A 20-section interactive reference covering the shared math and science foundation behind every engineering discipline. Includes differential and integral calculus, first- and second-order differential equations, linear algebra (matrices, determinants, eigenvalues), trigonometry and the unit circle, statics and dynamics (Newton's laws, trusses, kinematics, friction, vibrations), general chemistry (moles, balancing equations, stoichiometry, gases), probability and statistics, and numerical methods (root-finding, integration, ODE solvers).
All seven core STEM domains covered by this studio are included: Calculus (limits, differentiation rules, related rates, optimization, integration techniques, first- and second-order differential equations with the characteristic equation), Linear Algebra (vectors, matrix transformations, determinants, inverses, solving Ax = b, eigenvalues and modal analysis), Trigonometry (the unit circle, core identities, and engineering applications), Mechanics (Newton's three laws, free body diagrams, statics equilibrium and truss analysis, dynamics kinematics and projectile motion, friction, work-energy methods, and vibrations), Chemistry (the mole concept, balancing equations, stoichiometry, limiting reactants, molarity and dilution, the ideal gas law), Probability & Statistics (descriptive statistics, probability rules, binomial and normal distributions, z-scores, hypothesis testing), and Numerical Methods (bisection, Newton-Raphson, trapezoidal and Simpson's rule, Euler and Runge-Kutta ODE solvers, Gaussian elimination).
Use the Prev / Next buttons at the bottom, or press the arrow keys on your keyboard. Click the ☰ menu button in the top-right corner to open the table of contents and jump directly to any of the 20 sections. The pink progress bar at the top tracks your position through all sections. Scroll the mouse wheel to advance or go back.
Every engineering discipline on this site builds on this same base. A structural engineer's stiffness matrix, a controls engineer's state-space model, and a chemical engineer's material balance are all applications of the linear algebra, calculus, and chemistry covered here. Probability and statistics is tested on every single version of the NCEES FE exam regardless of discipline, and statics, dynamics, and differential equations underlie the mechanical, civil, and aerospace FE exams directly. Working through this handbook once builds the vocabulary that every other studio on this site assumes you already have.
Every section pairs a core formula with at least one fully worked numerical example, so you can check your own work step by step rather than just re-deriving a result from a bare equation. Section 20 collects every key formula, physical constant, and an FE exam topic-area alignment table in one place for fast pre-exam review. Pair any section with its matching calculator in the STEM Learning Studio — derivative & limit evaluator, matrix operations, quadratic solver, projectile motion, Newton's second law, simple harmonic motion, stoichiometry, molarity/dilution, and the trig identity & unit circle tool — to turn a formula into a live, checkable number.
20 sections spanning calculus (limits, derivatives, integrals, first- and second-order differential equations), linear algebra (matrices, determinants, eigenvalues), trigonometry (unit circle and identities), mechanics (Newton's laws, statics, dynamics, vibrations), chemistry (moles, stoichiometry, gases), probability & statistics (distributions, hypothesis testing), and numerical methods (root-finding, integration, ODE solvers) — the shared foundation behind every engineering discipline.
A matrix is singular when its determinant equals zero — it has no inverse, and the system of equations it represents has either no solution or infinitely many. In engineering terms, a singular stiffness or system matrix almost always signals an unsupported structure, a redundant constraint, or a mechanism that can move freely without resistance, which is why solvers flag it as an error before returning any displacement or voltage results.
The sign of the discriminant in the system's characteristic equation (or equivalently, its damping ratio ζ). A positive discriminant (ζ > 1) gives two real roots and an overdamped, non-oscillating response; a discriminant of exactly zero (ζ = 1) gives critical damping, the fastest possible return to equilibrium with no overshoot; a negative discriminant (ζ < 1) gives complex conjugate roots and an underdamped, decaying oscillation. The same math describes a mechanical mass-spring-damper and an electrical RLC circuit — only the symbols (m, c, k versus L, R, C) change.
NCEES treats probability and statistics as a foundational engineering competency because every discipline deals with variability, measurement uncertainty, and data-driven decisions — material strength scatter in structural engineering, defect rates in manufacturing, signal noise in electrical systems. Rather than being discipline-specific, it's part of the common engineering toolkit, which is why every version of the FE exam — mechanical, electrical, civil, chemical, and every other discipline — devotes a section to it.
Gaussian elimination solves systems of linear equations, Ax = b — displacements from a stiffness matrix, node voltages from a conductance matrix, any problem where every unknown appears only to the first power. Newton-Raphson (or bisection) solves a single nonlinear equation, f(x) = 0, where the unknown may appear inside a logarithm, an exponential, or raised to a power — a pipe-friction factor, a transcendental natural-frequency equation, or a nonlinear circuit element. If your unknowns form a coupled linear system, reach for elimination; if you have one equation that can't be rearranged to isolate the unknown algebraically, reach for a root-finder.
Disclaimer: This reference guide summarizes general engineering mathematics and science principles for educational purposes only. Formulas and worked examples are simplified for clarity — always consult a full textbook, your course materials, or the official NCEES FE Reference Handbook for exam preparation and design-critical calculations.