stokes (version 1.2-0)

scalar: Scalars and losing attributes

Description

Scalars: 0-forms and 0-tensors

Usage

scalar(s,kform=TRUE,lose=FALSE)
is.scalar(M)
`0form`(s=1,lose=FALSE)
`0tensor`(s=1,lose=FALSE)
# S3 method for kform
lose(M)
# S3 method for ktensor
lose(M)

Value

The functions documented here return an object of class

kform or ktensor, except for is.scalar(), which returns a Boolean.

Arguments

s

A scalar value; a number

kform

Boolean with default TRUE meaning to return a kform and FALSE meaning to return a ktensor

M

Object of class ktensor or kform

lose

In function scalar(), Boolean with TRUE meaning to return a normal scalar, and default FALSE meaning to return a formal 0-form or 0-tensor

Author

Robin K. S. Hankin

Details

A k-tensor (including k-forms) maps k vectors to a scalar. If k=0, then a 0-tensor maps no vectors to a scalar, that is, mapping nothing at all to a scalar, or what normal people would call a plain old scalar. Such forms are created by a couple of constructions in the package, specifically scalar(), kform_general(1,0) and contract(). These functions take a lose argument that behaves much like the drop argument in base extraction. Functions `0form()` and `0tensor()` are wrappers for `scalar()`.

Function lose() takes an object of class ktensor or kform and, if of arity zero, returns the coefficient.

Note that function kform() always returns a kform object, it never loses attributes.

There is a slight terminological problem. A k-form maps k vectors to the reals: so a 0-form maps 0 vectors to the reals. This is what anyone on the planet would call a scalar. Similarly, a 0-tensor maps 0 vectors to the reals, and so is a scalar. Mathematically, there is no difference between 0-forms and 0-tensors, but the package makes a distinction:

> scalar(5,kform=TRUE)
An alternating linear map from V^0 to R with V=R^0:
     val
  =    5
> scalar(5,kform=FALSE)
A linear map from V^0 to R with V=R^0:
     val
  =    5
> 

Compare zero tensors and zero forms. A zero tensor maps V^k to the real number zero, and a zero form is an alternating tensor mapping V^k to zero (so a zero tensor is necessarily alternating). See zero.Rd.

See Also

zeroform

Examples

o <- scalar(5)
o
lose(o)

kform_general(1,0)  
kform_general(1,0,lose=FALSE)