Symbolic form

Arguments

M

Object of class kform or ktensor; a map from $V^k$ to $\mathbb{R}$, where $V=\mathbb{R}^n$

symbols

A character vector giving the names of the symbols

d

String specifying the appearance of the differential operator

Details

Spivak (p89), in archetypically terse writing, states:

A function $f$ is considered to be a 0-form and $f\cdot\omega$ is also written $f\wedge\omega$. If $f\colon\mathbb{R}^n\longrightarrow\mathbb{R}$ is differentiable, then $Df(p)\in\Lambda^1\left(\mathbb{R}^n\right)$. By a minor modification we therefore obtain a 1-form $\mathrm{d}f$, defined by $$\mathrm{d}f(p)\left(v_p\right)=Df(p)(v).$$

Let us consider in particular the 1-forms $\mathrm{d}\pi^i$. It is customary to let $x^i$ denote the function $\pi^i$ (On $\mathbb{R}^3$ we often denote $x^1$, $x^2$, and $x^3$ by $x$, $y$, and $z$). This standard notation has obvious disadvantages but it allows many classical results to be expressed by formulas of equally classical appearance. Since $\mathrm{d}x^i(p)(v_p)=\mathrm{d}\pi^i(p)(v_p)=D\pi^i(p)(v)=v^i$, we see that $\mathrm{d}x^1(p),\ldots,\mathrm{d}x^n(p)$ is just the dual basis to $(e_1)_p,\ldots,(e_n)_p$. Thus every k-form $\omega$ can be written

$$\omega=\sum_{i_1 < \cdots < i_k}\omega_{i_1,\ldots,i_k} \mathrm{d}x^{i_1}\wedge\cdots\wedge\mathrm{d}x^{i_k}.$$

Function as.symbolic() uses this format. For completeness, we add (p77) that $k$-tensors may be expressed in the form

$$\sum_{i_1,\ldots, i_k=1}^n a_{i_1,\ldots,i_k}\cdot \phi_{i_1}\otimes\cdots\otimes\phi_{i_k}.$$

and this form is used for $k$-tensors. The print method for tensors, print.ktensor(), writes d1 for $\phi_1$, d2 for $\phi_2$ [where $\phi_i(x^j)=\delta_i^j$].

Value

Returns a “noquote” character string.

Author

Robin K. S. Hankin

See also

Examples

(o <- kform_general(3,2,1:3))
#> An alternating linear map from V^2 to R with V=R^3:
#>          val
#>  2 3  =    3
#>  1 3  =    2
#>  1 2  =    1
as.symbolic(o,d="d",symbols=letters[23:26])
#> [1]  +3 dx^dy +2 dw^dy + dw^dx

(a <- rform(n=50))
#> An alternating linear map from V^3 to R with V=R^50:
#>               val
#>   4 14 22  =   -6
#>   3 36 40  =   -4
#>   1 20 44  =   -5
#>   5 23 49  =    9
#>  25 28 34  =    3
#>   3 20 38  =    8
#>  28 38 50  =   -2
#>   6 11 22  =    7
#>   9 16 28  =   -1
as.symbolic(a,symbols=state.abb)
#> [1]  -6 AR^IN^MI -4 AZ^OK^SC -5 AL^MD^UT +9 CA^MN^WI +3 MO^NV^ND +8 AZ^MD^PA -2 NV^PA^WY +7 CO^HI^MI - FL^KS^NV