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I am trying to teach myself tensor calculus but have reached a stumbling block - expressing the magnitude of a cross product in indicial notation. I know that one can express a cross product of two vectors $\vec{A}$ and $\vec{B}$ in indicial notation as follows:

$$ \vec{A} \times \vec{B} = \epsilon_{ijk}a_j b_k \hat{e}_i$$

But I am not sure how to express the magnitude of the resulting vector using indicial notation. My guess is

$$ \mid \; \vec{A} \times \vec{B} \mid^2 \; = (\vec{A} \times \vec{B})_m(\vec{A} \times \vec{B})_m = \epsilon_{ijk}a_j b_k \hat{e}_i \; \epsilon_{ijk}a_j b_k \hat{e}_i$$

but I seem to recall reading that having an index occur more than twice is undefined. How would I write the magnitude of the cross product using correct notation?

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The problem is that you used the same indices to sum over the elements of the first and second $A\times B$ in your dot product. The product should actually read

$$ \mid \; \vec{A} \times \vec{B} \mid^2 \; = (\vec{A} \times \vec{B})\cdot(\vec{A} \times \vec{B}) = \epsilon_{ijk}a_j b_k \hat{e}_i \cdot\; \epsilon_{pqr}a_q b_r \hat{e}_p$$

From here on you can use the orthogonality of the $\hat{e}_i$s and then use the following relation to simplify your problem

\begin{align}\varepsilon _{ijk}\varepsilon _{imn}=\delta _{jm}\delta _{kn}-\delta _{jn}\delta _{km}\end{align}

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  • $\begingroup$ I think there are a couple of mistakes here. The expression $\mid \; \vec{A} \times \vec{B} \mid$ should be $\mid \; \vec{A} \times \vec{B} \mid^2$, and $(\vec{A} \times \vec{B})_m\cdot(\vec{A} \times \vec{B})_m$ should be $(\vec{A} \times \vec{B})_m(\vec{A} \times \vec{B})_m$ or $(\vec{A} \times \vec{B})\cdot(\vec{A} \times \vec{B})$. I won't make an edit because I can't decide the best way to correct them. $\endgroup$ – Oscar Cunningham Aug 29 '16 at 7:34
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You're right that

$$\mid \; \vec{A} \times \vec{B} \mid^2 \; = (\vec{A} \times \vec{B})_m(\vec{A} \times \vec{B})_m$$

We can write $(\vec{A} \times \vec{B})_m$ as

$$\epsilon_{mij}a_ib_j$$

and since we need to use different indices we'll write the second $(\vec{A} \times \vec{B})_m$ as

$$\epsilon_{mkl}a_kb_l$$

(by changing $i$ for $k$ and $j$ for $l$).

Putting these together gives

$$\mid \; \vec{A} \times \vec{B} \mid^2 \; =\epsilon_{mij}a_ib_j\epsilon_{mkl}a_kb_l$$

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