as.1form {stokes}

Coerce vectors to 1-forms

Description

Given a vector, return the corresponding 1-form; the exterior derivative of a 0-form (that is, a scalar function). Function grad() is a synonym.

Usage

as.1form(v)
grad(v)

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(s)

Robin K. S. Hankin

See Also

kform

Examples

as.1form(1:9)  # note ordering of terms


as.1form(rnorm(20))

grad(c(4,7)) ^ grad(1:4)