may offer engineering services to the public.
- Newton's first law
- A body at rest stays at rest, and a body in motion stays in motion, unless acted on by a net external force (inertia).
- Moment of a force
- M = F × d, where d is the perpendicular distance from the pivot to the line of action of the force.
- Couple
- Two equal, opposite, parallel forces that produce a pure moment (no net force) independent of the reference point.
- Free-body diagram
- A sketch of a body isolated from its surroundings showing all external forces and moments acting on it.
- Reactions at a pin support
- Two force components (horizontal and vertical); a pin resists translation but allows rotation, so no moment reaction.
- Reaction at a roller support
- A single force perpendicular to the rolling surface.
- Fixed (built-in) support reactions
- Two force components plus a moment — it resists translation in both directions and rotation.
- Two-force member
- A member loaded at only two points carries force only along the line joining those points (pure tension or compression).
- Method of joints (trusses)
- Apply ΣFx = 0 and ΣFy = 0 at each joint to solve for member forces; start at a joint with two unknowns.
- Method of sections (trusses)
- Cut through the members of interest and apply equilibrium to one portion to find specific member forces directly.
- Centroid
- The geometric center of an area or volume — where the first moment of area equals zero.
- Static friction force
- F ≤ μₛN. It opposes impending motion and reaches a maximum of μₛN just before sliding starts.
- Kinetic vs static friction
- Kinetic friction (object sliding) is usually less than the maximum static friction (object about to slide).
- Distributed load resultant
- Replaced by a single force equal to the area under the load curve, acting at the centroid of that area.
- Zero-force member
- A truss member carrying no load; common at unloaded joints with two collinear members and one non-collinear member.
- Moment of inertia (area)
- A geometric property describing resistance to bending; depends on cross-sectional shape and distance from the neutral axis.
- Parallel axis theorem
- I = I_c + Ad², shifting a moment of inertia from the centroidal axis to a parallel axis a distance d away.
- Newton's second law
- F = ma. Net force equals mass times acceleration.
- Difference between kinematics and kinetics
- Kinematics describes motion (position, velocity, acceleration) without forces; kinetics relates motion to the forces causing it.
- Velocity from constant acceleration
- v = v₀ + at.
- Position from constant acceleration
- s = s₀ + v₀t + ½at².
- Velocity-displacement equation (constant a)
- v² = v₀² + 2a(s − s₀).
- Projectile motion key idea
- Horizontal and vertical motions are independent; horizontal velocity is constant while vertical motion accelerates at g.
- Centripetal acceleration
- a = v² / r, directed toward the center of the circular path.
- Kinetic energy
- KE = ½mv².
- Gravitational potential energy
- PE = mgh, relative to a chosen reference height.
- Work-energy principle
- The net work done on a body equals its change in kinetic energy: W = ΔKE.
- Linear momentum
- p = mv. A vector quantity conserved when no net external force acts.
- Impulse-momentum theorem
- Impulse (force × time) equals the change in momentum: FΔt = Δ(mv).
- Conservation of momentum
- In a collision with no external forces, total momentum before equals total momentum after.
- Elastic vs inelastic collision
- Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve momentum but lose kinetic energy.
- Coefficient of restitution
- The ratio of relative separation velocity to relative approach velocity; 1 for perfectly elastic, 0 for perfectly plastic.
- Angular velocity (ω) relation to linear velocity
- v = ωr for a point at radius r on a rotating body.
- Rotational analog of F = ma
- Torque τ = Iα, where I is the mass moment of inertia and α is angular acceleration.
- Simple harmonic motion frequency (spring-mass)
- Angular frequency ω = √(k/m); period T = 2π√(m/k).
- Conservative force
- A force whose work is path-independent (e.g., gravity, springs); energy is conserved when only conservative forces act.
- Normal stress
- σ = P / A — force divided by the cross-sectional area resisting it.
- Normal strain
- ε = ΔL / L — the change in length divided by the original length (dimensionless).
- Hooke's law
- σ = Eε. Within the elastic region, stress is proportional to strain; E is the modulus of elasticity.
- Modulus of elasticity (Young's modulus)
- The slope of the linear elastic portion of the stress-strain curve; a measure of stiffness.
- Shear stress
- τ = V / A — force acting parallel to a surface divided by the area.
- Poisson's ratio
- The ratio of lateral strain to axial strain (typically 0.25–0.35 for metals); describes transverse contraction under axial load.
- Yield strength
- The stress at which a material begins to deform permanently (plastically).
- Ultimate tensile strength
- The maximum stress a material can withstand before failure.
- Elastic vs plastic deformation
- Elastic deformation is recoverable when the load is removed; plastic deformation is permanent.
- Axial deformation formula
- δ = PL / (AE) — elongation of a bar under axial load.
- Bending (flexure) stress
- σ = Mc / I, where M is the bending moment, c the distance to the outer fiber, and I the moment of inertia.
- Torsional shear stress (circular shaft)
- τ = Tr / J, where T is torque, r the radius, and J the polar moment of inertia.
- Thermal strain
- ε = αΔT, where α is the coefficient of thermal expansion and ΔT the temperature change.
- Factor of safety
- The ratio of a material's failure (or yield) strength to the actual applied stress.
- Euler buckling load
- P_cr = π²EI / (KL)², the critical axial load that buckles a slender column; K depends on end conditions.
- Stress concentration
- A local rise in stress at a geometric discontinuity such as a hole, notch, or fillet.
- Ductile vs brittle failure
- Ductile materials deform significantly before fracture (warning); brittle materials fracture suddenly with little deformation.
- Mohr's circle use
- A graphical method to find principal stresses and maximum shear stress from a known stress state.
- Neutral axis in bending
- The line in a beam's cross-section where bending stress is zero; it passes through the centroid for symmetric sections.
- Phase diagram
- A map showing the stable phases of a material as a function of temperature and composition.
- Ferrous vs nonferrous metals
- Ferrous metals contain iron (steel, cast iron); nonferrous do not (aluminum, copper, titanium).
- Effect of carbon content on steel
- Higher carbon increases hardness and strength but reduces ductility and weldability.
- Annealing
- A heat treatment that softens metal, relieves internal stresses, and improves ductility by slow cooling.
- Quenching and tempering
- Rapid cooling (quench) hardens steel; reheating (temper) restores some toughness and reduces brittleness.
- Alloy
- A material made by combining a metal with one or more elements to improve properties such as strength or corrosion resistance.
- Hardness
- A material's resistance to localized plastic deformation, such as indentation or scratching.
- Toughness
- The ability to absorb energy and deform plastically before fracturing; the area under the stress-strain curve.
- Fatigue failure
- Failure under repeated cyclic loading at stresses below the static strength; cracks initiate and propagate over many cycles.
- Endurance (fatigue) limit
- The stress amplitude below which some materials (notably steel) can endure essentially infinite load cycles.
- Creep
- Slow, permanent deformation under constant load over time, accelerated by high temperature.
- Corrosion
- The gradual degradation of a material, usually metal, by chemical or electrochemical reaction with its environment.
- Galvanic corrosion
- Accelerated corrosion when two dissimilar metals are in electrical contact in an electrolyte; the more active metal corrodes.
- Grain size effect on strength
- Smaller grains generally increase strength and hardness (Hall-Petch relationship).
- Polymer vs ceramic vs metal
- Metals are ductile conductors; ceramics are hard, brittle, heat-resistant insulators; polymers are light, flexible, low-melting.
- Composite material
- A material combining two or more constituents (e.g., fiber + matrix) to achieve properties neither could alone.
- Fluid density vs specific weight
- Density ρ = mass/volume. Specific weight γ = ρg = weight/volume.
- Viscosity
- A fluid's resistance to shear or flow; the proportionality between shear stress and velocity gradient (τ = μ du/dy).
- Hydrostatic pressure with depth
- P = ρgh. Gauge pressure increases linearly with depth in a static fluid.
- Pascal's principle
- A pressure change applied to an enclosed fluid is transmitted undiminished to every point in the fluid.
- Buoyant force (Archimedes' principle)
- F_b = ρ_fluid × g × V_displaced — equal to the weight of the fluid displaced.
- Continuity equation
- For incompressible flow, A₁V₁ = A₂V₂; flow rate is conserved, so velocity increases where area decreases.
- Bernoulli's equation
- Along a streamline, P/ρ + V²/2 + gz = constant — pressure, kinetic, and potential energy trade off in ideal flow.
- Reynolds number
- Re = ρVD/μ. A dimensionless ratio of inertial to viscous forces that predicts laminar vs turbulent flow.
- Laminar vs turbulent flow (pipe)
- Laminar (smooth, layered) for Re below ~2300; turbulent (chaotic, mixing) for Re above ~4000.
- Volumetric flow rate
- Q = AV — cross-sectional area times average velocity.
- Mass flow rate
- ṁ = ρAV = ρQ.
- Head loss in a pipe
- Energy lost to friction, found with the Darcy-Weisbach equation h_f = f(L/D)(V²/2g).
- Manometer principle
- Measures pressure difference from the height difference of a liquid column: ΔP = ρg Δh.
- Ideal vs real fluid
- An ideal fluid has no viscosity and no friction losses; real fluids have viscosity and dissipate energy.
- Pump power
- P = ρgQH / η, where H is the head added and η the pump efficiency.
- Specific gravity
- The ratio of a substance's density to the density of water (1000 kg/m³); dimensionless.
- Stagnation (total) pressure
- The pressure when a flowing fluid is brought to rest; static pressure plus dynamic pressure (½ρV²).
- Zeroth law of thermodynamics
- If two systems are each in thermal equilibrium with a third, they are in equilibrium with each other — the basis of temperature.
- Second law of thermodynamics
- The total entropy of an isolated system never decreases; heat flows spontaneously from hot to cold.
- Entropy
- A measure of a system's disorder or unavailable energy; it increases in any real (irreversible) process.
- Carnot efficiency
- η = 1 − T_cold/T_hot (temperatures in kelvin) — the maximum possible efficiency of a heat engine between two reservoirs.
- Why use kelvin in Carnot efficiency?
- Carnot efficiency uses absolute temperature; only the kelvin scale starts at absolute zero, so ratios are physically meaningful.
- Enthalpy
- H = U + PV. A property combining internal energy with flow work; useful for constant-pressure processes.
- Ideal gas law
- PV = nRT (or PV = mRT). Relates pressure, volume, and absolute temperature for an ideal gas.
- Specific heat
- The energy required to raise the temperature of a unit mass by one degree: Q = mcΔT.
- cp vs cv
- cp is specific heat at constant pressure; cv at constant volume. For ideal gases cp − cv = R, and cp > cv.
- Isothermal vs adiabatic process
- Isothermal: constant temperature (heat exchanged). Adiabatic: no heat transfer (Q = 0).
- Latent heat
- Energy absorbed or released during a phase change (melting, vaporization) at constant temperature.
- Heat engine
- A device that converts heat into work by operating in a cycle between a hot and a cold reservoir.
- Coefficient of performance (COP)
- For refrigerators/heat pumps, the ratio of useful heat moved to the work input; can exceed 1.
- Three modes of heat transfer
- Conduction (through a solid), convection (fluid motion), and radiation (electromagnetic waves).
- Fourier's law of conduction
- q = −kA(dT/dx) — heat flows down the temperature gradient; k is thermal conductivity.
- Newton's law of cooling (convection)
- q = hA(T_surface − T_fluid), where h is the convective heat transfer coefficient.
- Stefan-Boltzmann law (radiation)
- Radiated power ∝ εσAT⁴; emitted energy rises with the fourth power of absolute temperature.
- Thermal conductivity meaning
- A material property describing how readily it conducts heat; metals are high, insulators low.
- Electric current
- The rate of flow of electric charge: I = Q/t, measured in amperes (coulombs per second).
- Voltage (potential difference)
- The energy per unit charge between two points: V = energy/charge, measured in volts (joules per coulomb).
- Electrical power
- P = VI = I²R = V²/R, measured in watts.
- Resistors in series
- Add directly: R_total = R₁ + R₂ + … The same current flows through each.
- Resistors in parallel
- 1/R_total = 1/R₁ + 1/R₂ + … The same voltage appears across each; total resistance is less than the smallest.
- Kirchhoff's current law (KCL)
- The sum of currents entering a node equals the sum leaving it (charge is conserved).
- Kirchhoff's voltage law (KVL)
- The sum of voltage rises and drops around any closed loop equals zero (energy is conserved).
- Capacitor charge relation
- Q = CV. A capacitor stores energy in an electric field; energy = ½CV².
- Inductor behavior
- Stores energy in a magnetic field and opposes changes in current: V = L(di/dt); energy = ½LI².
- Capacitive reactance
- X_C = 1/(2πfC). It decreases as frequency increases; a capacitor blocks DC, passes high-frequency AC.
- Inductive reactance
- X_L = 2πfL. It increases with frequency; an inductor passes DC, opposes high-frequency AC.
- RMS value of a sinusoid
- V_rms = V_peak / √2 ≈ 0.707 × V_peak — the equivalent DC value delivering the same power.
- Power factor
- The cosine of the phase angle between voltage and current; the ratio of real power to apparent power.
- Impedance
- The total opposition to AC current, combining resistance and reactance: Z = √(R² + X²).
- Real vs reactive vs apparent power
- Real (W) does useful work; reactive (VAR) oscillates in reactive elements; apparent (VA) is their vector sum.
- Energy stored vs dissipated elements
- Resistors dissipate energy as heat; capacitors and inductors store and return it.
- Ideal transformer voltage relation
- V₁/V₂ = N₁/N₂ — the voltage ratio equals the turns ratio.
- Mole
- The SI unit for amount of substance; one mole contains Avogadro's number (6.022 × 10²³) of particles.
- Stoichiometry
- The study of the quantitative relationships between reactants and products in a chemical reaction.
- Balancing a chemical equation
- Adjust coefficients so each element has equal atoms on both sides — conservation of mass.
- pH scale
- pH = −log[H⁺]. Below 7 is acidic, 7 is neutral, above 7 is basic; each unit is a tenfold change.
- Acid vs base
- An acid donates protons (H⁺) or accepts electrons; a base accepts protons or donates hydroxide (OH⁻).
- Molarity
- Concentration in moles of solute per liter of solution (mol/L).
- Ideal gas law (chemistry form)
- PV = nRT, relating pressure, volume, moles, and absolute temperature of a gas.
- Exothermic vs endothermic
- Exothermic reactions release heat (negative ΔH); endothermic reactions absorb heat (positive ΔH).
- Oxidation vs reduction
- Oxidation is loss of electrons; reduction is gain of electrons (OIL RIG).
- Limiting reactant
- The reactant that is fully consumed first, determining the maximum amount of product formed.
- Ionic vs covalent bond
- Ionic bonds transfer electrons between metal and nonmetal; covalent bonds share electrons between nonmetals.
- Catalyst
- A substance that speeds a reaction by lowering activation energy and is not consumed in the process.
- Atomic number vs mass number
- Atomic number = number of protons (defines the element); mass number = protons + neutrons.
- Avogadro's number
- 6.022 × 10²³ — the number of particles in one mole of a substance.
- Conservation of mass in reactions
- Matter is neither created nor destroyed; the total mass of reactants equals the total mass of products.
