as.1form | R Documentation |
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.
as.1form(v)
grad(v)
v |
A vector with element \mjseqni being \mjeqn\partial f/\partial x_idf/dxi |
The exterior derivative of a \mjseqnk-form \mjeqn\phiphi is a \mjseqn(k+1)-form \mjeqn\mathbfd\phid phi given by
\mjdeqn \mathbfd\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 \mathbfd\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
Robin K. S. Hankin
kform
as.1form(1:9) # note ordering of terms
as.1form(rnorm(20))
grad(c(4,7)) ^ grad(1:4)
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