# contract() and related functions in the stokes package In stokes: The Exterior Calculus

knitr::opts_chunk$set(echo = TRUE) library("stokes") set.seed(0)  ![](r system.file("help/figures/stokes.png", package = "stokes")){width=10%} contract contract_elementary  ## Contractions Given a$k$-form$\phi\colon V^k\longrightarrow\mathbb{R}$and a vector$\mathbf{v}\in V$, the contraction$\phi_\mathbf{v}$of$\phi$and$\mathbf{v}$is a$k-1$-form with [ \phi_\mathbf{v}\left(\mathbf{v}^1,\ldots,\mathbf{v}^{k-1}\right) = \phi\left(\mathbf{v},\mathbf{v}^1,\ldots,\mathbf{v}^{k-1}\right) ] provided$k>1$; if$k=1$we specify$\phi_\mathbf{v}=\phi(\mathbf{v})$. Function contract_elementary() is a low-level helper function that translates elementary$k$-forms with coefficient 1 (in the form of an integer vector corresponding to one row of an index matrix) into its contraction with$\mathbf{v}$; function contract() is the user-friendly front end. We will start with some simple examples. I will use phi and$\phi$to represent the same object. (phi <- as.kform(1:5))  Thus$k=5$and we have$\phi=dx^1\wedge dx^2\wedge dx^3\wedge dx^4\wedge dx^5$. We have that$\phi$is a linear alternating map with $$\phi\left(\begin{bmatrix}1\0\0\0\0\end{bmatrix}, \begin{bmatrix}0\1\0\0\0\end{bmatrix}, \begin{bmatrix}0\0\1\0\0\end{bmatrix}, \begin{bmatrix}0\0\0\1\0\end{bmatrix}, \begin{bmatrix}0\0\0\0\1\end{bmatrix}, \right)=1$$. The contraction of$\phi$with any vector$\mathbf{v}$is thus a$4$-form mapping$V^4$to the reals with$\phi_\mathbf{v}\left(\mathbf{v}^1,\mathbf{v}^2,\mathbf{v}^3,\mathbf{v}^4\right)=\phi\left(\mathbf{v},\mathbf{v}^1,\mathbf{v}^2,\mathbf{v}^3,\mathbf{v}^4\right)$. Taking the simplest case first, if$\mathbf{v}=(1,0,0,0,0)$then v <- c(1,0,0,0,0) contract(phi,v)  that is, a linear alternating map from$V^4$to the reals with $$\phi_\mathbf{v}\left( \begin{bmatrix}0\1\0\0\0\end{bmatrix}, \begin{bmatrix}0\0\1\0\0\end{bmatrix}, \begin{bmatrix}0\0\0\1\0\end{bmatrix}, \begin{bmatrix}0\0\0\0\1\end{bmatrix}\right)=1$$. (the contraction has the first argument of$\phi$understood to be$\mathbf{v}=(1,0,0,0,0)$). Now consider$\mathbf{w}=(0,1,0,0,0)$: w <- c(0,1,0,0,0) contract(phi,w)  $$\phi_\mathbf{w}\left( \begin{bmatrix}0\0\1\0\0\end{bmatrix}, \begin{bmatrix}1\0\0\0\0\end{bmatrix}, \begin{bmatrix}0\0\0\1\0\end{bmatrix}, \begin{bmatrix}0\0\0\0\1\end{bmatrix}\right)=1 \qquad\mbox{or}\qquad \phi_\mathbf{w}\left( \begin{bmatrix}1\0\0\0\0\end{bmatrix}, \begin{bmatrix}0\0\1\0\0\end{bmatrix}, \begin{bmatrix}0\0\0\1\0\end{bmatrix}, \begin{bmatrix}0\0\0\0\1\end{bmatrix}\right)=-1$$. Contraction is linear, so we may use more complicated vectors: contract(phi,c(1,3,0,0,0)) contract(phi,1:5)  We can check numerically that the contraction is linear in its vector argument:$\phi_{a\mathbf{v}+b\mathbf{w}}= a\phi_\mathbf{v}+b\phi_\mathbf{w}$. a <- 1.23 b <- -0.435 v <- 1:5 w <- c(-3, 2.2, 1.1, 2.1, 1.8) contract(phi,a*v + b*w) == a*contract(phi,v) + b*contract(phi,w)  We also have linearity in the alternating form:$(a\phi+b\psi)\mathbf{v}=a\phi\mathbf{v} + b\psi_\mathbf{v}$. (phi <- rform(2,5)) (psi <- rform(2,5)) a <- 7 b <- 13 v <- 1:7 contract(a*phi + b*psi,v) == a*contract(phi,v) + b*contract(psi,v)  It is of course possible to contract a contraction. If$\phi$is a$k$-form, then$\left(\phi_\mathbf{v}\right)_\mathbf{w}$is a$k-2$form with $$\left(\phi_\mathbf{u}\right)_\mathbf{v}\left(\mathbf{w}^1,\ldots,\mathbf{w}^{k-2}\right)=\phi\left(\mathbf{u},\mathbf{v},\mathbf{w}^1,\ldots,\mathbf{w}^{k-2}\right)$$ And this is straightforward to realise in the package: (phi <- rform(2,5)) u <- c(1,3,2,4,5,4,6) v <- c(8,6,5,3,4,3,2) contract(contract(phi,u),v)  But contract() allows us to perform both contractions in one operation: if we pass a matrix$M$to contract() then this is interpreted as repeated contraction with the columns of$M$: M <- cbind(u,v) contract(contract(phi,u),v) == contract(phi,M)  We can verify directly that the system works as intended. The lines below strip successively more columns from argument V and contract with them: (o <- kform(spray(t(replicate(2, sample(9,4))), runif(2)))) V <- matrix(rnorm(36),ncol=4) jj <- c( as.function(o)(V), as.function(contract(o,V[,1,drop=TRUE]))(V[,-1]), # scalar as.function(contract(o,V[,1:2]))(V[,-(1:2),drop=FALSE]), as.function(contract(o,V[,1:3]))(V[,-(1:3),drop=FALSE]), as.function(contract(o,V[,1:4],lose=FALSE))(V[,-(1:4),drop=FALSE]) ) print(jj) max(jj) - min(jj) # zero to numerical precision  and above we see agreement to within numerical precision. If we pass three columns to contract() the result is a$0$-form: contract(o,V)  In the above, the result is coerced to a scalar which is returned in the form of a disord object; in order to work with a formal$0$-form (which is represented in the package as a spray with a zero-column index matrix) we can use the lost=FALSE argument: contract(o,V,lose=FALSE)  thus returning a$0$-form. If we iteratively contract a$k$-dimensional$k\$-form, we return the determinant, and this may be verified as follows:

o <- as.kform(1:5)
V <- matrix(rnorm(25),5,5)
LHS <- det(V)
RHS <- contract(o,V)
c(LHS,RHS,LHS-RHS)


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stokes documentation built on Jan. 18, 2022, 1:11 a.m.