kform and ktensor objectsOps.RdAllows arithmetic operators to be used for \(k\)-forms and \(k\)-tensors such as addition, multiplication, etc, where defined.
# S3 method for class 'kform'
Ops(e1, e2 = NULL)
# S3 method for class 'ktensor'
Ops(e1, e2 = NULL)The functions Ops.kform() and Ops.ktensor() pass unary
and binary arithmetic operators (“+”, “-”,
“*”, “/” and “^”) to the
appropriate specialist function by coercing to spray objects.
For wedge products of \(k\)-forms, use wedge() or
%^% or ^; and for tensor products of
\(k\)-tensors, use tensorprod() or %X%.
All functions documented here return an object of class
kform or ktensor.
A plain asterisk, “*” behaves differently for ktensors
and kforms. Given two ktensors T1, T2, then
“T1*T2” will return the their tensor product. This on
the grounds that the idiom has only one natural interpretation. But
its use is discouraged (use %X% or tensorprod()
instead). An asterisk can also be used to multiply a tensor by a
scalar, as in T1*5.
An asterisk cannot be used to multiply two kforms K1,
K2, as in K1*K2, which will always return an error.
This on the grounds that it has no sensible interpretation in general
and you probably meant to use a wedge product, K1^K2. Note
that multiplication by scalars is acceptable, as in K1*6.
Further note that K1*K2 returns an error even if one or both is
a 0-form (or scalar), as in K1*scalar(3). This behaviour may
change in the future.
In the package the caret (“^”) evaluates the wedge
product; note that %^% is also acceptable. Powers simply do
not make sense for alternating forms: S %^% S = S^S is zero
identically. Here the caret is interpreted consistently as a wedge
product, and if one of the factors is numeric it is interpreted as a
zero-form (that is, a scalar). Thus S^2 = wedge(S,2) = 2^S =
S*2 = S+S, and indeed S^n==S*n. Caveat emptor! If S
is a kform object, it is very tempting [but incorrect] to
interpret “S^3” as something like “S to
the power 3”. See also the note at Ops.clifford in the
clifford package.
Powers are not implemented for ktensors on the grounds that a ktensor to the power zero is not defined.
Note that one has to take care with order of operations if we mix
^ with *. For example, dx ^ (6*dy) is perfectly
acceptable; but (dx ^ 6)*dy) will return an error, as will the
unbracketed form dx ^ 6 * dy. In the second case we attempt to
use an asterisk to multiply two k-forms, which triggers the error.
## dx_1 ^ dx_2 + 6dx_5 ^ dx_6:
as.kform(1) ^ as.kform(2) + 6*as.kform(5) ^ as.kform(6)
#> An alternating linear map from V^2 to R with V=R^6:
#> val
#> 1 2 = 1
#> 5 6 = 6
k1 <- kform_general(4,2,rnorm(6))
k2 <- kform_general(4,2,rnorm(6))
E <- matrix(rnorm(8),4,2)
as.function(k1+k2)(E)
#> [1] -1.278449
## verify linearity, here 2*k1 + 3*k2:
as.function(2*k1+3*k2)(E)-(2*as.function(k1)(E) + 3*as.function(k2)(E))
#> [1] -1.221245e-15
## should be small