Page 35 - Vector Analysis

§1.9 The Einstein Summation Convention 31

the values of the index. For example, with this convention, the inner product u ¨ v of two

vectors u, v P Rn, where u = (u1, ¨ ¨ ¨ , un) and v = (v1, ¨ ¨ ¨ , vn), can be expressed as uivi,

and the i-th component of the cross product u ˆ v of two vectors u, v P R3 can be expressed

as εijkujvk.

In this book, we make a further convention that repeated Latin indices are summed

from 1 to n, and repeated Greek indices are summed from 1 to n ´ 1, where n is the space

n

dimension. In other words, we use the symbol figi to denote the sum ř figi, and the symbol

n´1 i=1

fαgα to denote the sum ř fαgα. Starting from the next Chapter, we use such summation

i=1

convention for notational simplicity.