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๐Ÿงฎ Math & Engineering Core Matrix

Interactive Universal Formula Dashboard & Computational Engine

๐Ÿ”น Basic Core Formulas & Calculators

Circle Geometry

Basic Math
\[A = \pi r^2 \quad \text{and} \quad C = 2\pi r\]

Variable & SI Unit Definitions:

  • A: Area โ€” Square Meters (\(\text{m}^2\)) | Spatial 2D surface metric.
  • C: Circumference โ€” Meters (\(\text{m}\)) | Linear perimeter path.
  • r: Radius โ€” Meters (\(\text{m}\)) | Center to boundary distance.

Enter the radius to compute total Area (\(A\)) and Circumference (\(C\)).

Output awaits input...

Ohm's Law (DC Circuits)

Electrical
\[V = I \times R\]

Variable & SI Unit Definitions:

  • V: Voltage โ€” Volt (\(\text{V}\)) | Electrical potential difference.
  • I: Current โ€” Ampere (\(\text{A}\)) | Direct current (DC) flow rate.
  • R: Resistance โ€” Ohm (\(\Omega\)) | Opposition to current flow.

Leave the field blank that you wish to solve for. Fill the other two.

Output awaits input...

AC Single-Phase Power

Electrical
\[P = V \times I \times \cos(\theta)\]

Variable & SI Unit Definitions:

  • P: Active Power โ€” Watt (\(\text{W}\)) | Real work-producing active electrical power.
  • V: Voltage โ€” Volt (\(\text{V}\)) | Root-Mean-Square (RMS) alternative voltage.
  • I: Current โ€” Ampere (\(\text{A}\)) | Root-Mean-Square (RMS) alternative current.
  • \(\cos(\theta)\): Power Factor โ€” Dimensionless (\(\text{pf}\)) | Ratio of real to apparent power.

Calculate True Power (Watts) and Apparent Power (VA) using the Power Factor.

Output awaits input...

Kinetic Energy

Mechanics
\[E_k = \frac{1}{2} m v^2\]

Variable & SI Unit Definitions:

  • \(E_k\): Kinetic Energy โ€” Joule (\(\text{J}\)) | Work energy due to physical motion.
  • m: Mass โ€” Kilogram (\(\text{kg}\)) | Total structural quantity of matter.
  • v: Velocity โ€” Meters per Second (\(\text{m/s}\)) | Speed rate of spatial displacement.

Calculate total dynamic kinetic energy stored within a moving physical mass velocity profile.

Output awaits input...

Fluid Density

Mechanics
\[\rho = \frac{m}{V}\]

Variable & SI Unit Definitions:

  • \(\rho\): Density โ€” Kilograms per Cubic Meter (\(\text{kg/m}^3\)) | Volumetric mass concentration.
  • m: Total Mass โ€” Kilogram (\(\text{kg}\)) | Quantity weight definition of the substance.
  • V: Total Volume โ€” Cubic Meter (\(\text{m}^3\)) | Three-dimensional spatial footprint filled.

Find the concentration volumetric density (\(\rho\)) of a uniform body structural compound.

Output awaits input...

AC Three-Phase Power

Advanced Electrical
\[P = \sqrt{3} \times V_L \times I_L \times \cos(\theta)\]

Variable & SI Unit Definitions:

  • P: Active Power โ€” Watt (\(\text{W}\)) / Kilowatt (\(\text{kW}\)) | Total balanced line output.
  • V_L: Line-to-Line Voltage โ€” Volt (\(\text{V}\)) | RMS electrical potential between any two phases.
  • I_L: Line Current โ€” Ampere (\(\text{A}\)) | RMS current vector per line.
  • \(\cos(\theta)\): Power Factor โ€” Dimensionless | Operating efficiency scalar.

Compute true active power drawn by balanced three-phase systems.

Output awaits input...

Motor Load Analysis

Industrial Machinery
\[I = \frac{\text{HP} \times 746}{\text{Volts} \times \text{PF} \times \text{Eff} \times \text{Phase Factor}}\]

Variable & SI Unit Definitions:

  • HP: Motor Rating โ€” Horsepower (\(1\text{ HP} = 746\text{ W}\)) | Mechanical output power.
  • Volts: Operating Voltage โ€” Volt (\(\text{V}\)) | Input line rating.
  • Eff / PF: Efficiency / Power Factor โ€” Decimals (0.01 to 1.00) | Loss metrics.

Determine active full-load operating current drawn by single or three-phase configurations.

Output awaits input...

โšก Advanced Engineering & Physics Matrices

Building Infrastructure & Fault Analytics

Power Distribution
\[I_{sc} = \frac{I_{FLA\_XFMR}}{Z_{\%}} \times 100 \quad \text{and} \quad I_{\text{breaking}} \rightarrow \text{Standard Tier [kA]}\]

Variable & SI Unit Definitions:

  • Connected Load: Kilowatt (\(\text{kW}\)) | Total combined power structural footprint.
  • Main Breaker Selection: Ampere (\(\text{A}\)) | Continuous overcurrent trip protection.
  • Short Circuit Current: Kiloampere (\(\text{kA}\)) | Ultimate breaking threshold capacity.
  • Transformer Impedance (\(Z_{\%}\)): Percentage | Internal voltage drop metric under full short circuit.

Enter aggregate building load parameters to scale infrastructure, breakers, cables, and interrupting thresholds.

Output awaits input...

Capacitor Rating Required (PFC)

Advanced Electrical
\[Q_C = P \times (\tan\theta_1 - \tan\theta_2)\]

Variable & SI Unit Definitions:

  • \(Q_C\): Capacitor Rating โ€” **kVAR** (Kilovolt-Amperes Reactive) | Necessary net reactive power boost.
  • P: Active Power โ€” **kW** (Kilowatts) | Total real baseline active load consumption.
  • \(\theta_1 / \theta_2\): Phase Angles โ€” Radians/Degrees | Angles calculated via current and target power factor vectors.

Calculate required kVAR rating to improve an inductive load's overall operational Power Factor level.

Output awaits input...

RLC Circuit Resonance

Advanced Electrical
\[f_0 = \frac{1}{2\pi \sqrt{L \times C}}\]

Variable & SI Unit Definitions:

  • \(f_0\): Resonant Frequency โ€” Hertz (\(\text{Hz}\)) | Natural frequency at zero circuit phase shift.
  • L: Inductance โ€” Henry (\(\text{H}\)) | Inductive component magnetic inertia metric.
  • C: Capacitance โ€” Farad (\(\text{F}\)) | Capacitive component electrostatic storage potential.

Compute the natural undamped resonant frequency peak (\(f_0\)) of a reactive electrical network layout.

Output awaits input...

Planck-Einstein Quantum Energy

Quantum Mechanics
\[E = h \times f\]

Variable & SI Unit Definitions:

  • E: Photon Energy โ€” Joule (\(\text{J}\)) | Discrete quantized electromagnetic package value.
  • h: Planck Constant โ€” Joule-Seconds (\(\text{J}\cdot\text{s}\)) | Static atomic anchor at \(6.62607015 \times 10^{-34}\).
  • f: Frequency โ€” Hertz (\(\text{Hz}\)) | Operating wave frequency profile of light node.

Isolate discrete energy quanta values (\(E\)) carried over specific monochromatic electromagnetic wave nodes.

Output awaits input...

Maxwell's Field Equations

Core Reference
\[\begin{aligned} \nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} & \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \cdot \mathbf{B} &= 0 & \nabla \times \mathbf{B} &= \mu_0 \left( \mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right) \end{aligned}\]

Field Elements & SI Units:

  • \(\mathbf{E}\): Electric Field Vector โ€” Volt per Meter (\(\text{V/m}\))
  • \(\mathbf{B}\): Magnetic Flux Density Vector โ€” Tesla (\(\text{T}\))
  • \(\rho\): Volumetric Charge Density โ€” Coulomb per Cubic Meter (\(\text{C/m}^3\))
  • \(\mathbf{J}\): Current Density Vector โ€” Ampere per Square Meter (\(\text{A/m}^2\))

๐Ÿ’ก Classical Field Theory Anchor Node: Differential system governing the entire spatial generation, propagation, and interactive parameters of combined Electromagnetic fields.

Navier-Stokes Equations

Core Reference
\[\rho \left( \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} \right) = -\nabla p + \mu \nabla^2 \mathbf{u} + \mathbf{f}\]

Fluid Elements & SI Units:

  • \(\mathbf{u}\): Flow Velocity Vector โ€” Meters per Second (\(\text{m/s}\))
  • p: Thermodynamic Pressure โ€” Pascal (\(\text{Pa}\))
  • \(\mu\): Dynamic Fluid Viscosity โ€” Pascal-Second (\(\text{Pa}\cdot\text{s}\))
  • \(\mathbf{f}\): Volumetric Body Force vectors โ€” Newtons per Cubic Meter (\(\text{N/m}^3\))

๐Ÿ’ก Incompressible Continuum Flow Anchor: Resolves velocity vectors (\(\mathbf{u}\)) and pressure properties across complex boundary conditions over variable viscous parameters.

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