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Supersonic Mach Cone Angle & Subsonic Prandtl-Glauert Rule
Calculate the Mach-wave half-angle for supersonic flow and a separate Prandtl-Glauert pressure-coefficient approximation. Distinguish weak waves, finite-strength shocks and the sonic limit.
Read the guide →EngineeringSupersonic Normal Shock Wave: Downstream Mach & Rankine-Hugoniot Jumps
Calculate supersonic normal shock downstream Mach number M_2 = SQRT((1 + ((g-1)/2)*M1^2) / (g*M1^2 - (g-1)/2)) and static pressure jump P2/P1 = 1 + (2*g/(g+1))*(M1^2 - 1) for compressible gas flows.
Read the guide →StatisticalSupply Chain Bullwhip Effect Ratio: order variance amplification & lead-time bound
Measure supply-chain order distortion via Bullwhip Variance Ratio VAR.S(Orders) / VAR.S(Demand), CV Ratio, and Chen-Simchi-Levi moving-average lower bound.
Read the guide →Precision Abrasive Machining, Surface/Creep-Feed Grinding & Malkin Grinding BurnSurface Grinding Specific Energy, MRR & Burn Power Formula
Calculate Volumetric Grinding Removal Rate Qw (mm³/s), Specific Removal Rate Per Wheel Width Q'w (mm³/(mm·s)), Grinding Spindle Power (kW), and Workpiece Heat-Partition Burn Load (kW).
Read the guide →FinancialSustainable Growth Rate (SGR) & Internal Growth Rate (IGR) corporate finance formulas
Explore constant-ratio financing models for growth, distinguishing beginning-equity ROE from ending-equity ROE and internal funding from constant-leverage funding.
Read the guide →EngineeringSutherland's Law for Gas Dynamic Viscosity vs Temperature
Calculate gas dynamic viscosity with Sutherland’s temperature correlation, using a documented air parameter set, raw outputs and guards for Kelvin temperatures and reference inputs.
Read the guide →Engineering & ScientificSutherland’s Law for Temperature-Dependent Gas Dynamic Viscosity
Model gas dynamic viscosity mu(T) across wide Kelvin temperature spans using Sutherland’s three-coefficient molecular-potential law for HVAC, flue-gas, and aerodynamic Reynolds numbers.
Read the guide →EngineeringSwamee-Jain Explicit Pipe Friction Factor formula (Moody Chart)
Estimate a Darcy pipe-friction factor with explicit Swamee–Jain arithmetic in a declared turbulent range, use 64/Re for laminar flow, and leave the transition region unresolved.
Read the guide →Engineering & ScienceSweet-Parker Magnetic Reconnection Rate, Lundquist Number (S) & Sheet Thickness
Calculate the dimensionless MHD Lundquist number S = (L * vA) / eta_m, Sweet-Parker reconnection inflow speed v_in = vA / SQRT(S), and resistive current-sheet half-thickness delta = L / SQRT(S).
Read the guide →TextSwitch "Last, First" names to "First Last" (and back)
Flip HR and payroll comma-formatted names ("Rivera, Elena M.") into natural "Elena M. Rivera" order while leaving non-comma names untouched.
Read the guide →Lookup & ReferenceSwitch Financial Scenarios (Bear, Base, Bull) with CHOOSECOLS & INDEX
Switch among three named cash-flow scenarios, select a full column with CHOOSECOLS, and expose each year with INDEX while validating the selector and selected data.
Read the guide →Logical checks and errorsSWITCH function: map exact status codes to readable labels
Evaluate a cell once against a list of exact code-to-label pairs, include a final default fallback for unexpected codes, and know when to use IFS instead.
Read the guide →Lookup & ReferenceSwitch Lookup Tables Dynamically with CHOOSE & INDEX / MATCH
Route a two-way INDEX/MATCH lookup across Retail, Wholesale, or Enterprise rate tables dynamically based on a selected tier code using CHOOSE.
Read the guide →High-Speed Multilayer PCB Stackup, IPC-2141 Symmetric Stripline & Signal IntegritySymmetric Stripline PCB Impedance Z0 & Delay Formula
Calculate Single-Side Dielectric Height h = (b - t)/2 (mm), IPC-2141 Symmetric Stripline Characteristic Impedance Z0 (ohms), Pure TEM Propagation Delay (ps/mm), and Line Capacitance (pF/cm).
Read the guide →Power Electronics & Voltage Regulator DesignSynchronous Buck Converter Ripple & CCM Formula
Calculate Step-Down Buck Converter Duty Cycle D = V_out / V_in, Peak-to-Peak Inductor Current Ripple (V_in - V_out)*D / (f_MHz * L_uH), Peak Switch Current, and Critical CCM Inductance.
Read the guide →Engineering & ScienceSynchrotron Radiation: Critical X-Ray Photon Energy & Energy Loss per Turn
Calculate bending-magnet synchrotron critical photon energy E_crit = 0.665 * E_GeV^2 * B_Tesla (in keV) and relativistic electron radiative energy loss per storage-ring revolution U0 = 88.46 * E_GeV^4 / rho_m (in keV/turn).
Read the guide →Healthcare & Clinical CalculatorsSYNTAX Score Coronary Complexity Tertiles (<= 22, 23–32, >= 33 PCI vs. CABG)
Sum angiographic lesion complexity weights across >= 50% stenoses in >= 1.5 mm coronary vessels and classify ESC/AHA SYNTAX tertiles guiding multivessel PCI vs. CABG.
Read the guide →Healthcare & ClinicalSystemic & Pulmonary Vascular Resistance (SVR & PVR in dyn·s·cm⁻⁵)
Calculate ICU hemodynamic Systemic Vascular Resistance SVR = 80 * (MAP - CVP) / CO and Pulmonary Vascular Resistance PVR = 80 * (mPAP - PCWP) / CO in dyn·s·cm⁻⁵ and Wood units.
Read the guide →Quality & Six Sigma SPCTabular CUSUM Control Chart Formula: C+ / C- Cumulative Sum & h-Sigma Shift Alarm
Detect sustained 0.5-sigma to 1.5-sigma process drifts 3x–4x faster than Shewhart 3-sigma charts using Page’s Two-Sided Tabular CUSUM recursion C_i+ = MAX(0, x_i - (mu_0 + k*sigma) + C_{i-1}+).
Read the guide →Quality Engineering & Lean ManufacturingTaguchi Quadratic Quality Loss Function Formula
Replace binary goalpost tolerance thinking with Genichi Taguchi’s quadratic loss L(y) = (A_0 / Delta_0^2)*(y - m)^2 and expected batch loss k * (sigma^2 + (mu - m)^2).
Read the guide →Distributed Systems & Performance EngineeringTail-at-Scale Fan-Out Latency & Hedged Request Formula
Calculate Jeffrey Dean & Luiz André Barroso’s (2013) microservice fan-out tail probability 1 - (1 - p)^K and prove how r-way hedged requests collapse p99 amplification to 1 - (1 - p^r)^K.
Read the guide →Queueing Theory, Computer Systems Latency & Operations ResearchTakagi M/G/1 Server-Vacation Queue Waiting-Time Formula
Calculate Server Utilization rho = lambda*E[S], Pollaczek-Khinchine Non-Vacation M/G/1 Waiting Time W_q0, Mean Residual Vacation Life E[V]*(1 + CV_V^2)/2, and Takagi / Fuhrmann-Cooper Decomposed Queue Delay.
Read the guide →Dynamic arraysTAKE function: slice the first N or last N rows from a table
Keep a contiguous block of rows or columns from the start or end of a range—or pair TAKE with SORT for clean Top-N summaries.
Read the guide →Healthcare & Clinical CalculatorsTalcott Risk Classification for Febrile Neutropenia (Groups I–IV Triage)
Classify chemotherapy-induced febrile neutropenia into Talcott Groups I–IV to identify Group IV low-risk outpatients (< 5% serious medical complication rate).
Read the guide →Mechanical Engineering & TribologyTallian Specific Film Thickness Lambda and Stribeck Regime Formula
Calculate composite RMS surface roughness SQRT(Rq_1^2 + Rq_2^2), Tallian’s specific oil-film ratio Lambda, and Stribeck lubrication regime (Boundary, Mixed, or Full Film).
Read the guide →MathTAN function: calculate mast or roof height from horizontal run and angle
Multiply horizontal ground distance by TAN(RADIANS(angle)) to find vertical height, and guard 90-degree vertical asymptotes.
Read the guide →Bioprocess Engineering & Downstream PurificationTangential-Flow Diafiltration Diavolume Formula
Calculate Constant-Volume Diafiltration Residual Impurity C_end = C_0 * EXP(-N_dv * S), Required Diavolumes LN(C_0 / C_target) / S with Non-Ideal Sieving S, Buffer Volume, and Spec Gate.
Read the guide →Biostatistics & Survival AnalysisTarone-Ware and Gehan-Breslow Weighted Log-Rank Survival Formula
Compare standard Mantel-Haenszel (w=1), Tarone-Ware (w=SQRT(n_j)), and Gehan-Breslow (w=n_j) weighted log-rank Z statistics across discrete failure risk sets.
Read the guide →Statistical & Clinical AnalysisTarone’s Homogeneity Test Correction: Exact Asymptotic χ² Adjustment to Breslow-Day
Correct the conservative inefficiency of the Mantel-Haenszel Breslow-Day test using Robert Tarone’s (1985) score adjustment χ²_Tarone = χ²_BD - [Σ(a_k - E_k)]² / ΣVar_k.
Read the guide →Engineering & ScienceTauc Plot Optical Bandgap Calculator: Photon Energy & (alpha*h*nu)^n
Transform UV-Vis spectrophotometer wavelength (nm) and absorption coefficient alpha (cm^-1) into Tauc abscissa h*nu = 1239.842 / lambda_nm (eV) and direct ((alpha * h*nu)^2) vs. indirect (SQRT(alpha * h*nu)) ordinates.
Read the guide →Portfolio Management, Tax-Loss Harvesting & IRC Section 1091Tax-Loss Harvesting & IRC §1091 Partial Wash-Sale Formula
Calculate Realized Harvested Capital Loss ($), IRC Section 1091 Share-Matched Disallowed Wash-Sale Loss ($), Deductible Current-Year Loss ($), and Adjusted Replacement Lot Basis ($/sh).
Read the guide →FinancialTaylor Rule formula: central bank policy interest rate & ZLB floor
Explore a Taylor-style policy-rule benchmark with supplied neutral rate, inflation target and output gap, plus an optional zero-floor scenario.
Read the guide →CNC Machining & Manufacturing Process EngineeringTaylor Tool Life & Minimum-Cost Cutting Speed Formula
Calculate F.W. Taylor’s Cutting Tool Life T = (C / V)^(1/n), Gilbert’s Minimum-Cost Economic Tool Life T_opt = ((1 - n)/n) * (t_chg + C_edge / R_mach), and Optimal Cutting Speed V_opt.
Read the guide →CNC Machining Economics & Carbide Tool Wear, F.W. Taylor (1906) & Gilbert Minimum-Cost SpeedTaylor Tool-Life & Gilbert Minimum-Cost Cutting Speed
Calculate Predicted Carbide Insert Edge Life T (min), Gilbert Economic Minimum-Cost Tool Life Topt (min), Optimum Minimum-Cost Cutting Speed Vopt (m/min), and Overspeed Penalty (%).
Read the guide →Engineering & ScienceTaylor-Couette Rotating Annulus: Taylor Number & Toroidal Vortex Onset
Calculate the root Taylor Number Ta_1/2 = ((Omega * R_i * d) / nu) * SQRT(d / R_i) for concentric journal bearings and chemical Taylor-vortex reactors against G. I. Taylor’s 1923 critical instability threshold (Ta_1/2 >= 41.3).
Read the guide →Agency MBS Trading & Repo FinancingTBA Mortgage Dollar Roll Drop & Specialness Formula
Calculate Agency TBA Dollar Roll Price Drop (P_front - P_back), Breakeven Carry Drop, Net Roll Advantage Points, and Implied Dollar-Roll Financing Rate ("Trading Special").
Read the guide →EngineeringTCP Bandwidth-Delay Product (BDP) & Mathis Throughput Formula
Calculate bandwidth-delay product in decimal MB and explore a simplified Reno-style loss model, with explicit units and limits rather than a universal TCP speed prediction.
Read the guide →EngineeringTelescope Rayleigh Diffraction Limit, Dawes' Limit & Focal Ratio formulas
Calculate Airy-disk angular resolution in arcseconds theta = DEGREES(1.22 * lambda / D) * 3600, Dawes' empirical double-star limit 116 / D_mm, f-number, and magnification.
Read the guide →Process Thermal Engineering & Heat ExchangersTEMA Heat Exchanger Fouling & Dirty U-Value Formula
Calculate Outer-Area-Referenced Total Fouling Resistance R_f_tot = R_f_in * (D_o / D_i) + R_f_out, Service Dirty Overall Conductance U_dirty, Cleanliness Factor, and Surface Overdesign Fraction.
Read the guide →Engineering & ScienceTemkin & Dubinin-Radushkevich (D-R) Adsorption Isotherm Energy Formula
Distinguish physical micropore filling (< 8 kJ/mol) from ion-exchange / chemisorption (8–16+ kJ/mol) using Dubinin-Radushkevich mean free adsorption energy E = 1 / SQRT(2 * B_DR) and Temkin surface uptake q_e = ((R * T) / b_T) * LN(A_T * C_e).
Read the guide →Geotechnical Engineering, Cofferdam Dewatering, Terzaghi Quicksand & Piping HeaveTerzaghi Critical Hydraulic Gradient & Quicksand Formula
Calculate Terzaghi Critical Hydraulic Gradient i_cr = (Gs - 1)/(1 + e), Cofferdam Upward Exit Gradient dh/L, Piping Factor of Safety FS, and Sand-Boiling Heave Audit.
Read the guide →EngineeringTerzaghi Effective Soil Stress & Rankine Earth Pressure (Ka & Kp) formulas
Explore a simplified hydrostatic effective-stress calculation and Rankine coefficients for level cohesionless backfill.
Read the guide →EngineeringTerzaghi Shallow Foundation Bearing Capacity (q_u & Allowable q_all)
Calculate strip and square footing ultimate pressures from supplied bearing-capacity factors, then distinguish net and gross allowable shear pressures with an explicit safety-factor convention.
Read the guide →Text & CleaningTest Whether a Range Contains Any Substring from a Keyword List (COUNTIF)
Check whether any cell across an entire column or log range contains at least one keyword from a watchlist using wildcard COUNTIF checks, returning a clean TRUE/FALSE flag or total keyword hits.
Read the guide →Logical & ConditionalTest Whether a Range Contains One of Many Exact Values (Set Overlap Audit)
Determine whether an entire column or manifest range contains at least one exact code from a restricted or priority list using whole-cell equality or COUNTIF checks without partial-prefix false positives.
Read the guide →Dates & SchedulesTest Whether a Schedule Range Contains a Specific Date (Date & Timestamp Safe)
Check whether a target calendar date appears inside a holiday or blackout range using COUNTIFS whole-day bounds so fractional timestamps never cause missed matches.
Read the guide →Cleaning imported dataTEXTBEFORE and TEXTAFTER: extract variable-length text around delimiters
Pull city prefixes and trailing status tags from hyphenated codes of any length without nesting FIND inside LEFT or RIGHT, and handle rows missing a delimiter.
Read the guide →Apparel Sourcing, Textile Engineering & Commodity TradeTextile Yarn Tex, Denier, Ne & Moisture Regain Formula
Convert Direct Linear Density Tex (g/1,000 m) into Synthetic Filament Denier (9 * Tex) and Indirect English Cotton Count Ne (590.5 / Tex), and calculate ASTM/BISFA Commercial Moisture Regain Invoice Weight.
Read the guide →Cleaning imported dataTEXTJOIN function: combine optional address fields without double commas
Join street, optional unit, and city with a comma-space delimiter while automatically skipping genuinely blank cells—and check how zero and space-only cells behave.
Read the guide →Cleaning imported dataTEXTSPLIT function: split semicolon or comma strings into columns
Break a delimited import string across columns dynamically with TEXTSPLIT, and control whether consecutive delimiters keep blank slots or collapse.
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