Einstein Notation

Superscript and subscript are for contravariant and covariant components, we do not use them so far. We only use subscript as indices.

Vector dot product. \(a\cdot b=a_ib_i\)

a=c(1,2,3)
b=c(4,5,6)
a%*%b
##      [,1]
## [1,]   32

Dyadic vector product \(a⊗b=a_ib_j\)

a=c(1,2,3)
b=c(4,5,6)
a%*%t(b)
##      [,1] [,2] [,3]
## [1,]    4    5    6
## [2,]    8   10   12
## [3,]   12   15   18

Vector cross products, length 3 case \(a\times b=\epsilon_{ijk} a_j b_k\) \[\epsilon_{1jk}=\left[ {\begin{array} *{0}&{0}&0\\ {0}&{0}&-1\\{0}&{1}&0 \end{array}} \right] \epsilon_{2jk}=\left[ {\begin{array} *{0}&{0}&1\\ {0}&{0}&0\\{-1}&{0}&0 \end{array}} \right] \epsilon_{3jk}=\left[ {\begin{array} *{0}&{-1}&0\\ {1}&{0}&0\\{0}&{0}&0 \end{array}} \right]\]

a=c(1,2,3)
b=c(4,5,6)

e1=matrix(c(0,0,0,0,0,-1,0,1,0),nrow=3)
e2=matrix(c(0,0,1,0,0,0,-1,0,0),nrow=3)
e3=matrix(c(0,-1,0,1,0,0,0,0,0),nrow=3)

sum(e1*a%*%t(b))
## [1] -3
sum(e2*a%*%t(b))
## [1] 6
sum(e3*a%*%t(b))
## [1] -3
library(pracma)
cross(a, b)
## [1] -3  6 -3

Transpose, \((C_{ij})^T=C_{ji}\)

T=matrix(c(1,2,3,4,5,6,7,8,9),nrow=3)
t(T)
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    4    5    6
## [3,]    7    8    9

Same position product of matrix, \(T : S=T_{ij}S_{ij}\)

T=matrix(c(1,2,3,4,5,6,7,8,9),nrow=3)
S=matrix(c(1,2,3,4,5,6,7,8,9),nrow=3)
T*S
##      [,1] [,2] [,3]
## [1,]    1   16   49
## [2,]    4   25   64
## [3,]    9   36   81