_Laplacian
laplacian(f, x, y, …) — the Laplacian of a scalar field, a scalar.
Defined as div(grad(f)) rather than as its own sum of second derivatives, so the two can never disagree. That definition is why the curvilinear systems needed no change here: composing the curvilinear grad and div yields (1/J)·Σᵢ ∂/∂qᵢ (J/hᵢ² · ∂f/∂qᵢ), which is exactly the general orthogonal-curvilinear Laplacian — a formula written out separately would have been a second definition to keep in step, and this one cannot drift.
A vector argument stays symbolic, and now for a load-bearing reason rather than a cautious one: outside Cartesian coordinates the vector Laplacian is not the component-wise scalar Laplacian (it is grad(div F) − curl(curl F)), so answering it by mapping this scalar formula over the components would be wrong in exactly the systems the curvilinear systems add.
Value parameters
- coords
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the ordered coordinate tuple
- e
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the scalar field
- system
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the coordinate system (only
Cartesianevaluates)
Attributes
- Graph
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- Supertypes
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trait Serializabletrait Producttrait Equalstrait _VectorOperatortrait _Expressionclass Objecttrait Matchableclass AnyShow all
Members list
Value members
Concrete methods
Reduces this expression in the given environment.
Reduces this expression in the given environment.
Value parameters
- env
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variable bindings and display precision
Attributes
- Returns
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Right(v)when all free variables resolved to concrete values;Left(e)when reduction is partial or impossible - Definition Classes
Reconstructs the same node shape with replacement sub-expressions.
Reconstructs the same node shape with replacement sub-expressions.
Value parameters
- newChildren
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replacements in the same order and count as
children
Attributes
- Definition Classes
Inherited methods
Sub-expressions subject to recursive structural traversal. Binder positions (e.g. the differentiation variable) are excluded.
Sub-expressions subject to recursive structural traversal. Binder positions (e.g. the differentiation variable) are excluded.
Attributes
- Definition Classes
- Inherited from:
- _VectorOperator
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.
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.
Attributes
- Inherited from:
- _VectorOperator
∂x/∂coords(i), simplified.
The scale factors, or an empty vector when the request is ill-posed (guarded by wellPosed at every use site).
The scale factors, or an empty vector when the request is ill-posed (guarded by wellPosed at every use site).
Attributes
- Inherited from:
- _VectorOperator
Orientation of the ordered basis: +1 right-handed, -1 left-handed.
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.
Attributes
- Inherited from:
- _VectorOperator
The Jacobian factor h₁·h₂·…, the volume element's coefficient.
The Jacobian factor h₁·h₂·…, the volume element's coefficient.
Attributes
- Inherited from:
- _VectorOperator
Attributes
- Inherited from:
- Product
Attributes
- Inherited from:
- Product
The scale factors (h₁, h₂, …) for this coordinate system and tuple.
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.
Attributes
- Inherited from:
- _VectorOperator
Returns a string representation of the object.
Returns a string representation of the object.
The default representation is platform dependent.
Attributes
- Returns
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a string representation of the object.
- Definition Classes
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_VectorOperator -> Any
- Inherited from:
- _VectorOperator
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.
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.
Attributes
- Inherited from:
- _VectorOperator
Inherited fields
Cached set of free variable names; O(1) after the first access.
Cached set of free variable names; O(1) after the first access.
Attributes
- Inherited from:
- _Expression