stokes (version 1.0-8)

as.1form: 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 \(\partial f/\partial x_i\)

Value

A one-form

Details

The exterior derivative of a \(k\)-form phi is a \((k+1)\)-form dd phi given by

d ( P_x(v_i,…,v_k+1) ) = _h 01h^k+1_ P_x(hv_1,…,hv_k+1) omitted; see latex

We can use the facts that

d(f\,dx_i_1 dx_i_k)= df dx_i_1 dx_i_k omitted; see latex

and

df=_j=1^n(D_j f)\,dx_j omitted; see latex

to calculate differentials of general \(k\)-forms. Specifically, if

=_1 i_i < < i_k n a_i_1… i_kdx_i_1 dx_i_k omitted; see latex

then

d= _1 i_i < < i_k n [_j=1^nD_ja_i_1… i_kdx_j] dx_i_1 dx_i_k. omitted; see latex

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

See Also

kform

Examples

# NOT RUN {
as.1form(1:9)  # note ordering of terms


as.1form(rnorm(20))

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




# }