Math & Numbers

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

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

Compute remaining radionuclide activity A(t) = A₀·2^(−t/T½) with λ = ln2/T½ in Bq/Ci/dpm units, for Tc-99m, F-18 and other isotopes.

Math & Numbers

Carrier-free specific activity from the half-life: A = ln2·N_A/T½ in Bq/mol and Ci/mmol, with optional Bq/g and Ci/g from molar mass.

Math & Numbers

Convert angular velocity (angular speed) between radian per second (rad/s, SI base), revolutions per minute (rpm = 2π/60 rad/s), degrees per second (deg/s = π/180 rad/s), and hertz (Hz, used as revolution per second = 2π rad/s). Converts via rad/s to the target unit and lists all four equivalents. Note: 1 Hz in angular-frequency context means one full revolution per second, so 1 Hz = 2π rad/s ≈ 6.283185307 rad/s. Reference: vinyl LP 33⅓ rpm ≈ 3.49 rad/s, car engine idle ~800 rpm ≈ 83.8 rad/s.

Math & Numbers

Convert illuminance between lux (lx = lm/m², SI base), kilolux (1 = 1000 lx), footcandle (1 = 10.7639104 lx), phot (1 = 10000 lx), and nox (1 = 0.001 lx). Converts via lux to the target unit and lists the equivalent value in all five units for reference. Reference values: full moon ≈ 0.25, office lighting ≈ 300–500, overcast day ≈ 1000, direct sunlight ≈ 100000 lux.

Math & Numbers

Convert sound power between linear units (W, mW, µW, pW) and the decibel sound power level dB Lw (Lw = 10·log₁₀(W / 1 pW), reference P₀ = 1 pW = 1e-12 W, per ISO 3744 / IEC 60027-3). Supports bidirectional linear↔logarithmic conversion and reports the equivalent value in all five units. Reference values: whisper ≈ 1e-9 W (Lw ≈ 30), speech ≈ 1e-5 W (Lw ≈ 70), jet engine at 1 m ≈ 1e4 W (Lw ≈ 160).

Math & Numbers

Convert sound pressure between linear units (Pa, µPa, µbar, mbar) and the decibel sound pressure level dB SPL (Lp = 20·log₁₀(p / 20 µPa), reference p₀ = 20 µPa = 2e-5 Pa RMS, per ISO 1683). Supports bidirectional linear↔logarithmic conversion. Calibration anchor: 1 Pa → 20·log₁₀(1/2e-5) ≈ 94 dB SPL. Reference values: threshold of hearing 20 µPa (0 dB), speech ≈ 0.02 Pa (60 dB), rock concert ≈ 20 Pa (120 dB), pain threshold ≈ 200 Pa (140 dB).

Math & Numbers

Uniform corrosion rate by weight loss per ASTM G1: CR (mm/a) = 87.6 × ΔW / (ρ × A × t), with ΔW in mg, ρ in g/cm³, A in cm², t in hours. Also reports mpy (mils/year = mm/a × 39.37) and an indicative NACE-style carbon-steel rating (excellent/good/fair/poor). Used for immersion tests and coupon exposure.

Math & Numbers

Convert convective heat transfer coefficient (film coefficient h) between W/(m²·K) (watt per square metre-kelvin, SI base, = W/(m²·°C)) and kcal/(m²·h·°C) (1 = 1.1627778 W/(m²·K)). Converts via W/(m²·K) and lists both equivalents. h depends on flow regime and surface, not material. Typical ranges: natural-convection air ≈ 5–25, forced air ≈ 25–250, still water ≈ 100–1000, boiling/condensing water ≈ 2500–50000+ W/(m²·K).

Math & Numbers

Convert specific heat capacity between J/(kg·K) (joule per kilogram-kelvin, SI base, = J/(kg·°C)), kcal/(kg·°C) (1 = 4186.8 J/(kg·K)), and BTU/(lb·°F) (1 = 4186.8 J/(kg·K)). Key identity: 1 kcal/(kg·°C) = 1 BTU/(lb·°F) — numerically equal because the energy/mass·temperature factors cancel. Converts via J/(kg·K) and lists all three equivalents. Reference values: water ≈ 4186, air cp ≈ 1005, aluminum ≈ 900, iron ≈ 450 J/(kg·K).

Math & Numbers

Specific strength = σ/ρ and specific modulus = E/ρ (optional E). Enter strength in MPa and density in g/cm³; output in N·m/kg (also as kN·m/kg). The figure of merit for lightweight design — a material with high strength but high density (e.g. tungsten) often loses to a lighter alloy. Typical values: Ti-6Al-4V ≈ 210 kN·m/kg, Al 7075 ≈ 180, carbon steel ≈ 50–90, CFRP ≈ 1000+.