emmy.calculus.form-field — org.mentat/emmy 0.32.0

This namespace implements a form field operator and a number of functions for creating and working with form fields.

A form-field of rank n is an operator that takes n vector fields to a real-valued function on the manifold. A one-form field takes a single vector field.

The namespace also contains two definitions of the wedge product (alt-wedge is the second), plus the exterior-derivative.

Alt clj/s

(Alt form)

Returns the alternation of the supplied differential form.

alt-wedge clj/s

(alt-wedge & args)

Alternative definition of wedge in terms of alternation.

basis-components->oneform-field clj/s

(basis-components->oneform-field components oneform-basis)

Given a structure of components functions defined on manifold points and and a matching oneform-basis (of identical structure),

Returns a new one-form field that

• passes its vector-field argument to oneform-basis, returning a new equivalent structure with each slot populated by functions from a manifold point to the directional derivative (using the vector field) in that coordinate direction

• contracts the result of that operation with the result of applying each component in components to the manifold point.

NOTE:

• This is for any basis, not just a coordinate basis
• The components are evaluated at a manifold point, not its coordinates
• Given a dual basis, you can retrieve the original components with oneform-field->basis-components

components->oneform-field clj/s

(components->oneform-field components coordinate-system)

(components->oneform-field components coordinate-system name)

Takes:

• a down tuple of components of the one-form field relative to coordinate-system
• the coordinate-system

And returns a full one-form field.

A one-field field is an operator that takes a vector field to a real-valued function on the manifold.

coordinate-basis-oneform-field clj/s

(coordinate-basis-oneform-field coordinate-system name & indices)

Given some coordinate-system, a symbolic name and a sequence of indices into the structure of the coordinate system's representation, returns a one-form field.

The returned one-form field at each structural spot takes a vector field and returns a function that takes the directional derivative in that coordinate's direction using the vector field.

coordinate-system->oneform-basis clj/s

(coordinate-system->oneform-basis coordinate-system)

Given some coordinate-system, returns a structure of coordinate-basis-oneform-field instances.

The one-form field at each structural spot takes a vector field and returns a function that takes the directional derivative in that coordinate's direction using the vector field.

When applied as a function, the structure behaves equivalently to

(coordinate-basis-oneform-field <coordinate-system> 'ignored-name)

With no indices supplied.

d clj/s

differential-of-function clj/s

Alias for function->oneform-field. One of the two incompatible definitions of differential.

This differential is a special case of exterior derivative. The other one lives at map/differential.

exterior-derivative clj/s

ff:zero clj/s

(ff:zero _)

Returns a form field that returns, for any supplied vector field vf, a manifold function manifold/zero-manifold-function that maps every input manifold point to the scalar value 0.

form-field? clj/s

(form-field? ff)

Returns true if the supplied f is a form field operator, false otherwise.

function->oneform-field clj/s

(function->oneform-field f)

One of the two incompatible definitions of differential.

This differential is a special case of exterior derivative. The other one lives at map/differential.

get-rank clj/s

(get-rank f)

Returns the rank of the supplied differential form f. Functions are treated as differential forms of rank 0.

Throws for any non differential form supplied.

literal-oneform-field clj/s

(literal-oneform-field name coordinate-system)

Given a symbolic name sym and a coordinate-system, returns a one-form field consisting of literal real-valued functions from the coordinate system's dimension for each coordinate component.

These functions are passed to components->oneform-field, along with the supplied coordinate-system and symbolic name sym.

For coordinate systems of dimension 1, literal-form-field's component functions will accept a single non-structural argument.

nform-field? clj/s

(nform-field? f n)

Returns true if the supplied f is an form field of rank n, false otherwise.

A form-field of rank n is an operator that takes n vector fields to a real-valued function on the manifold.

oneform-field->basis-components clj/s

(oneform-field->basis-components w vector-basis)

Given a structure w of and a vector field basis vector-basis, returns a new structure generated by applying the full vector basis to each element of w.

Here's an example of how to use this function to round trip a structure of basis components:

(let [vb    (vf/coordinate-system->vector-basis coordsys)
      basis (coordinate-system->oneform-basis coordsys)
      components (down d:dx d:dy)]
  (= components
     (-> components
         (basis-components->oneform-field basis)
         (oneform-field->basis-components vb))))

oneform-field->components clj/s

(oneform-field->components form coordinate-system)

Given a one-form field form and a coordinate-system, returns a function from the coordinate representation of a manifold point to a coordinate representation of the coordinatized components of the form field at that point.

For example:

(let-coordinates [[x y] R2-rect]
  (let [f (literal-oneform-field 'f R2-rect)]
    ((oneform-field->components f R2-rect)
     (up 'x0 'y0))))

;;=> (down (f_0 (up x0 y0))
;;         (f_1 (up x0 y0)))

oneform-field? clj/s

(oneform-field? f)

Returns true if the supplied f is a One-form, false otherwise.

A One-form takes a single vector field to a real-valued function on the manifold.

wedge clj/s

(wedge)

(wedge f)

(wedge f & fs)

Computes the wedge product of the sequence fs of one-forms.

Higher rank forms can be constructed from one-forms by wedging them together. This antisymmetric tensor product is computed as a determinant. The purpose of this is to allow us to use the construction dx^dy to compute the area described by the vectors that are given to it.

See Spivak p275 v1 of 'Differential Geometry' to see the correct definition. The key is that the wedge of the coordinate basis forms had better be the volume element.