Module vector 

Vector calculus: gradient, divergence, curl, laplacian. Vector calculus: gradient, divergence, curl, Laplacian (in Cartesian, cylindrical and spherical coordinates), directional derivatives, line integrals and scalar potentials.

Scalar fields are Ex; vector fields are n×1 column-vector Matrix values whose components are expressed in the coordinate basis of the chosen CoordinateSystem (e.g. (F_r, F_φ, F_z) for cylindrical coordinates).

Coordinate conventions

System	vars order	Scale factors (h₁, h₂, h₃)
Cartesian	(x, y, z) (any dimension)	(1, 1, 1)
Cylindrical	(r, φ, z)	(1, r, 1)
Spherical	(r, θ, φ) — physics convention: θ polar angle from +z, φ azimuth	(1, r, r·sin θ)

The Cartesian functions (gradient, divergence, curl, laplacian) work in any dimension (curl: 3-D only). The _in variants take an explicit coordinate system; the curvilinear systems are 3-D and require exactly three variables.

Shape preconditions

Wrong field dimensions / variable counts are programming errors and panic (like slice indexing), consistent with the rest of this module. Mathematical failure (e.g. a non-conservative field passed to scalar_potential) returns Err.

Enums

CoordinateSystem

Orthogonal coordinate systems supported by the vector-calculus operators.

Functions

curl

Curl of a 3-D vector field in Cartesian coordinates: ∇×F.

curl_in

Curl in the given coordinate system:

directional_derivative

Directional derivative ∇f · d of f along the (Cartesian) direction vector d.

divergence

Divergence of a vector field in Cartesian coordinates: ∇·F = Σᵢ ∂Fᵢ/∂xᵢ.

divergence_in

Divergence in the given coordinate system: ∇·F = (1/(h₁h₂h₃)) Σᵢ ∂/∂qᵢ ( (h₁h₂h₃/hᵢ) Fᵢ ).

gradient

Gradient of a scalar field in Cartesian coordinates: ∇f = [∂f/∂x₁, …, ∂f/∂xₙ]ᵀ (an n×1 column vector).

gradient_in

Gradient in the given coordinate system: (∇f)ᵢ = (1/hᵢ) ∂f/∂qᵢ.

hessian

Hessian matrix H[i][j] = ∂²f / ∂vars[i] ∂vars[j].

is_conservative

Is the 3-D vector field conservative (curl-free)? Three-valued.

is_irrotational

Alias of is_conservative (irrotational ⇔ curl-free).

is_solenoidal

Is the vector field solenoidal (divergence-free)? Three-valued, like is_conservative.

laplacian

Laplacian of a scalar field in Cartesian coordinates: ∇²f = Σᵢ ∂²f/∂xᵢ².

laplacian_in

Laplacian ∇²f = ∇·(∇f) in the given coordinate system.

line_integral_scalar

Scalar line integral ∫_C f ds = ∫ₐᵇ f(r(t)) ‖r′(t)‖ dt.

line_integral_vector

Vector line integral (work) ∫_C F·dr = ∫ₐᵇ F(r(t)) · r′(t) dt.

scalar_potential

Scalar potential φ with ∇φ = F for a conservative Cartesian field.