I am currently studying

*Classical Dynamics of Particles and Systems* by Marion/Thornton. In the first chapter matrices, vectors and vector calculus are discussed, and in the context of that the vector product in three dimensions is introduced like so :

with the justification that

The symbol

is herein called the

*Levi-Civita density*. Purely algebraically this all checks out ( I have done the sums myself ), but I was wondering the following :

1. Why is

called a

*density* ? Is there any significance to that ?

2. While this definition of the vector product works algebraically, it is not at all intuitive due to the presence of aforementioned density. Can someone explain the deeper significance of the Levi-Civita symbol ? Is there any way to visualise this geometrically ? I get that this is a definition in terms of coordinate basis ( unit ) vectors, and that makes sense. I just don't really get the presence of the Levi-Civita symbol, and its true meaning.

I also don't really understand this bit :

Again, algebraically there is no problem, but I don't really get the deeper reasoning and significance behind this.