9.14 DeRhamComplex
The domain constructor DeRhamComplex creates the class of
differential forms of arbitrary degree over a coefficient ring. The
De Rham complex constructor takes two arguments: a ring, coefRing, and a list of coordinate variables.
This is the ring of coefficients.
Type: Domain
These are the coordinate variables.
lv : List Symbol := [x,y,z]
Type: List Symbol
This is the De Rham complex of Euclidean three-space using coordinates
x, y and z.
der := DERHAM(coefRing,lv)
|
|
Type: Domain
This complex allows us to describe differential forms having
expressions of integers as coefficients. These coefficients can
involve any number of variables, for example, f(x,t,r,y,u,z).
As we've chosen to work with ordinary Euclidean three-space,
expressions involving these forms are treated as functions of
x, y and z with the additional arguments t, r
and u regarded as symbolic constants.
Here are some examples of coefficients.
Type: Domain
f : R := x**2*y*z-5*x**3*y**2*z**5
Type: Expression Integer
g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2
Type: Expression Integer
h : R :=x*y*z-2*x**3*y*z**2
Type: Expression Integer
We now define the multiplicative basis elements for the exterior
algebra over R.
Type: DeRhamComplex(Integer,[x,y,z])
Type: DeRhamComplex(Integer,[x,y,z])
Type: DeRhamComplex(Integer,[x,y,z])
This is an alternative way to give the above assignments.
[dx,dy,dz] := [generator(i)$der for i in 1..3]
Type: List DeRhamComplex(Integer,[x,y,z])
Now we define some one-forms.
alpha : der := f*dx + g*dy + h*dz
|
|
Type: DeRhamComplex(Integer,[x,y,z])
beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy
Type: DeRhamComplex(Integer,[x,y,z])
A well-known theorem states that the composition of
exteriorDifferentialexteriorDifferentialDeRhamComplex with itself is the
zero map for continuous forms. Let's verify this theorem for alpha.
exteriorDifferential alpha
|
|
Type: DeRhamComplex(Integer,[x,y,z])
We see a lengthy output of the last expression, but nevertheless, the
composition is zero.
Type: DeRhamComplex(Integer,[x,y,z])
Now we check that exteriorDifferentialexteriorDifferentialDeRhamComplex
is a ``graded derivation'' D, that is, D satisfies:
D(a*b) = D(a)*b + (-1)**degree(a)*a*D(b)
|
|
Type: DeRhamComplex(Integer,[x,y,z])
We try this for the one-forms alpha and beta.
exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha * exteriorDifferential(beta))
Type: DeRhamComplex(Integer,[x,y,z])
Now we define some ``basic operators'' (see
OperatorXmpPage ).
Type: BasicOperator
Type: BasicOperator
Type: BasicOperator
We also define some indeterminate one- and two-forms using these
operators.
sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz
|
|
Type: DeRhamComplex(Integer,[x,y,z])
theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz
|
|
Type: DeRhamComplex(Integer,[x,y,z])
This allows us to get formal definitions for the ``gradient'' ...
totalDifferential(a(x,y,z))$der
|
|
Type: DeRhamComplex(Integer,[x,y,z])
the ``curl'' ...
exteriorDifferential sigma
|
|
Type: DeRhamComplex(Integer,[x,y,z])
and the ``divergence.''
exteriorDifferential theta
|
|
Type: DeRhamComplex(Integer,[x,y,z])
Note that the De Rham complex is an algebra with unity. This element
1 is the basis for elements for zero-forms, that is, functions
in our space.
Type: DeRhamComplex(Integer,[x,y,z])
To convert a function to a function lying in the De Rham complex,
multiply the function by ``one.''
g1 : der := a([x,t,y,u,v,z,e]) * one
Type: DeRhamComplex(Integer,[x,y,z])
A current limitation of Axiom forces you to write functions with more
than four arguments using square brackets in this way.
h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one
Type: DeRhamComplex(Integer,[x,y,z])
Now note how the system keeps track of where your coordinate functions
are located in expressions.
|
|
Type: DeRhamComplex(Integer,[x,y,z])
|
|
Type: DeRhamComplex(Integer,[x,y,z])
In this example of Euclidean three-space, the basis for the De Rham complex
consists of the eight forms: 1, dx, dy, dz,
dx*dy, dx*dz, dy*dz, and dx*dy*dz.
coefficient(gamma, dx*dy)
|
|
Type: Expression Integer
Type: Expression Integer
Type: Expression Integer