_VectorOperator

The ordered coordinate tuple; free in the result, never substituted.

The scalar field or vector field the operator is applied to.

The operator's own name, for toString.

The coordinate system; only Cartesian evaluates today.

Sub-expressions subject to recursive structural traversal. Binder positions (e.g. the differentiation variable) are excluded.

Builds a column vector from its components, simplified.

Reads e as an n×1 vector field: its components, or None when it is not one.

A literal matrix is read without evaluating it, and that ordering is load-bearing: the components must be differentiated before any point is substituted. Evaluating first would collapse [[x^2], [y^3]] to a dense _MatrixValue as soon as the coordinates happen to be bound — every cell a constant, every derivative zero, and a confidently wrong answer rather than a refusal.

Only a non-literal argument is evaluated, which covers the variable-bound-to-a-matrix case (the general lesson that matrix-vs-scalar cannot always be decided at parse time). An Environment holds _Values, so such a binding is necessarily a dense _MatrixValue: its cells are constants and their derivatives are legitimately zero.

A field is a column (n×1); a row vector is a different object and is refused.

∂x/∂coords(i), simplified.

The scale factors, or an empty vector when the request is ill-posed (guarded by wellPosed at every use site).

Orientation of the ordered basis: +1 right-handed, -1 left-handed.

Only curl reads this, and it must. The curl formula below is derived for a right-handed (q₁, q₂, q₃); SphericalMaths exchanges the last two coordinates of Spherical, and swapping two basis vectors flips the orientation, so the same formula returns −curl there. grad, div and laplacian involve no cross product and are orientation-free, which is why the sign lives here rather than in each operator.

The Jacobian factor h₁·h₂·…, the volume element's coefficient.

The scale factors (h₁, h₂, …) for this coordinate system and tuple.

Every operator below is written once, in terms of these; Cartesian is the h = 1 case, which is why generalising the formulas left its results untouched. None when the system cannot describe this tuple — cylindrical and spherical are three-dimensional, and a request in any other arity is refused rather than answered in the wrong geometry.

Returns a string representation of the object.

The default representation is platform dependent.

Sums a list of terms, folding to 0 when empty.

True when the request is one this tier can answer at all: a coordinate tuple with no repeats, in an arity the coordinate system supports. grad(f, x, x) names no basis and is refused rather than silently producing a duplicated component.

Reduces this expression in the given environment.

Reconstructs the same node shape with replacement sub-expressions.

Cached set of free variable names; O(1) after the first access.