kinner {stokes}

Inner product of two kforms

Description

Given two k-forms \alpha and \beta, return the inner product \left\langle\alpha,\beta\right\rangle. Here our underlying vector space V is \mathcal{R}^n.

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

\left\langle\alpha,\beta\right\rangle=\det\left( \left\langle\alpha_i,\beta_j\right\rangle_{1\leq i,j\leq n}\right)

and secondly, we extend to the whole of \Lambda^k(V) 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