as.1form: Coerce vectors to 1-forms In stokes: The Exterior Calculus

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.

Arguments

 v A vector with element \mjseqni being \mjeqn\partial f/\partial x_idf/dxi

Details

The exterior derivative of a \mjseqnk-form \mjeqn\phiphi is a \mjseqn(k+1)-form \mjeqn\mathbfd\phid phi given by

\mjdeqn \mathbf

d\phi \left( P_\mathbfx\left(\mathbfv_i,...,\mathbfv_k+1\right) \right) = \lim_h\longrightarrow 0\frac1h^k+1\int_\partial P_\mathbfx\left(h\mathbfv_1,...,h\mathbfv_k+1\right)\phi omitted; see latex

We can use the facts that

\mjdeqn \mathbf

d\left(f\,dx_i_1\wedge\cdots\wedge dx_i_k\right)= \mathbfdf\wedge dx_i_1\wedge\cdots\wedge dx_i_k omitted; see latex

and \mjdeqn \mathbfdf=\sum_j=1^n\left(D_j f\right)\,dx_j omitted; see latex

to calculate differentials of general \mjseqnk-forms. Specifically, if

\mjdeqn \phi

=\sum_1\leq i_i < \cdots < i_k\leq n a_i_1... i_kdx_i_1\wedge\cdots\wedge dx_i_k omitted; see latex

then \mjdeqn \mathbfd\phi= \sum_1\leq i_i < \cdots < i_k\leq n [\sum_j=1^nD_ja_i_1... i_kdx_j]\wedge dx_i_1\wedge\cdots\wedge dx_i_k. omitted; see latex

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

A one-form

Author(s)

Robin K. S. Hankin