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Yes. Flashcards use active recall — retrieving an answer from memory — which research shows is one of the most effective ways to make information stick, especially for the many formulas, laws, and definitions the FE exam tests across a dozen topics.

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The broadly-shared engineering core: Mathematics, Probability and Statistics, Engineering Economics, Ethics and Professional Practice, Statics, Dynamics, Strength of Materials, Materials Science, Fluid Mechanics, Thermodynamics and Heat Transfer, Electrical Engineering, and Chemistry.

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Mix the modes: flip to learn, type to test recall, match for speed, and quiz to check yourself. Start early and review daily, and focus on the largest areas — fluid mechanics, statics, dynamics, mechanics of materials, and thermodynamics.

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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).

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.

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.

eˣ. The exponential function is its own derivative.

d/dx f(g(x)) = f′(g(x)) · g′(x). Differentiate the outer function, then multiply by the derivative of the inner function.

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.

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.

c² = a² + b² − 2ab·cos C. Generalizes the Pythagorean theorem to any triangle.

e raised to the power iθ equals cos θ + i·sin θ. It links the exponential and trigonometric functions for complex numbers.

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 (∇·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 is the arithmetic average (sum ÷ count). Median is the middle value when data are ordered; it resists outliers.

The variance's positive root (√variance); a measure of how spread out data are around the mean.

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).

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.

(x − mean) / standard deviation. The number of standard deviations a value lies from the mean.

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FE Exam Flashcards

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.

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By Career EmployerUpdated Sep 12, 2026

200+ free FE Exam flashcards — 4 ways to study

FE Exam Flashcard of the DayNew daily

The classic card. Do you know this one?

Click Study Flashcards above to open the flashcard hub — 200+ FE exam cards you can flip, match, type, or quiz yourself on. Every card covers the shared engineering core the FE tests, so you study exactly what the exam asks.^[1] Pair them with our free practice test and study guide.

FE Exam Flashcard Study Modes

Flip mode is the plain study pass: read a front, recall it, turn the card. Type mode makes you produce the term from its definition, so Hooke’s law has to come from memory rather than recognition. Match races you to pair fronts and backs under time, and Quiz builds multiple-choice questions from the same 224 cards.

Free FE exam flashcards from Career Employer — active recall for the shared engineering core

Why Flashcards Work for the FE Exam

Mathematics is the biggest block in the deck at 31 cards, and it drills the calculus and trigonometry identities that show up inside every other subject. Chain rule, Product rule, and Quotient rule sit next to Law of cosines and Euler’s formula, along with the card that asks sin²θ + cos²θ = ? and the one for Integral of 1/x dx. Probability and Statistics follows with 21 cards on the descriptive and inferential vocabulary the exam assumes you already own, including Z-score, Standard deviation, and Type I vs Type II error.

Three engineering-science domains carry 19 cards each. Thermodynamics and Heat Transfer covers property and cycle terms such as Entropy, Enthalpy, and Carnot efficiency. Dynamics works through motion and energy relationships with Work-energy principle, Impulse-momentum theorem, and Centripetal acceleration. Strength of Materials handles stress-strain definitions and design language, including Hooke’s law, Poisson’s ratio, and Factor of safety.

Statics, Electrical Engineering, and Fluid Mechanics each hold 18 cards. The Statics set drills equilibrium reasoning through Free-body diagram, Zero-force member, and Two-force member. Electrical Engineering covers circuit and AC fundamentals like Ohm’s law, Power factor, and Inductive reactance. Fluid Mechanics runs from Reynolds number and Continuity equation to Head loss in a pipe.

Engineering Economics contributes 16 cards on cash-flow and decision terms such as MARR, Capitalized cost, and Straight-line depreciation. Materials Science also has 16 cards, covering structure and failure vocabulary including Annealing, Fatigue failure, and Phase diagram.

Chemistry closes out the science side with 15 cards on Molarity, Limiting reactant, and Avogadro’s number. Ethics and Professional Practice adds 14 cards on licensure conduct, including Acting as a faithful agent, Conflict of interest duty, and Plan stamping / sealing rule.

That matters for the FE exam, which is dense with formulas (the time value of money, F = ma, σ = Eε, Bernoulli’s equation) and rules that reward repetition. Used alongside our practice test and study guide, flashcards turn review time into measurable progress.^[3]

FE Exam Flashcards by Topic

The cards are organized by the shared engineering core the FE disciplines test. Weight your study toward the largest areas — fluid mechanics, statics, dynamics, mechanics of materials, and thermodynamics:^[2]

FE exam flashcards by topic (2026)
FE core topic	What it covers
Mathematics	Calculus, differential equations, linear algebra, vectors
Probability & Statistics	Descriptive stats, probability rules, distributions
Engineering Economics	Time value of money, NPV, IRR, depreciation
Ethics & Professional Practice	Codes of ethics, public safety, licensure
Statics	Equilibrium, trusses, friction, moment of inertia
Dynamics	Kinematics, kinetics, energy and momentum
Strength of Materials	Stress, strain, bending, torsion, buckling
Materials Science	Phase diagrams, properties, fatigue and corrosion
Fluid Mechanics	Pressure, continuity, Bernoulli, Reynolds number
Thermodynamics & Heat Transfer	The laws, ideal gas, Carnot, heat transfer modes
Electrical Engineering	Ohm's & Kirchhoff's laws, circuits, AC reactance
Chemistry	Stoichiometry, pH, bonding, reactions

How to Get the Most Out of These Flashcards

Start with Mathematics. At 31 cards it is the largest domain and its rules feed Dynamics, Fluid Mechanics, and Thermodynamics and Heat Transfer, so shaky calculus costs you points twice.
Type-drill the definitions you confuse. Force yourself to produce Poisson’s ratio and Type I vs Type II error from the definition side, since recognition alone hides the gaps.
Use Match for short label cards. Terms like Entropy, Viscosity, and Sunk cost pair quickly, which makes the timed game a good warm-up for Engineering Economics and Materials Science.
Move to the practice test once recall holds. When Quiz on Statics and Electrical Engineering stops surprising you, switch to full timed sets and use the study guide for the gaps they expose.
Rotate domains instead of cramming one. Two or three domains per session across a 224-card deck keeps Chemistry and Ethics and Professional Practice from becoming the sets you never review.

FE Exam Flashcards FAQ

Hundreds of free FE exam flashcards, organized across the shared engineering core the FE disciplines test — math, statistics, ethics, economics, statics, dynamics, materials, fluids, thermodynamics, and electrical. They're free to use with no account required.

References

1.NCEES. “FE Exam — Fundamentals of Engineering.” NCEES. ↑
2.NCEES. “FE Other Disciplines — CBT Exam Specifications.” NCEES. ↑
3.Institute of Education Sciences (U.S. Dept. of Education). “Organizing Instruction and Study to Improve Student Learning (Practice Guide).” What Works Clearinghouse, IES. ↑

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