sicmutils.calculus.form-field

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.

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]].

raw docstring

Alt^clj/s

(Alt form)

Returns the alternation of the supplied differential form.

Returns the alternation of the supplied differential `form`.

source raw docstring

alt-wedge^clj/s

(alt-wedge & args)

Alternative definition of wedge in terms of alternation.

Alternative definition of [[wedge]] in terms of alternation.

source raw docstring

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

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]]

source raw docstring

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.

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.

source raw docstring

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.

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.

source raw docstring

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.

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

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

With no indices supplied.

source raw docstring

d^clj/s

source

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.

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]].

source raw docstring

exterior-derivative^clj/s

source

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.

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.

source raw docstring

form-field?^clj/s

(form-field? ff)

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

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

source raw docstring

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.

One of the two incompatible definitions of differential.

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

source raw docstring

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.

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.

source raw docstring

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.

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.

source raw docstring

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.

Returns true if the supplied `f` is an [form field of rank
n](https://en.wikipedia.org/wiki/Differential_form), false otherwise.

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

source raw docstring

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))))

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:

```clojure
(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))))
```

source raw docstring

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)))

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:

```clojure
(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)))
```

source raw docstring

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.

Returns true if the supplied `f` is
a [One-form](https://en.wikipedia.org/wiki/One-form), false
otherwise.

A [One-form](https://en.wikipedia.org/wiki/One-form) takes a single vector
field to a real-valued function on the manifold.

source raw docstring

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.

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.

source raw docstring

cljdoc is a website building & hosting documentation for Clojure/Script libraries

Keyboard shortcuts Report a problem cljdoc on GitHub

× close