Ops: Arithmetic Ops Group Methods for 'kform' and 'ktensor'...

In RobinHankin/wedge: The Exterior Calculus

Arithmetic Ops Group Methods for kform and ktensor objects

Description

Allows arithmetic operators to be used for k-forms and k-tensors such as addition, multiplication, etc, where defined.

Usage

## S3 method for class 'kform'
Ops(e1, e2 = NULL)
## S3 method for class 'ktensor'
Ops(e1, e2 = NULL)

Arguments

e1, e2

Objects of class kform or ktensor

Details

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%.

Value

All functions documented here return an object of class kform or ktensor.

Note

A plain asterisk, “*” behaves differently for ktensors and kforms.

Given two ktensors T1 and T2, then “T1*T2” will return their tensor product, tensorprod(T1,T2). 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, in which the scalar is interpreted as a 0-tensor.

Given two kforms K1 and K2, asterisks behave differently. An asterisk cannot be used to multiply K1 and K2: K1*K2 will return an error. Multiplication by scalars is OK, as in K1*6.

Powers simply do not make sense for alternating forms: if we wish to calculate S^2 (that is, the square of S), then the only natural interpretation of this is S^S, but S %^% S = S^S is zero identically. In the package, therefore, the caret (“^”) evaluates the wedge product; note that %^% is also acceptable.

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 ot 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.

Author(s)

Robin K. S. Hankin

Examples

## dx_1 ^ dx_2 + 6dx_5 ^ dx_6:
as.kform(1) ^ as.kform(2) + 6*as.kform(5) ^ as.kform(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)

## verify linearity, here 2*k1 + 3*k2:
as.function(2*k1+3*k2)(E)-(2*as.function(k1)(E) + 3*as.function(k2)(E))
## should be small

RobinHankin/wedge documentation built on June 9, 2025, 2:53 p.m.