print.RdPrint methods for objects with options for printing in matrix form or multivariate polynomial form
Printing is dispatched to print.ktensor() and
print.kform() depending on its argument. Special dispensation is
given for the zero object.
Although \(k\)-forms are alternating tensors and thus mathematically are tensors, they are handled differently.
The default print method uses the spray print methods,
and as such respects the polyform option. However, setting
polyform to TRUE can give misleading output, because
spray objects are interpreted as multivariate polynomials not
differential forms (and in particular uses the caret to signify
powers).
It is much better to use options ktensor_symbolic_print or
kform_symbolic_print instead: the bespoke print methods
print.kform() and print.ktensor() are sensitive to these
options.
For kform objects, if option kform_symbolic_print is
non-null, the print method uses as.symbolic() to give an
alternate way of displaying \(k\)-tensors and \(k\)-forms. The
generic non-null value for this option would be “x”
which gives output like “dx1 ^ dx2”. However, it has
two special values: set kform_symbolic_print to
“dx” for output like “dx ^ dz” and
“txyz” for output like “dt ^ dx”, useful
in relativistic physics with a Minkowski metric. See the examples.
For ktensor objects, if option ktensor_symbolic_print is
TRUE, a different system is used. Given a tensor
\(3\phi_4\otimes\phi_1 -5\phi_2\otimes\phi_2\), for example (where
\(\phi_i(x^j)=\delta_i^j\)), the method will give output that looks
like “+3 d4*d1 -5 d2*d2”. I am not entirely happy with
this and it might change in future.
More detail is given at symbolic.Rd and the dx vignette.
Returns its argument invisibly.
For both kform and ktensor objects,
the print method asserts that its argument is a map from \(V^k\)
to \(\mathbb{R}\) with \(V=\mathbb{R}^n\). Here,
\(n\) is the largest element in the index matrix. However, such
a map naturally furnishes a map from \((\mathbb{R}^m)^k\)
to \(\mathbb{R}\), provided that \(m\geq n\) via the
natural projection from \(\mathbb{R}^n\) to
\(\mathbb{R}^m\). Formally this would be
\(\left(x_1,\ldots,x_n\right)\mapsto\left(x_1,\ldots,x_n,0,\ldots,0\right)\in\mathbb{R}^m\). In the case of the zero \(k\)-form or
\(k\)-tensor, “n” is to be interpreted as
“any \(n\geq 0\)”. See also dovs().
a <- rform()
a
#> An alternating linear map from V^3 to R with V=R^7:
#> val
#> 1 2 5 = -8
#> 1 2 3 = -6
#> 1 3 6 = 7
#> 4 5 6 = 9
#> 4 5 7 = -2
#> 2 4 5 = 1
#> 2 4 7 = -3
#> 2 3 5 = 1
options(kform_symbolic_print = "x")
a
#> An alternating linear map from V^3 to R with V=R^7:
#> -8 dx1^dx2^dx5 -6 dx1^dx2^dx3 +7 dx1^dx3^dx6 +9 dx4^dx5^dx6 -2 dx4^dx5^dx7 + dx2^dx4^dx5 -3 dx2^dx4^dx7 + dx2^dx3^dx5
options(kform_symbolic_print = "dx")
kform(spray(kform_basis(3,2),1:3))
#> An alternating linear map from V^2 to R with V=R^3:
#> +3 dy^dz +2 dx^dz + dx^dy
kform(spray(kform_basis(4,2),1:6)) # runs out of symbols
#> An alternating linear map from V^2 to R with V=R^4:
#> +6 dz^dNA +5 dy^dNA +4 dx^dNA +3 dy^dz +2 dx^dz + dx^dy
options(kform_symbolic_print = "txyz")
kform(spray(kform_basis(4,2),1:6)) # standard notation
#> An alternating linear map from V^2 to R with V=R^4:
#> +6 dy^dz +5 dx^dz +4 dt^dz +3 dx^dy +2 dt^dy + dt^dx
options(kform_symbolic_print = NULL) # revert to default
a
#> An alternating linear map from V^3 to R with V=R^7:
#> val
#> 1 2 5 = -8
#> 1 2 3 = -6
#> 1 3 6 = 7
#> 4 5 6 = 9
#> 4 5 7 = -2
#> 2 4 5 = 1
#> 2 4 7 = -3
#> 2 3 5 = 1