as.1form {stokes} | R Documentation |
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 ∂f/∂xi
|
Details
The exterior derivative of a k
-form ϕ
is a
(k+1)
-form dϕ
given by
dϕ(Px(vi,…,vk+1))=limh⟶0hk+11∫∂Px(hv1,…,hvk+1)ϕ
We can use the facts that
d(fdxi1∧⋯∧dxik)=df∧dxi1∧⋯∧dxik
and
df=∑j=1n(Djf)dxj
to calculate differentials of general k
-forms. Specifically, if
ϕ=∑1≤ii<⋯<ik≤nai1…ikdxi1∧⋯∧dxik
then
dϕ=∑1≤ii<⋯<ik≤n[∑j=1nDjai1…ikdxj]∧dxi1∧⋯∧dxik.
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)
[Package
stokes version 1.2-1
Index]