Chapter 20
Gradient, Divergence, and Curl as Field Operators

Flow Algebra

flow "Gradient, Divergence, and Curl as Field Operators"

# ---------- CONCEPT ----------
# The gradient, divergence, and curl are the fundamental operators that describe how fields change, accumulate, and rotate.
# The gradient measures directional rate of change (scalar → vector).
# The divergence measures how much a vector field expands or contracts (vector → scalar).
# The curl measures how a vector field twists or circulates (vector → vector).
# Together they define the local geometry of flow — the differential anatomy of coherence.

# ---------- STATES ----------
state FieldSystem(F):
  ScalarField:
    description: "f(x,y,z) — a scalar field such as potential, temperature, or pressure."
  VectorField:
    description: "𝐅(x,y,z) = (P, Q, R) — a vector field representing velocity, flux, or force."
  Gradient:
    description: "∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) — converts scalar field to directional rate of change."
  Divergence:
    description: "∇·𝐅 = ∂P/∂x + ∂Q/∂y + ∂R/∂z — measures net outflow of coherence from a point."
  Curl:
    description: "∇×𝐅 = (∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y) — captures rotational coherence of the field."
  Laplacian:
    description: "∇²f = ∇·(∇f) — second-order coherence density (curvature of potential)."

# ---------- PARAMETERS ----------
params:
  variables: ["x", "y", "z"]
  field_types: ["scalar", "vector"]
  closure: true
  orientation: "Right-hand coordinate convention"
  smoothness: "f, P, Q, R ∈ C²"
  dimensionality: 3  # Extendable to nD via differential operators

# ---------- INVARIANTS ----------
invariants:
  - "Gradient transforms scalar → vector: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)."
  - "Divergence transforms vector → scalar: ∇·𝐅 = ∂P/∂x + ∂Q/∂y + ∂R/∂z."
  - "Curl transforms vector → vector: ∇×𝐅 = (∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y)."
  - "Laplacian expresses scalar field curvature: ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²."
  - "If ∇×𝐅 = 0, field is irrotational ⇒ conservative potential exists (𝐅 = ∇φ)."
  - "If ∇·𝐅 = 0, field is solenoidal ⇒ incompressible flow (no net source/sink)."
  - "Identity: ∇·(∇×𝐅) = 0 — divergence of curl always zero (coherence conservation)."

# ---------- TRANSFORMS ----------
transforms:
  ComputeGradient(f):
    rule: "∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z) — directional derivative field."
  ComputeDivergence(F):
    rule: "∇·𝐅 = ∂P/∂x + ∂Q/∂y + ∂R/∂z — coherence flux density."
  ComputeCurl(F):
    rule: "∇×𝐅 = (∂R/∂y - ∂Q/∂z, ∂P/∂z - ∂R/∂x, ∂Q/∂x - ∂P/∂y)."
  ComputeLaplacian(f):
    rule: "∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z² — second-order coherence curvature."
  VerifyConservationIdentities(F):
    rule: "Check ∇·(∇×𝐅)=0 and ∇×(∇f)=0 — internal consistency of coherence operators."
  ConvertToPotential(F):
    condition: "If ∇×𝐅=0."
    rule: "Integrate 𝐅 to find scalar potential φ where 𝐅 = ∇φ."
  AssessFlowType(F):
    rule: "If ∇·𝐅 ≠ 0 ⇒ source/sink; if ∇×𝐅 ≠ 0 ⇒ rotational flow."

# ---------- MONITORS ----------
monitors:
  GradientMagnitude:
    measure: "|∇f|"
    meaning: "Intensity of local rate of change."
  DivergenceDensity:
    measure: "∇·𝐅"
    meaning: "Net creation (+) or absorption (−) of coherence at a point."
  CurlIntensity:
    measure: "|∇×𝐅|"
    meaning: "Strength of local rotational coherence."
  ConservationCheck:
    meaning: "Verify ∇·(∇×𝐅)=0 (true for all differentiable fields)."
  PotentialExistence:
    meaning: "Checks if field is irrotational (∇×𝐅=0) ⇒ conservative potential φ exists."

# ---------- EQUATIONS ----------
equations:
  - "∇f = \\left( \\frac{∂f}{∂x}, \\frac{∂f}{∂y}, \\frac{∂f}{∂z} \\right)"
  - "∇·𝐅 = \\frac{∂P}{∂x} + \\frac{∂Q}{∂y} + \\frac{∂R}{∂z}"
  - "∇×𝐅 = \\left( \\frac{∂R}{∂y} - \\frac{∂Q}{∂z}, \\frac{∂P}{∂z} - \\frac{∂R}{∂x}, \\frac{∂Q}{∂x} - \\frac{∂P}{∂y} \\right)"
  - "∇²f = \\frac{∂²f}{∂x²} + \\frac{∂²f}{∂y²} + \\frac{∂²f}{∂z²}"
  - "∇·(∇×𝐅) = 0, ∇×(∇f) = 0"
  - "𝐅 = ∇φ (if ∇×𝐅 = 0)"
  - "∇·𝐅 = 0 (incompressible or divergence-free field)"

# ---------- NOTES ----------
notes:
  - "These three operators describe how coherence behaves within continuous fields: gradient as directional rise, divergence as expansion, and curl as rotation."
  - "The gradient identifies slope and flow direction of potential; divergence measures local sources and sinks; curl reveals rotational structure."
  - "Together they define the local dynamics of any field — how coherence moves, accumulates, or circulates."
  - "The Laplacian adds a second-order layer — the curvature of coherence, or diffusion rate of potential."
  - "In Flow Algebra, these operators form the language of spatial coherence: gradient (flow), divergence (flux), curl (rotation) — the recursive anatomy of continuous systems."