1.1 Domain: DeRhamComplex¶

The module DeRhamComplex is contained in the file …/algebra/derham.spad which contains two other domains, ExtAlgBasis, AntiSymm and a category LeftAlgebra. The description field in the corresponding headers tells us:

LeftAlgebra
The category of all left algebras over an arbitrary ring.
ExtAlgBasis
A domain used in the construction of the exterior algebra on a set X over a ring R. This domain represents the set of all ordered subsets of the set X, assumed to be in correspondance with {1,2,3…}. The ordered subsets are themselves ordered lexicographically and are in bijective correspondance with an ordered basis of the exterior algebra. In this domain we are dealing strictly with the exponents of basis elements which can only be 0 or 1. Thus we really have L({0, 1}). The multiplicative identity element of the exterior algebra corresponds to the empty subset of X. A coerce from List Integer to an ordered basis element is provided to allow the convenient input of expressions. Another exported function forgets the ordered structure and simply returns the list corresponding to an ordered subset.
AntiSymm
The domain of antisymmetric polynomials.
DeRhamComplex
The DeRham complex of Euclidean space, that is, the class of differential forms of arbitary degree over a coefficient ring. See Flanders, Harley, Differential Forms, with Applications to the Physical Sciences, New York, Academic Press, 1963.

The author of these modules, Larry A. Lambe, is also the author of many research papers and the co-author of the book [5]. His current website is http://pages.bangor.ac.uk/~mas019/.

In order to enter a differential form (1) one needs some preparations which are best illustrated by some examples:

Example

X:=DeRhamComplex(Integer, [x,y,z])

$\mathrm{DeRhamComplex(Integer,[x,y,z])}$

Type: Type

this defines the space X of differential forms in the coordinates x,y,z. That is the underlying manifold (in a broad sense) is three dimensional. The coefficient ring Integer may be misleading, but we will come back to this point. Next we define the generators:

[dx,dy,dz]:=[generator(i)\$X for i in 1..3]

$[dx,dy,dz]$

Type: List(DeRhamComplex(Integer,[x,y,z]))

Now we can enter a differential form: $$dx+2 dy+3 dz$$:

dx+2*dy+3*dz

$3\,dz + 2\,dy + dx$

Type: DeRhamComplex(Integer,[x,y,z])

which represents a differential 1-form. When we multiply (i.e. $$\wedge$$) the result above by $$dy$$ from the right we should get the 2-form

$(dx+2\,dy+3\,dz)\wedge dy = dx\wedge dy - 3\, dy\wedge dz$
%*dy

$- 3\,dy\,dz + dx\,dy$

Type: DeRhamComplex(Integer,[x,y,z])

One recognizes that the terms are automatically ordered according to the order given by the generators on the one hand (there usually is never a term $$dy\wedge dx$$ in the output, unless deliberately generated) and lexicographic ordering on the other. This might be a source of confusion if the variables and generators are carelessly chosen. Surprisingly enough the following works as well:

3/5*dx

$-\frac{3}{5} \,dx$

Type: DeRhamComplex(Integer,[x,y,z])

x*dx

$x \, dx$

Type: DeRhamComplex(Integer,[x,y,z])

x^2*dx

$x^2 \, dx$

Type: Polynomial(DeRhamComplex(Integer,[x,y,z]))

so the coefficient ring is certainly bigger than Integer ($$\mathbb{Z}$$). Actually the coefficient ring is Expression(Integer) which is a huge space (with all the properties of a field), however, it is not compatible to the type Polynomial(Integer). The output above shows that the return type does not correspond to the defined space X. This is because $$x^2$$ has type Polynomial(Integer) instead of Expression(Integer). This means that we have to coerce certain expressions:

R ==> Expression(Integer)
Type: Void

(x^2)::R*dx

$x^2\,dx$

Type: DeRhamComplex(Integer,[x,y,z])

Warning. Polynomial expressions must be coerced to the coefficient ring (R). The return type should always coincide with the initially defined space (X in our example). Other functions are usually no problem, e.g. $$\sin(x+y),dx\wedge dy$$. It is of course possible to work with general undetermined functions using the BasicOperator type (note that BOP is the official abbreviation for the latter):

a : BOP := operator('a);

Type: BasicOperator

b : BOP := operator('b);

Type: BasicOperator

c : BOP := operator('c);

Type: BasicOperator


Now we can enter the most general 2-form in X:

$\sigma=a(x,y,z)\,dx\wedge dy + b(x,y,z)\,dx\wedge dz + c(x,y,z)\, dy\wedge dz.$
sigma:=a(x,y,z)* dx*dy + b(x,y,z)* dx*dz + c(x,y,z)* dy*dz

$c(x,y,z)\,dy\,dz + b(x,y,z)\,dx\,dz + a(x,y,z)\,dx\, dy$

Type: DeRhamComplex(Integer,[x,y,z])

This BOP method is very convenient and gives great flexibility. We can also define forms which depend on parameters, i.e. form valued functions $$\sigma:T \rightarrow X$$ as follows:

sigma(t) == a(x,y,z,t)*dx*dy + b(x,y,z,t)*dx*dz + c(x,y,z,t)*dy*dz


Although $$\sigma(t)$$ is not a form, it will become one as soon as it is evaluated:

 sigma(tau)

Compiling function sigma with type Variable(tau) -> DeRhamComplex(
Integer,[x,y,z])

c(x,y,z,tau)dy dz + b(x,y,z,tau)dx dz + a(x,y,z,tau)dx dy
Type: DeRhamComplex(Integer,[x,y,z])


Contrary to most introductional expositions, by differential forms we here mean the graded differential algebra, that is we can add forms of different degrees:

$1+dx + dy\wedge dz + dx\wedge dy \wedge dz$

FriCAS :: DeRhamComplex