1.2 Exported methods

Let us briefly disuss the exported methods of X=DERHAM:

leadingCoefficient

Returns the leading coefficient, obviously, but what is the leading coefficient?

leadingCoefficient(a*dx + b*dy + c*dz)

\[c\]

_{Type: Expression(Integer)}

It seems that it is the first coefficient given in the OutputForm. It is not quite clear what happens when this order is changed by some method, if possible at all (todo).

leadingBasisTerm

Returns the basis form (corresponding to the leadingCoefficient).

leadingBasisTerm(a*dx + b*dy +c*dz)

\[dz\]

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

reductum

Returns df minus the leading term of σ if σ has two or more terms, and 0 otherwise.

reductum(a*dx + b*dy + c*dz)

\[b dy + a dx\]

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

One can see that this works like a list of cons’: head/tail where head can be split into coefficient and basis term and so on, allowing traverse the term recursively.

coefficient

coefficient(σ, u), where σ is a differential form, returns the coefficient of df containing the basis term u if such a term exists, and 0 otherwise. If we want the coefficient of the basis term d x∧d y for example:

coefficient(a*dx + b*dy + c*dz, dy)

\[b\]

_{Type: Expression(Integer)}

generator

generator(i) returns the i^th basis 1-form for X. For instance

generator(1)$X

\[dx\]

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

The description given in derham.spad seems to be not quite suitable.

homogeneous?

homogeneous?(σ) tests if all of the terms of differential form σ have the same degree. The degree is defined as p if σ is a pure p-form.

homogeneous?(a*dx + b*dy + c*dz)

\[\mathrm{true}\]

_{Type: Boolean}

Adding a term like dx*dy above would result in false.

retractable?

retractable?(σ) tests if differential form σ is a 0-form, i.e., if degree(σ) = 0.

retractable? (1::X)

\[\mathrm{true}\]

_{Type: Boolean}

retractable? dx

\[\mathrm{false}\]

_{Type: Boolean}

degree

degree(σ) returns the homogeneous degree of differential form σ.

degree(dx+dx*dz)

>> Error detected within library code:
not a homogeneous element

degree(dx*dy)

\[2\]

_{Type: PositiveInteger}

map

map(f, σ) replaces each coefficient c of a differential form σ by f(c). For example let f:R→R⁣,x↦x^2, then all coefficients should be squared:

R ==> Expression Integer
f:R->R
f(x)==x^2

map(f,a*dx + b*dy + c*dz)

\[c^2\, dz + b^2\, dy + a^2\, dx\]

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

totalDifferential

totalDifferential(f) returns the total differential (gradient) form for a function f.

g:=operator 'g

totalDifferential(g(x,y,z))$X

\[{{{g _ {{,3}}} \left( {x, \: y, \: z} \right)} \ dz}+{{{g _ {{,2}}} \left( {x, \: y, \: z} \right)} \ dy}+{{{g _ {{,1}}} \left( {x, \: y, \: z} \right)} \ dx}\]

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

Here, polynomial expressions are allowed as well:

totalDifferential(x^m)$X

\[m\,x^{m-1}\,dx\]

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

exteriorDifferential

exteriorDifferential(σ) returns the exterior derivative of the differential form σ. This is the well known d with d∘d=0. We will abbreviate it by

d ==> exteriorDifferential

d(x*dy + z*dx*dy)

\[{dx \ dy}+{dx \ dy \ dz}\]

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

FriCAS :: DeRhamComplex