# In what sense is the cross-product not a tensor?

Using the definition of tensors as tensor products, let $$\operatorname{CROSS}$$ be a type $$(1,2)$$ tensor defined by $$\operatorname{CROSS} =\sum_i\sum_j\sum_k (e_k \cdot (e_i \times e_j))\,\,e_i^*\otimes e_j^*\otimes e_k$$

which is a linear combination of other tensors, therefore a tensor.

Observe that $$\operatorname{CROSS}_{ij}^k u^iv^j = (u \times v)^k$$. Hence, the cross product actually is a tensor. ([EDIT] Wikipedia agrees).

What do people mean when they say the cross product is not a tensor? And what do they mean by "the" cross product when they say that? Because the above proves it is.

 I understand on an intuitive level what this article on pseudo-vectors is trying to say about the cross product. But I don't understand how to formalise that "problem". I'm not talking about formalising the "solution", which involves exterior algebra or Clifford algebra.

( Attempt at formalising: The examples they give don't involve changing the coordinate system, but involving transforming the actual physical system by some reflection. For instance, if in the cars example, the coordinate system were reflected in the way they described, then the angular momentum vector would point in the opposite direction, contrary to what they're saying. More generally, if by reflecting a left-handed cross product, you end up with another left-handed cross product, then your reflection was not a change of basis, but a change that affected physical objects.)

See here for people claiming that the cross product is not a tensor. It seems like what's really happening is that they're defining the cross product to be the exterior product. In other words, when they say cross product, they don't actually mean cross product.

Also this article on Levi-Civita symbols seems to be saying something relevant.

• What makes you think there is a problem with it? – Eric Wofsey Mar 2 at 23:12
• @EricWofsey I seem to be getting confused. Some sources (see here) seem to be saying that the "cross product" is not a tensor. I'm thinking now that they're talking about the "result" of the cross product, which is supposed to be something called a pseudovector, and not the cross-product operation itself – man and laptop Mar 2 at 23:20
• Ok, now I see. The problem is just the definition of tensor. Usually a tensor is an element of a tensor product of tangent and cotangent spaces, so a vector is an example of tensor. The most important propriety, is that coordinate changes "tensorially". So, the cross product of two vector field is a a vector field too and so it is a tensor. – user84976 Mar 2 at 23:30