Coerce vectors to 1-forms

Arguments

v

A vector with element $i$ being $\partial f/\partial x_i$

Details

The exterior derivative of a $k$-form $\phi$ is a $(k+1)$-form $\mathrm{d}\phi$ given by

$$\mathrm{d}\phi \left( P_\mathbf{x}\left(\mathbf{v}_i,\ldots,\mathbf{v}_{k+1}\right) \right) = \lim_{h\longrightarrow 0}\frac{1}{h^{k+1}}\int_{\partial P_\mathbf{x}\left(h\mathbf{v}_1,\ldots,h\mathbf{v}_{k+1}\right)}\phi$$

We can use the facts that

$$\mathrm{d}\left(f\,\mathrm{d}x_{i_1}\wedge\cdots\wedge\mathrm{d}x_{i_k}\right)= \mathrm{d}f\wedge\mathrm{d}x_{i_1}\wedge\cdots\wedge\mathrm{d}x_{i_k}$$

and $$\mathrm{d}f=\sum_{j=1}^n\left(D_j f\right)\,\mathrm{d}x_j$$

to calculate differentials of general $k$-forms. Specifically, if

$$\phi=\sum_{1\leq i_i < \cdots < i_k\leq n} a_{i_1\ldots i_k}\mathrm{d}x_{i_1}\wedge\cdots\wedge\mathrm{d}x_{i_k}$$

then $$\mathrm{d}\phi= \sum_{1\leq i_i < \cdots < i_k\leq n} [\sum_{j=1}^nD_ja_{i_1\ldots i_k}\mathrm{d}x_j]\wedge\mathrm{d}x_{i_1}\wedge \cdots\wedge\mathrm{d}x_{i_k.}$$

The entry in square brackets is given by grad(). See the examples for appropriate R idiom.

Value

A one-form

Author

Robin K. S. Hankin

See also

Examples

as.1form(1:9)  # note ordering of terms
#> An alternating linear map from V^1 to R with V=R^9:
#>        val
#>  9  =    9
#>  8  =    8
#>  7  =    7
#>  6  =    6
#>  5  =    5
#>  4  =    4
#>  3  =    3
#>  2  =    2
#>  1  =    1


as.1form(rnorm(20))
#> An alternating linear map from V^1 to R with V=R^20:
#>                val
#>   1  =  -0.8607243
#>  14  =  -0.4321952
#>   2  =   0.4212304
#>  15  =  -0.6675648
#>   3  =   1.4505432
#>  16  =   1.3895059
#>   4  =   0.1943924
#>  13  =  -0.7820777
#>  12  =  -0.7721921
#>  11  =   0.9110785
#>  10  =  -0.7160587
#>   9  =  -1.7810619
#>   8  =  -0.9441017
#>   7  =   2.7361084
#>  20  =  -0.7893881
#>  19  =   2.5844322
#>   6  =   1.3398599
#>  18  =   0.2053894
#>   5  =  -0.6912054
#>  17  =   0.9118739

grad(c(4,7)) ^ grad(1:4)
#> An alternating linear map from V^2 to R with V=R^4:
#>          val
#>  1 2  =    1
#>  1 3  =   12
#>  2 3  =   21
#>  1 4  =   16
#>  2 4  =   28