DifferentialForms

 coordSymbols()$M returns a list of the coordinates as symbols. This
 is useful, for example, if the differential operators "D" are to be
 used.

 coordVector()$M returns a list of the coordinates in the space
 M=DFORM(Ring,Coordinates).

 zeroForms()$M returns a vector of the basis zero forms, i.e. the
 coordinates.

 oneForms()$M returns a vector of the basis one forms, i.e. the
 differentials of the coordinate functions.

 zero()$M gives the zero form, i.e. 0@DeRhamComplex.

 one()$M gives 1@DeRhamComplex, i.e. "1" as a differential form. This
 is useful to intern elements of the function ring (just multiply them
 by  one()$DFORM).

 d computes the total differential of a function.

 d(form)$M computes the exterior derivative and is just an
 abbreviation for the fucntion "exteriorDifferential"
 defined in the domain "DeRhamComplex".

 d computes the exterior derivative of each component.

 d computes the exterior derivative of each component.

 w1*w2 computes the sum of the exterior products
 w1_i * w2_i, where w1,w2 are differential forms.

 indices(m)$M returns a list of elements which are oredered lists
 of integers representing the the m-element subsets of {1..n}.

 basisForms()$M returns a list of all base forms in the
 space M=DFORM(Ring,Coordinates).

 vectorField(V) creates a vector (actually a list) whose
 components are given by V[j](x[1],...,x[n]), j=1..n,
 whereby "x" are the space coordinates (possibly not the
 same symbol).

 scalarField(s) creates a scalar function s(x[1],...,[n]),
 whereby "x" are the space coordinates (possibly not the
 same symbol).

 covectorField(Y) creates a covector (actually a list)
 whose components are given by w[j](x[1],...,x[n]),
 j=1..n.

 zeroForm(s) creates a zero form with symbol "s". This
 is the same as scalarField(s)*one().

 volumeForm(g) returns the volume form with respect to
 the (pseudo-) metric "g".

 proj(p,form) yields the projection to homogeneous terms of degree p

 coefficients(p,form) returns a list of the coefficients of
 the p-forms in the argument 
.

 degrees(form,[]) returns a list of the degrees in .

 dual(form) returns the conjugate form, i.e. the Hodge star
 with respect to the standard metric.

 dot(form,form) computes the inner product of two differential forms
 with ++ respect to the standard metric (i.e. g=unit matrix).

 dot(g,form,form) computes the inner product of two differential forms
 with  respect to the metric g.

 atomizeBasisTerm  returns a list of the generators (atoms),
 i.e. [dx,dy,du] if given a basis term dx*dy*du, for example.

 interiorProduct(v,form) calculates the interior product i_X(a) of
 the vector field X with the differential form a.

 lieDerivative(v,form) calculates the Lie derivative L_X(a) of the
 differential form a with respect to the vector field X.

 genericMetric(symbol) return a square matrix whose elements
 are basic operators of the form symbol[i,j](x,y,..). The
 indices are covariant.

 genericInverseMetric(symbol) return a square matrix whose elements
 are basic operators of the form symbol[i,j](x,y,...). The indices
 are contravariant.

 hodgeStar(g,form) computes the Hodge dual of the differential form
 with respect to a metric g.

 invHodgeStar(g,form) denotes the inverse of hodgeStar.

 codifferential(g,x), also known as "delta", computes the
 co-differential of a form.

 hodgeLaplacian(g,x) also known as "Laplace-de Rham operator" is
 defined on any manifold equipped with a (pseudo-) Riemannian
 metric and is given by d codifferential(g,x)+ codifferential(g, d x).
 Note that in the Euclidean case hodgeLaplacian = - Laplacian.

 volumeForm(g) returns the volume form with respect to
 the (pseudo-) metric "g".

 s(g) determines the sign of determinant(g) and is related to the
 signature of g (n=p+q,t=p-q,s=(-)^(n-t)/2 => s=(-)^q).

 dotg(squareMatrix,form,form) returns the dot product of two
 forms with respect to the metric given by squareMatrix.

 convert a matrix whose elements belong to the base ring to a
 matrix with the same elements as memebrs of DeRhamComplex.

 retract a matrix whose elements belong to DeRhamComplex to a
 matrix with the same elements as memebrs of the base ring. This
 is of course only possible if the degree of all the forms is
 zero.

 frameVectors(A), A in GL(n), returns a list of the frame vectors
 g=A*e, where e is the standard frame e_1,...,e_n.

 movingFrame(A), A in GL(n), returns a matrix whose rows are the
 frame vectors (see frameVectors(A)).

 sigma(A), A in GL(n), returns the basis one forms of the corresponding
 moving frame (see movingFrame(A)).

 Omega(A), A in GL(n), returns the basis connection forms of the
 corresponding moving frame (see movingFrame(A)).

 Gramian(A), A in GL(n), returns the Gramian of the corresponding
 moving frame (see movingFrame(A)).