vector - How to define an orthogonal basis in the right way?

I am trying to work with the vector notation without defining vector components explicitly.

  $Assumptions = (x | y | z) \[Element] Vectors[3]

The vectors x, y and z are unit and orthogonal. To set this I'm using the UpSet command:

x.x^=1; x.y^=0; x.z^=0;
y.x^=0; y.y^=1; y.z^=0;
z.x^=0; z.y^=0; z.z^=1;

Most of the needed vector operations work correctly in this case, for example:

TensorProduct[x, y].(2 y + z) // TensorExpand

evaluates correctly to 2 x. However, (x,y,z) is not a "complete" coordinate system without defining "handness", so the cross product doesn't work.

Cross[x, y] should evaluate to z, Cross[z,y] to -x and so on for a "right hand" coordinate system. How do I implement this?

I am using Mathematica 9, but I can upgrade to 10 if needed.

edit:

As the response to one the comments: I'm working with the vector fields, say s and p, defined in the terms of x,y,z. I need to calculate matrix elements such as s.T.t where T, for example an operator e.z, with e as a Levi Civita tensor. For 'e.z' I can simply define the operator as T == -TensorProduct[x,y] + TensorProduct[y,x], but for more complex expressions it can be complicated.

Answer

You can combine the best of both worlds: symbolic tensors and vectors on one hand, and explicit vectors on the other. Explicit vectors are necessary in most vector algebra operations, unless you want to rely heavily on UpValues defined for all those operations and all the symbols you're using. It's cleaner to let Mathematica's matrix algebra take over whenever symbolic simplifications don't get anywhere.

So here is what I'd suggest:

First keep the $Assumptions that you defined, in order for TensorExpand to give simplifications whenever possible.

Then I define just two functions that can take care of all the rest: tensorExpand (with lower case spelling) is an extension of TensorExpand that post-processes the result by temporarily replacing x, y, and z, by their canonical unit vector counterparts.

This allows for things like Cross and Dot to work without any UpSet definitions. When that's complete, you have a simplified expression that contains 3D vectors, matrices and potentially higher rank tensors. Whatever those may be, I can always convert each of them to SparseArrays and extract their ArrayRules. From those, it's easy to read off how it can be rewritten as a linear combination of tensor products of the basis vectors. This is done in the function toSymbols.

Clear[x, y, z];
$Assumptions = (x | y | z) ∈ Vectors[3];

tensorExpand[expr_] := 
 Simplify[TensorExpand[expr] /. 

     Thread[{x, y, z} -> IdentityMatrix[3]]] /. 
   l_List :> SparseArray[l] /. s_SparseArray :> toSymbols[ArrayRules@s]

toSymbols[ruleList_] := 
 Total[ruleList /. 
   HoldPattern[Rule[a_, b_]] :> 
    Times[Apply[TensorProduct, a /. Thread[{1, 2, 3} -> {x, y, z}]], 
     b]]

Here are some tests:

tensorExpand[TensorProduct[x, y]]

(* ==> x⊗y *)

tensorExpand[TensorProduct[x, y].(2 y + z)]

(* ==> 2 x *)

tensorExpand[Cross[x, y]]


(* ==> z *)

tensorExpand[Cross[x, y].(2 y + z)]

(* ==> 1 *)

tensorExpand[Cross[x, {a, b, c}]]

(* ==> -c y + b z *)

As you can see, the use of component vectors in the intermediate calculations is completely invisible in the end results. You always get back an expression that is written in terms of the symbolic basis vectors, so you can do further manipulations on them if needed.

Edit adding LeviCivitaTensor etc.

The handling of LeviCivitaTensor requested in the question also works without any problems:

ϵ = LeviCivitaTensor[3];

tensorExpand[ϵ]

(*
==> x⊗y⊗z - x⊗z⊗y - y⊗x⊗z + y⊗z⊗x + z⊗x⊗y - z⊗y⊗x
*)


tensorExpand[ϵ.z]

(* ==> x⊗y - y⊗x *)

By the way, this tells me that you had a sign error in your version of T = ϵ.z.

When mixing symbolic vectors and explicit arrays, one has to watch out for cases where the symbols get pulled into the array when you don't use the Dot multiplication. But I don't think this issue arises in your question.