Chapter 40
Euler’s Formula — Bridge Between Curvature and Phase

Flow Algebra

flow "Euler’s Formula — Bridge Between Curvature and Phase"

# ---------- CONCEPT ----------
# Euler’s formula unifies exponential growth and circular rotation.
# It expresses curvature and oscillation as one continuous phenomenon — phase evolving along a boundary of constant magnitude.
# e^{iθ} = cos θ + i sin θ links linear accumulation (exponentials) to angular motion (trigonometric rotation).

# ---------- STATES ----------
state EulerSystem(E):
  ComplexExponential:
    description: "e^{iθ} = cos θ + i sin θ — projection of circular rotation from exponential flow."
  UnitCircle:
    description: "|e^{iθ}| = 1 — constant-magnitude boundary representing perfect curvature."
  CartesianProjection:
    description: "Re(e^{iθ}) = cos θ, Im(e^{iθ}) = sin θ — orthogonal components of rotation."
  OscillatoryFlow:
    description: "Phase θ evolves linearly; position e^{iθ} traces a circular path in ℂ."
  DifferentialRelation:
    description: "d/dθ (e^{iθ}) = i e^{iθ} — 90° rotation per unit phase change."
  CurvatureIdentity:
    description: "Second derivative returns −e^{iθ} — closed curvature: acceleration orthogonal to velocity."

# ---------- PARAMETERS ----------
params:
  variable: "θ ∈ ℝ (phase)"
  amplitude: "r (for general form r e^{iθ})"
  frequency: "ω (for time evolution e^{iωt})"
  closure: true
  units: "radians, curvature radius = r"

# ---------- INVARIANTS ----------
invariants:
  - "e^{iθ} = cos θ + i sin θ."
  - "Magnitude invariant: |e^{iθ}| = 1."
  - "Additive phase: e^{i(θ₁+θ₂)} = e^{iθ₁} e^{iθ₂}."
  - "Differentiation rotates by i: (d/dθ)e^{iθ} = i e^{iθ}."
  - "Integration reverses rotation: ∫ e^{iθ} dθ = (1/i)e^{iθ} + C."
  - "Acceleration orthogonal to motion: (d²/dθ²)e^{iθ} = −e^{iθ}."
  - "Real and imaginary components obey cos²θ + sin²θ = 1 — unity of curvature."
  - "For r e^{iθ}, radius fixed ⇒ total curvature κ = 1/r."

# ---------- TRANSFORMS ----------
transforms:
  ToTrigonometric(θ):
    rule: "Use Euler’s identity to expand e^{iθ} = cos θ + i sin θ."
  FromTrigonometric(c,s):
    rule: "Recombine cos θ + i sin θ = e^{iθ}."
  MultiplyPhasors(θ₁,θ₂):
    rule: "e^{iθ₁} e^{iθ₂} = e^{i(θ₁+θ₂)} — composition of rotations."
  Differentiate(e^{iθ}):
    rule: "(d/dθ)e^{iθ} = i e^{iθ} — rotation rate equals 90° turn per unit curvature."
  GeneraliseAmplitude(r):
    rule: "z = r e^{iθ} ⇒ |z| = r constant, phase evolving."
  ConvertToTimeDomain(ω,t):
    rule: "e^{iωt} = cos(ωt) + i sin(ωt) — harmonic oscillation."
  DecomposeSignal(s(t)):
    rule: "Represent any periodic s(t) as Σ A_k e^{i(kωt+φ_k)} — Fourier synthesis."

# ---------- MONITORS ----------
monitors:
  MagnitudeCheck:
    measure: "|e^{iθ}|"
    target: 1
    meaning: "Preserve curvature radius; no expansion or contraction."
  Orthogonality:
    measure: "Re(e^{iθ})·Im(e^{iθ})"
    target: 0
    meaning: "Real and imaginary components orthogonal in phase."
  PhaseContinuity:
    check: "Ensure θ continuous modulo 2π for smooth rotation."
  CurvatureRate:
    measure: "|d²e^{iθ}/dθ²|"
    meaning: "Constant ⇒ perfect circular curvature."
  EnergyConservation:
    meaning: "Constant magnitude ⇒ conserved rotational energy."

# ---------- EQUATIONS ----------
equations:
  - "e^{iθ} = cos θ + i sin θ"
  - "\\frac{d}{dθ}e^{iθ} = i e^{iθ}"
  - "\\frac{d²}{dθ²}e^{iθ} = -e^{iθ}"
  - "e^{i(θ₁+θ₂)} = e^{iθ₁} e^{iθ₂}"
  - "cos θ = \\frac{e^{iθ} + e^{-iθ}}{2}, \\; sin θ = \\frac{e^{iθ} - e^{-iθ}}{2i}"
  - "|e^{iθ}| = 1, \\; |r e^{iθ}| = r"
  - "e^{iωt} = cos(ωt) + i sin(ωt)"
  - "∫ e^{iθ} dθ = (1/i)e^{iθ} + C"
  - "κ = 1/r ⇒ curvature inversely proportional to radius"

# ---------- NOTES ----------
notes:
  - "Euler’s formula unifies two geometries: exponential growth and circular rotation — straight flow and curved flow expressed as one equation."
  - "The operator i induces curvature: multiplication by i rotates a vector 90° in the complex plane."
  - "Differentiation acts as continuous rotation; each step in phase turns the vector while conserving magnitude."
  - "In physics, e^{iωt} represents harmonic oscillation — constant energy curvature of motion in phase space."
  - "In mathematics, e^{iθ} parameterises the unit circle — curvature encoded as complex exponential."
  - "In Flow Algebra, Euler’s formula defines curvature as rotational phase flow: an exponential path whose magnitude is fixed and whose direction continuously rotates — the perfect bridge between growth, motion, and form."