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In the book An introduction to manifolds by Tu, Loring W, it says that

The cross product $v\times w$ on $\mathbb R^3$ is alternating.

However in the definition of alternating, it says that

A $k$-linear function $f:V^k\to\mathbb R$ is symmetric, if \begin{align*} f\left(v_{\sigma(1)}, \ldots, v_{\sigma(k)}\right)=f\left(v_{1}, \ldots, v_{k}\right) \end{align*} for all permutations $\sigma\in S_k$; it is alternating, if \begin{align*} f\left(v_{\sigma(1)}, \ldots, v_{\sigma(k)}\right)=(\operatorname{sgn} \sigma) f\left(v_{1}, \ldots, v_{k}\right) \end{align*} for all $\sigma\in S_k$.

However, I can't understand why the cross product is alternating. I think that the cross product $\times$ can be viewed as a function $\mathbb R^3\times \mathbb R^3\to\mathbb R^3$, but not any function that maps to $\mathbb R$. So why this is alternating?

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You're correct that using the given definition strictly, this doesn't make sense. However, there are a couple small changes in interpretation you could choose to make it work:

  1. Extend the definition of a symmetric and alternating functions to apply to $k-$linear functions $V^k \to W$ for any vector space $W$. (You can use the same defining formulae.) In the case of the cross product, it's alternating in this sense simply because $u \times v = - v \times u.$

  2. "Lower an index" on the cross product via the metric, i.e. consider $f(u,v,w) = (u \times v) \cdot w.$ This is then a 3-tensor on $\mathbb R^3$ that co-incides with the determinant.

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  • $\begingroup$ Could you please expand on via the metric? Except from dot product, what can also be used to "Lower an index"? $\endgroup$
    – FFjet
    Nov 24 '20 at 1:41
  • $\begingroup$ @FFjet: in Riemannian geometry there's a generalization of the dot product called the Riemannian metric, and it is quite common to implicitly use it to change the type of a tensor. This is also known raising/lowering indices or as the musical isomorphisms. If you're just beginning to learn differential geometry then I wouldn't worry too much about that (and I'm surprised Tu's book seems to assume you're familiar with the idea). $\endgroup$ Nov 24 '20 at 1:45
  • $\begingroup$ Ok, thanks a lot :) $\endgroup$
    – FFjet
    Nov 24 '20 at 1:48
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If you expand the definition of alternating to linear functions from $\mathbb{R}^n$ to $\mathbb{R}^m$, one has the following.

If $u$ and $v$ are linearly independant vectors, then $u\times v$ is defined to be the unique vector such that

  • $\|u\times v\| = \|u\|\|v\||\sin(u,v)|$
  • $u\times v$ is orthogonal to the plane generated by $u$ and $v$
  • $(u,v,u\times v)$ is a positively oriented basis

Thus, if $(u,v,u\times v)$ is positively oriented, then $(v,u,u\times v)$ is negatively oriented. Hence, $(v,u,-(u\times v))$ is again positively oriented. The norm of $-(u\times v)$ is the good one, and it is orthogonal to $u$ and $v$. By uniqueness, $-(u\times v) = v\times v$.

As the cross product is bilinear, and the only permutations on a set of two elements are the identity or the transposition, it follows that the cross product is alternating.

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