tensor notation surprise

Question

I'm trying to study tensors from several textbooks. One early example completely confuses me:

Islam, Tensors and their Applications, in the "Preliminaries" chapter, gives this example (page 3, using summation convention):

$a_{ij} x^i x^j = a_{11} (x^1)^2 + a_{22} (x^2)^2 + \cdots + a_{nn} x^n x^n$

i.e. there are no "cross" factors. I would have thought instead it would be

$a_{ij} x^i x^j = a_{11} (x^1)^2 + a_{12} x^1 x^2 + \cdots + a_{21} x^2 x^1 + a_{22} (x^2)^2 + \cdots$

i.e. like a quadratic form (in linear algebra terms). The book looks appropriate for me, lots of worked examples, but sadly I'm stuck already!

Answer 1

I believe there is a error in the book. Note that the book performs the expansion in two steps:

$$ a_{ij} x^i x^j = a_{1j} x^1 x^j + a_{2j} x^2 x^j + \cdots + a_{nj} x^n x^j \\ = a_{11} x^1 x^1 + a_{22} x^2 x^2 + \cdots + a_{nn} x^n x^n $$

The first step has the cross products, as you correctly expected, but the second step just seems wrong unless you assume that $a_{ij} = \delta_{ij}$. Kind of a bad error to have in the preliminary explanation of the notation.