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The vector triple product is defined as $\mathbf{a}\times (\mathbf{b}\times \mathbf{c})$. This is often re-written in the following way: \begin{align*}\mathbf{a}\times (\mathbf{b}\times \mathbf{c}) = \mathbf{b}(\mathbf{a}\cdot\mathbf{c}) - \mathbf{c}(\mathbf{a}\cdot\mathbf{b})\end{align*} This is a very useful identity for integrating over vector fields and so on (usually in physics).

Every proof I have encountered splits the vectors into components first. This is understandable, because the cross product is purely a three dimensional construct. However, I'm curious as to whether or not there is a coordinate free proof of this identity. Although I don't know much differential geometry, I feel that tensors and so on may form a suitable framework for a coordinate free proof.

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  • $\begingroup$ By ”components“ do you mean coordinates relative to some basis or a decomposition of $a$ into components parallel to and perpendicular to the plane of $b$ and $c$? The latter can be without reference to coordinates by using orthogonal projection/rejection. The identity can be derived via these components of $a$. $\endgroup$ – amd Apr 11 '17 at 5:38
  • $\begingroup$ I was referring to the former. The second approach sounds interesting. I feel the overall nature is similar as they both involve splitting things into components but I'll give that a go when I have a bit more time on my hands! $\endgroup$ – Harambe Apr 11 '17 at 6:33
  • $\begingroup$ Coordinate-free proofs like this can "easily" be computed using differential forms and particularly the Hodge map. Of course, by "easily", I mean you need a knowledge and understanding of differential forms in the first place. $\endgroup$ – AloneAndConfused Apr 11 '17 at 7:05
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    $\begingroup$ Could this be what you're after ?math.stackexchange.com/questions/305285/… $\endgroup$ – Arnaud D. Apr 11 '17 at 7:43
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Let $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ be vector fields on $\mathbb{R}^{3}$ (we could extend to $\mathbb{R}^{n}$ if we wish!), considered as a Riemannian manifold equipped with metric $g$ and induced Hodge map $\star$. Let $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ have corresponding vector field representations $U,V,W$ on $\mathbb{R}^{n}$ respectively. Then

$$\begin{align} \mathbf{a}\times(\mathbf{b}\times \mathbf{c}) \,\equiv\, \star(\widetilde{U} \wedge \star(\widetilde{V} \wedge \widetilde{W})) \end{align}$$

where $\widetilde{X}$ denotes the metric dual of $X$ (i.e. $\widetilde{X}=g(X,-)$) and the equivalence is up to metric dual. Then

$$\begin{align} \star(\widetilde{U} \wedge \star(\widetilde{V} \wedge \widetilde{W})) &\,=\, \star(\widetilde{U} \wedge i_{W}\star \widetilde{V}) \,=\, \star ( i_{W}\widetilde{U} \wedge \star \widetilde{V} - i_{W} (\widetilde{U}\wedge \star \widetilde{V})) \\ &\,=\, (i_{W}\widetilde{U})\star\star \widetilde{V} - \star i_{W}(\widetilde{U}\wedge\star \widetilde{V}) \\ &\,=\, g(U,W)\widetilde{V} - \star(\widetilde{U}\wedge\star \widetilde{V}) \wedge \widetilde{W} \\ &\,=\, g(U,W)\widetilde{V} - g(U,V)\widetilde{W} \end{align}$$

where $i_{X}$ denotes the interior derivative with respect to $X$ and we have used the identities:

$$\begin{align} \star\star \alpha &\,=\, \alpha \\[0.2cm] \star(\widetilde{X}\wedge\star\widetilde{Y}) &\,=\, g(X,Y) \\[0.2cm] \star(\alpha \wedge \widetilde{X}) &\,=\, i_{X}\star \alpha \\[0.2cm] \widetilde{X} \wedge \star\alpha &\,=\, (-1)^{p+1}\star i_{W}\alpha \end{align}$$

for any $p$-form $\alpha$ and vector fields $X,Y$ (note that the first and second identities are specific to $\mathbb{R}^{3}$). Then note that

$$g(U,W)\equiv \mathbf{a}\cdot\mathbf{c}\quad\text{and}\quad g(U,V)\equiv\mathbf{a}\cdot\mathbf{b}$$

and so then the result follows after taking the metric dual of our expression.

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    $\begingroup$ For now I have no idea what this means, but it's absolutely beautiful. $\endgroup$ – Harambe Apr 11 '17 at 22:20
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    $\begingroup$ If you are interested, I found the following application-oriented introduction to differential forms extremely useful and it's where the notation can be found: A primer on exterior differential calculus by David A. Burton $\endgroup$ – AloneAndConfused Apr 12 '17 at 12:28
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    $\begingroup$ it's been over a year, but i've spent the past half year primarily learning differential geometry, and I just came back to this and realised I understood it all :) $\endgroup$ – Harambe Jun 19 '18 at 11:26
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Since $b\times c$ is normal to the plane $b,\,c$ span, $a\times (b\times c)$, which is orthogonal to this vector, is in said plane. The coefficients $B,\,C$ for which the result is $Bb+Cc$ are invariant under rotations, and clearly $B$ must be linear in $a,\,c$ while $C$ is linear in $a,\,b$, so constants $B',\,C'$ exist with $a\times (b\times c) =B' (a\cdot c) b + C' (a\cdot b) c$. Since the left-hand side is antisymmetric, $C'=-B'$. Since $B'$ must be a constant (since both sides are linear in each vector), we can use any vectors we like for which both sides are non-zero to compute $B'$. Example: $a=b=i,\,c=j$ so $a\times (b\times c) = i\times k = -j$ and $(a\cdot c) b - (a\cdot b) c = -j$ as required.

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Okay, I really hate the sign issues with the Hodge star, so I am gonna assume that $\star\star=1$, the end result will be good.

The fundamental relationship is that for $k$-vectors/forms, we have $\langle\omega,\eta\rangle\mu=\omega\wedge\star\eta$, where the angle brackets are the inner product on the exterior algebra and $\mu$ is the volume form/multivector.

Let's start with $x$ being an arbitrary vector and taking a look at $$ \langle x,a\times(b\times c)\rangle\mu=\langle x,\star(a\wedge\star(b\wedge c))\rangle\mu= \\=x\wedge(a\wedge\star(b\wedge c))=(x\wedge a)\wedge\star(b\wedge c)= \\=\langle x\wedge a,b\wedge c\rangle\mu=\det\left(\begin{matrix}\langle x,b\rangle & \langle x,c\rangle \\ \langle a,b\rangle & \langle a,c\rangle\end{matrix}\right)\mu= \\=(\langle x,b\rangle \langle a,c\rangle-\langle x,c\rangle \langle a,b\rangle)\mu, $$ comparing the LHS with the RHS we can "divide" (ofc not really divide) by $\mu$ since the coefficients need to agree, and because $x$ was arbitrary, and the inner product is nondegenerate, we can "" divide "" by $x$ and we have $$ a\times(b\times c)=b\langle a,c\rangle-c\langle a,b\rangle. $$

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  • $\begingroup$ I wonder why the inner product of wedge products is a determinant? $\endgroup$ – Frenzy Li Apr 11 '17 at 7:32
  • $\begingroup$ Also, may cancel be a more appropriate substitute for "divide"? $\endgroup$ – Frenzy Li Apr 11 '17 at 7:33
  • $\begingroup$ @FrenzyLi Maybe "cancel" would be ok. In the first case, it was because 3-forms in a 3-dim space form a one dimensional space, so equality of 3-forms $\Longleftrightarrow$ equality of the single coefficient, in the second case, I have essentially viewed the inner product as the action of a 1-form (due to Riesz isomorphism and all), and 1-forms agree if their actions on an arbitrary vector agree. As for the determinant, the inner product on the exterior algebra is defined as, if $\alpha=\alpha_1\wedge...\wedge\alpha_k$ and $\beta=\beta_1\wedge... \wedge\beta_k$ then... $\endgroup$ – Bence Racskó Apr 11 '17 at 7:38
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    $\begingroup$ @FrenzyLi $\langle \alpha,\beta\rangle=\det(\langle \alpha_i,\beta_j\rangle)$. See for example en.wikipedia.org/wiki/…. $\endgroup$ – Bence Racskó Apr 11 '17 at 7:39
  • $\begingroup$ Nice! And last thing, is $\star\star=1$ generally not true? $\endgroup$ – Frenzy Li Apr 11 '17 at 7:41
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Adapted from my previous proof of $\nabla \times (\vec{A} \times \vec{B})$: \begin{align} \vec a \times (\vec b \times \vec c) & = a_l \hat{e}_l \times (b_i c_j \hat{e}_k \epsilon_{ijk}) \\ & = a_l b_i c_j \epsilon_{ijk} \underbrace{ (\hat{e}_l \times \hat{e}_k)}_{(\hat{e}_l \times \hat{e}_k) = \hat{e}_m \epsilon_{lkm} } \\ & = a_l b_i c_j \hat{e}_m \underbrace{\epsilon_{ijk} \epsilon_{mlk}}_{\text{contracted epsilon identity}} \\ & = a_l b_i c_j \hat{e}_m \underbrace{(\delta_{im} \delta_{jl} - \delta_{il} \delta_{jm})}_{\text{They sift other subscripts}} \\ & = a_j (b_i c_j \hat{e}_i) - a_i (b_i c_j \hat{e}_j) \\ & = (b_i \hat{e}_i) (a_j c_j) - (c_j \hat{e}_j) (a_i b_i) \\ & = \vec b (\vec a\cdot\vec c) - \vec c(\vec a\cdot \vec b) \end{align}

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  • $\begingroup$ It uses tensors but in the index notation. Not sure if this is what the OP wants. $\endgroup$ – Frenzy Li Apr 11 '17 at 4:49
  • $\begingroup$ Yes I've seen a very similar proof before and it's not exactly what I wanted. However, due to what the cross product is, I wouldn't be surprised if all proofs involved index notation or explicit components. (Unless there are "suitable" generalisations!) $\endgroup$ – Harambe Apr 11 '17 at 5:11
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    $\begingroup$ @Shanye2020 I would actually like to see this proof using more compact notation. I do hope someone could enlighten us. $\endgroup$ – Frenzy Li Apr 11 '17 at 5:13
  • $\begingroup$ Hmm... An almost identical proof is also seen here. $\endgroup$ – Frenzy Li Apr 11 '17 at 5:28
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    $\begingroup$ @FrenzyLi Ok, I'm done, posted an answer. $\endgroup$ – Bence Racskó Apr 11 '17 at 7:28
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I just want to add a proof using my favourite method for this kind of thing: Penrose graphical notation. I think it is as coordinate free as you can go.

Every piece of the diagram has a meaning. Tensors are shapes with a lines going upwards or downwards, depending on the type of tensor. For example, vectors have 1 line going upwards, and covectors one line going downwards. Contracion is represented joining the lines.

For example: the first line on the right side, over the word "proof", is the statement $(b\times c)^{a} = b^{b}c^{c}g_{bd}g_{ce}\epsilon^{dea} = b^{b}c^{c}\epsilon_{bcd}g^{da}$ where the indices are proper abstract indices: they do not represent components, but the slots of the tensors.

At the bottom I repeat the properties I use.

Penrose graphical notation proof

  1. Antisymmetry of $\epsilon^{abc}$
  2. $\epsilon^{abc}\epsilon_{def} = \delta^{abc}_{def}$
  3. $\delta^{abc}_{dec} = \delta^{ab}_{de}$
  4. $\delta^{ab}_{cd}A_{ab} = 2A_{[cd]} = A_{cd} - A_{dc}$
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