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2.0 Package Details

Package Name:DifferentialForms
Abbreviation:DFORM
Source files:dform.spad
Dependent on:DeRhamComplex (DERHAM) 
DifferentialForms(R,v)

R: Join(Ring,Comparable)   -- e.g. Integer
v: List Symbol             -- e.g. [x,y,z] or [x[0],x[1],x[2]]

X ==> Expression R         -- function Ring

For the examples following, we choose R=Integer and v=[x[0],x[1],x[2],x[3]], and the abbreviation M:=DFORM(R,v).

2.1 The metric g

Some functions expect the metric g as a parameter. Generally this will be provided by an invertible square matrix g:SquareMatrix(#v,X).

For the examples following, we choose the Minkowski metric::

g := diagonalMatrix([-1,1,1,1])@SquareMatrix(4,Integer)
\[\begin{split}\left[ \begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right]\end{split}\]

Type: SquareMatrix(4,Integer)

2.2 Exported Functions

The function baseForms return the basis one forms, while coordVector returns a list of the coordinates. The function coordSymbols also returns the coordinates, however, as symbols only (convenient when used by D).

SMR ==> SquareMatrix(#v,X)
DRC ==> DeRhamComplex(R,v)

dx:=baseForms()$M     -- [dx[0],...,dx[3]]
x:=coordVector()$M    -- [x[0],...,x[3]]
xs:=coordSymbols()$M  -- as above but as List Symbol (for differentiation)

2.2.1 Volume Form

volumeForm
Given a metric \(g\) the function returns the corresponding volume element of the Riemannian (pseudo-) manifold.
volumeForm : SquareMatrix(#v,X) -> DeRhamComplex(R,v)

volumeForm(g)$M
\[{dx _ {0}} \ {dx _ {1}} \ {dx _ {2}} \ {dx _ {3}}\]

Type: DeRhamComplex(Integer,[x[0],x[1],x[2],x[3]])

2.2.1 Scalar Product

dot
Compute the inner product of two differential forms with respect to the metric g.
dot : (SMR,DRC,DRC) -> X

dot(g,dx.1*dx.2,dx.1*dx.2)$M   -- note dx.1 corresponds to dx[0].
\[- 1\]

Type: Expression(Integer)

2.2.2 Hodge Star Operator

hodgeStar
Compute the Hodge dual form of a differential form with respect to a metric g.
hodgeStar : (SMR,DRC) -> DRC

hodgeStar(g,dx.2 * dx.3)
\[{dx _ {0}} \ {dx _ {3}}\]

Type: DeRhamComplex(Integer,[x[0],x[1],x[2],x[3]])

2.2.3 Interior Product

interiorProduct
Calculate the interior product \(i_X(a)\) of the vector field X with the differential form a.
interiorProduct : (Vector(X),DRC) -> DRC

interiorProduct(vector x, dx.1*dx.3)$M
\[{{x _ {0}} \ {dx _ {2}}} -{{x _ {2}} \ {dx _ {0}}}\]

Type: DeRhamComplex(Integer,[x[0],x[1],x[2],x[3]])

2.2.4 Lie Derivative

lieDerivative
Calculates the Lie derivative \(\mathcal{L}_X(a)\) of the differential form a with respect to the vector field X.
lieDerivative : (Vector(X),DRC) -> DRC

lieDerivative(vector x, dx.1 * dx.3 * dx.4)
\[3 \ {dx _ {0}} \ {dx _ {2}} \ {dx _ {3}}\]

Type: DeRhamComplex(Integer,[x[0],x[1],x[2],x[3]])

2.2.5 Projection

proj
Project to homogeneous terms of degree p.
NNI ==> NonNegativeInteger
proj : (NNI,DRC) -> DRC

proj(2, 2*dx.1 + dx.2*dx.3 - dx.3*dx.4)
\[-{{dx _ {2}} \ {dx _ {3}}}+{{dx _ {1}} \ {dx _ {2}}}\]

Type: DeRhamComplex(Integer,[x[0],x[1],x[2],x[3]])

2.2.6 Monomials

monomials
List all monomials of degree p (p in 1..n). This is a basis for \(\Lambda_p^n\).
monomials : NNI -> List DRC

monomials(3)$M
\[\left[ {{dx _ {0}} \ {dx _ {1}} \ {dx _ {2}}}, \: {{dx _ {0}} \ {dx _ {1}} \ {dx _ {3}}}, \: {{dx _ {0}} \ {dx _ {2}} \ {dx _ {3}}}, \: {{dx _ {1}} \ {dx _ {2}} \ {dx _ {3}}} \right]\]

Type: List DeRhamComplex(Integer,[x[0],x[1],x[2],x[3]])

2.2.7 Atomize Basis Term

atomizeBasisTerm
Given a basis term, return a list of the generators (atoms).
atomizeBasisTerm : DRC -> List DRC

atomizeBasisTerm(dx.1 * dx.2 * dx.4)
\[\left[ {dx _ {0}}, \: {dx _ {1}}, \: {dx _ {3}} \right]\]

Type: List(DeRhamComplex(Integer,[x[0],x[1],x[2],x[3]]))

2.2.8 Conjugate Basis Term

conjBasisTerm
Return the complement of a basis term with respect to the Euclidean volume form.
conjBasisTerm : DRC -> DRC

conjBasisTerm dx.4
\[{dx _ {0}} \ {dx _ {1}} \ {dx _ {2}}\]

Type: DeRhamComplex(Integer,[x[0],x[1],x[2],x[3]])

2.2.9 Scalar and Vector Field

vectorField
Generate a generic vector field named by a given symbol.
covectorField
Generate a generic co-vector field named by a given symbol.
scalarField
Generate a generic scalar field named by a given symbol.
vectorField   : Symbol -> List X
covectorField : Symbol -> List DRC
scalarField   : Symbol -> X

vectorField(Q)$M
scalarField(f)$M
\[\left[ {{Q _ {1}} \left( {{x _ {0}}, \: {x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)}, \: {{Q _ {2}} \left( {{x _ {0}}, \: {x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)}, \: {{Q _ {3}} \left( {{x _ {0}}, \: {x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)}, \: {{Q _ {4}} \left( {{x _ {0}}, \: {x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)} \right]\]

Type: List(Expression(Integer))

\[f \left( {{x _ {0}}, \: {x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)\]

Type: Expression(Integer)

2.2.10 Miscellaneous Functions

A zero form with symbol s can be generated by

zeroForm : Symbol -> DRC

zeroForm(s)$M
\[s \left( {{x _ {0}}, \: {x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)\]

Type: DeRhamComplex(Integer,[x[0],x[1],x[2],x[3]])

A synonym for the exteriorDerivative is the common operator d:

d : DRC -> DRC

d zeroForm(f)$M
\[\begin{split}{{{f _ {{,4}}} \left( {{x _ {0}}, \: {x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)} \ {dx _ {3}}}+{{{f _ {{,3}}} \left( {{x _ {0}}, \: {x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)} \ {dx _ {2}}}+ \\ {{{f _ {{,2}}} \left( {{x _ {0}}, \: {x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)} \ {dx _ {1}}}+{{{f _ {{,1}}} \left( {{x _ {0}}, \: {x _ {1}}, \: {x _ {2}}, \: {x _ {3}}} \right)} \ {dx _ {0}}}\end{split}\]

The special zero forms 0 and 1 can be generated by

one :  -> DRC
zero : -> DRC

zero()$M
one()$M

There are also some special multiplication operators which allow to deal with a kind vector valued forms (actually lists):

_* : (List X, List DRC)   -> DRC
_* : (List DRC, List DRC) -> DRC

Note: the lists must have dimension #v.

For instance:

x * dx
\[{{x _ {3}} \ {dx _ {3}}}+{{x _ {2}} \ {dx _ {2}}}+{{x _ {1}} \ {dx _ {1}}}+{{x _ {0}} \ {dx _ {0}}}\]

An example for the second case:

dx*[hodgeStar(g,dx.j)$M for j in 1..4]
\[2 \ {dx _ {0}} \ {dx _ {1}} \ {dx _ {2}} \ {dx _ {3}}\]

Type: DeRhamComplex(Integer,[x[0],x[1],x[2],x[3]])