Is the Wedge Product of Vectors Equal to the Cross Product?

[Mandelbroth]

Consider $\vec{a}=\begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix}$ and $\vec{b}=\begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}$.

Is any part of the following NOT true?

$$\vec{a}\wedge\vec{b}=\frac{1}{2}(\vec{a}\otimes\vec{b}-\vec{b}\otimes\vec{a}) = \frac{1}{2}\begin{bmatrix} 0 & a_1b_2-a_2b_1 & a_1b_3-a_3b_1 \\ a_2b_1-a_1b_2 & 0 & a_2b_3-a_3b_2 \\ a_3b_1-a_1b_3 & a_3b_2-a_2b_3 & 0 \end{bmatrix}$$
(Edited for accuracy to make me feel better )

I wasn't sure if that is actually an equality or not. If it is an equality, it makes defining the cross product rather easy. It would just be the Hodge dual of the wedge product of a with b, because the inner product of the cross product of a and b with any basis of the matrix would be 0...

 

[WannabeNewton]

$v\wedge w = \frac{1}{2}(v\otimes w - w\otimes v)$ by definition i.e. $(v \wedge w)_{ab} = v_{[a}w_{b]} = \frac{1}{2}(v_{a}w_{b} - v_{b}w_{a})$.

 

[Mandelbroth]

$v\wedge w = \frac{1}{2}(v\otimes w - w\otimes v)$ by definition i.e. $(v \wedge w)_{ab} = v_{[a}w_{b]} = \frac{1}{2}(v_{a}w_{b} - v_{b}w_{a})$.

So it's one half of what I put for the second and third parts?

 

[WannabeNewton]

So it's one half of what I put for the second and third parts?

If by second and third parts you mean the two expressions following the wedge product (the alternating tensor product and the matrix) then yes.

 

[Ben Niehoff]

WBN has the factor of 2 wrong.

v \wedge w \equiv v \otimes w - w \otimes v

 

[WannabeNewton]

WBN has the factor of 2 wrong.

v \wedge w \equiv v \otimes w - w \otimes v

According to this: http://www.math.stonybrook.edu/~brweber/401s09/coursefiles/Lecture11.pdf page 4

 

[WannabeNewton]

Ok, based off of one of my texts, $(v\wedge w)_{ab} = 2v_{[a}w_{b]} = (v\otimes w)_{ab} - (v\otimes w)_{ba}$ which makes more sense because then we get that $^{*}(v\wedge w)_{a} = (v\times w)_{a}$ as usual but the notes linked above seem to say otherwise. Where are they getting the half factor from then?

 

[Mandelbroth]

Either way, the dot product of a basis of the matrix with the cross product is 0, so my direction is going to be fairly clear either way.

Ok, based off of one of my texts, $(v\wedge w)_{ab} = 2v_{[a}w_{b]} = (v\otimes w)_{ab} - (v\otimes w)_{ba}$ which makes more sense because then we get that $^{*}(v\wedge w)_{a} = (v\times w)_{a}$ as usual but the notes linked above seem to say otherwise. Where are they getting the half factor from then?

I think they forgot something. The second and third parts of the final equality (VERY bottom) are equivalent.

Suppose that $\omega: V^k \rightarrow K$ and $\eta:V^m\rightarrow K$ are two anti-symmetric maps where K is the base field. Then, the exterior product is of the form

$$\omega\wedge\eta = \frac{\Gamma(k+m+1)}{\Gamma(k+1)\Gamma(m+1)}\operatorname{Alt}(\omega \otimes \eta)\implies \vec{a}\wedge\vec{b}=\frac{\Gamma(3)}{\Gamma(2)^2}\operatorname{Alt} (\vec{a} \otimes \vec{b})$$

I could be egregiously wrong, but...

 

[Ben Niehoff]

It's easiest to think of $\omega \wedge \eta$ as an antisymmetric map. Then for any vector $X \in V$, $(\omega \wedge \eta)(X)$ is an antisymmetric map $V^{k+m-1} \to \mathbb{R}$ given by

(\omega \wedge \eta)(X) = \omega(X) \wedge \eta + (-1)^k \, \omega \wedge \eta(X)
where the notation $\eta(X)$ means that $X$ goes into the first "slot", leaving the remaining $m-1$ slots free.

You'll notice if you follow this idea through, then you don't get the factor of 1/2 that WBN gave.

 

[WannabeNewton]

Is this a convention issue? In Lee "Smooth Manifolds", for example, the wedge product is a map $\wedge :\Lambda ^{k}(V)\times \Lambda ^{l}(V)\rightarrow \Lambda ^{k+l}(V)$ defined by $\eta\wedge \omega = \frac{(k+l)!}{k!l!}\text{Alt}(\eta\otimes \omega)$. So in this case if $\eta,\omega\in \Lambda ^{1}(V)$ then the $\frac{1}{2}$ factor will not appear i.e. we will just get $\eta\wedge \omega = \eta\otimes \omega-\omega\otimes \eta$. But then why in that pdf, and in this thread: https://www.physicsforums.com/showthread.php?t=150289 do they have factors of $\frac{1}{2}$?

EDIT: Ok micromass just told me that it's a conventional issue in the definition. There is another convention where the $\frac{1}{2}$ does come in. See page 302 of Lee "Smooth Manifolds".

 

[micromass]

WBN has the factor of 2 wrong.

v \wedge w \equiv v \otimes w - w \otimes v

His answer isn't wrong. He simply uses a different convention. Many authors use the convention

\omega\wedge \eta = Alt(\omega \otimes \eta)

This is called the determinant convention, for example the paper wbn linked uses it.

Other authors, like Lee, use the Alt convention:

\omega\wedge \eta = \frac{(k+l)!}{k! l!}Alt(\omega \otimes \eta)

I guess it doesn't matter as long as you are consistent.

 

[Ben Niehoff]

I think the confusion comes from the distinction between an n-form and the components of an n-form. Generically, we can write the n-form $F$ as

F = \frac{1}{n!} F_{a_1 \ldots a_n} \, e^{a_1} \wedge \ldots \wedge e^{a_n}
and you'll notice that there is a $1/n!$ here. But this does NOT imply that the components of $F$ are $\frac{1}{n!} F_{a_1 \ldots a_n}$. Recall the definition of what "components" are, namely

F_{a_1 \ldots a_n} \equiv F(e_{a_1}, \ldots, e_{a_n})
where $e_i$ are the basis of $V$ given by

e^i (e_j) = \delta^i_j
(remember that $e^i$ are a basis of $V^*$). One could eliminate the $1/n!$ factor above by using the tensor product instead of the wedge product

F = F_{a_1 \ldots a_n} \, e^{a_1} \otimes \ldots \otimes e^{a_n}
assuming, of course, that $F_{a_1 \ldots a_n}$ is antisymmetric.

 

[Ben Niehoff]

His answer isn't wrong. He simply uses a different convention. Many authors use the convention

\omega\wedge \eta = Alt(\omega \otimes \eta)

What is the rule for distributing the interior product over wedge products in this convention? I.e., what is

i_X (\omega \wedge \eta)

 

[WannabeNewton]

Here's the relevant section in Lee, by the way, if you're interested:

 

[micromass]

What is the rule for distributing the interior product over wedge products in this convention? I.e., what is

i_X (\omega \wedge \eta)

The definitions on both cases seem to differ. In Lee, we have

i_X\omega(X_1,...,X_{n-1}) = \omega(X,X_1,...,X_{n-1})

while in Morita, we have

i_X\omega(X_1,...,X_{n-1}) = n \omega(X,X_1,...,X_{n-1})

I think that this causes that the distribution rule is the same in both cases:

i_X(\omega\wedge \eta) = i_X(\omega)\wedge \eta + (-1)^n \omega\wedge i_X(\eta)

 

[Ben Niehoff]

while in Morita, we have

i_X\omega(X_1,...,X_{n-1}) = n \omega(X,X_1,...,X_{n-1})

How perverse! Mathwonk seems to dislike this $1/n!$ convention as well, he goes on about it for some time in this thread: https://www.physicsforums.com/showthread.php?t=150289