Numeric convergence tests for Σ aₙ: nth-term, ratio, root, p-series comparison, and integral test; alternating series classified by Leibniz with a sum estimate.

Math & Numbers

Set Operations Calculator (Union/Intersection/Difference)

Union, intersection, difference, symmetric difference, and Cartesian product of two comma-separated sets, with cardinalities and capped pair display.

Math & Numbers

Runge-Kutta RK4 Solver (dy/dx = f(x, y))

Classical fourth-order Runge-Kutta for dy/dx = f(x, y): step table, k1–k4 stage details, final estimate, and optional exact-solution error column.

Math & Numbers

Second-Order ODE Solver (Characteristic Equation)

Solve a·y'' + b·y' + c·y = f(x): characteristic roots, three homogeneous cases, undetermined coefficients with resonance, and C1/C2 from initial conditions.

Math & Numbers

Safety Factor Calculator

Calculate the static safety factor n = σ_limit / σ_applied against yielding (σ_y) or ultimate tensile strength (σ_uts). Built-in typical strengths for carbon/alloy steel, aluminum, copper, austenitic stainless and titanium, plus a custom σ_limit option. Assessment: n<1 failure, 1≤n<1.5 marginal, n≥1.5 safe.

Math & Numbers

Sensible / Latent Heat Split (SHR = Sensible / Total)

Split an air-conditioning load into sensible and latent components. Total cooling capacity Qt = ṁ_da·(h1 − h2); sensible capacity Qs = ṁ_da·cp_ma·(T1 − T2) with cp_ma ≈ 1.006 + 1.86·W [kJ/(kg da·K)]; latent capacity Ql = Qt − Qs; Sensible Heat Ratio SHR = Qs / Qt. Entering/leaving states are described by dry-bulb temperature T and humidity ratio W; enthalpy h = 1.006·T + W·(2501 + 1.86·T) [kJ/kg da]. Three modes: full air-state split (T1,W1)→(T2,W2), from Qt & Qs (solve SHR + Ql), or from Qt & SHR (solve Qs, Ql).

Math & Numbers

Three-Phase Short-Circuit Current Calculator (IEC 60909)

Estimate the three-phase symmetric short-circuit current using the IEC 60909 far-from-generator method: I"_k = c·U_n/(√3·|Z|) and i_p = κ·√2·I"_k with κ = 1.02 + 0.98·e^(−3R/X). Source impedance is the series sum of transformer and line R/X.

Math & Numbers

Section Modulus Calculator

Calculate the elastic section modulus W = I/y_max and associated geometric properties. Supports rectangle, solid circle, hollow tube, and I-beam; computes I, distance c, area A, and radius of gyration r. Units in mm.

Math & Numbers

Shear Stress Calculator

Calculate the maximum shear stress τ_max = V·Q/(I·b) at the neutral axis of a beam section. Supports rectangle (1.5V/A), solid circle (4V/3A), hollow tube, and I-beam (exact V·Q_web/(I·tw)). Also returns the average shear stress V/A for comparison.

Math & Numbers

Shaft Critical Speed Calculator

Calculate the critical (resonance) speed of a rotating shaft using the single-DOF rotor model. ω_n = √(k/m), n_cr = (60/2π)·√(k/m) rpm. Also returns the natural frequency f_n in Hz.