kinner: Inner product of two kforms

In stokes: The Exterior Calculus

Inner product of two kforms

Description

\loadmathjax

Given two \mjseqnk-forms \mjeqn\alphaa and \mjeqn\betab, return the inner product \mjeqn\left\langle\alpha,\beta\right\rangle<a,b>. Here our underlying vector space \mjseqnV is \mjeqn\mathcalR^nR^n.

The inner product is a symmetric bilinear form defined in two stages. First, we specify its behaviour on decomposable \mjseqnk-forms \mjeqn\alpha=\alpha_1\wedge\cdots\wedge\alpha_komitted and \mjeqn\beta=\beta_1\wedge\cdots\wedge\beta_komitted as

\mjdeqn \left\langle\alpha

,\beta\right\rangle=\det\left( \left\langle\alpha_i,\beta_j\right\rangle_1\leq i,j\leq n\right) omitted

and secondly, we extend to the whole of \mjeqn\Lambda^k(V)omitted through linearity.

Usage

kinner(o1,o2,M)

Arguments

o1,o2	Objects of class kform
M	Matrix

Value

Returns a real number

Note

There is a vignette available: type vignette("kinner") at the command line.

Author(s)

Robin K. S. Hankin

See Also

hodge

Examples

a <- (2*dx)^(3*dy)
b <- (5*dx)^(7*dy)

kinner(a,b)
det(matrix(c(2*5,0,0,3*7),2,2))  # mathematically identical, slight numerical mismatch

stokes documentation built on Aug. 19, 2023, 1:07 a.m.