Levi-Civita symbol for cross products?

1. Aug 24, 2014

Nikitin

Hi. Is using the Levi-Civita symbol to calculate cross-product combos like A x (B x C) allot faster than just using the good old determinant method?

I ask because my lecturer in electrodynamics 2 told us it is better, but it seems to me that it's going to cost me time to learn to use this method efficiently (it took me over 20 minutes to do a triple cross product with it!). Will the pay-off be worth the practice?

Last edited: Aug 24, 2014
2. Aug 24, 2014

ShayanJ

It may not seem useful to you right now, but it really is. Try to learn it.

3. Aug 24, 2014

Fredrik

Staff Emeritus
Once you've learned how to handle $\varepsilon_{ijk}\varepsilon_{ijl}$ and $\varepsilon_{ijk}\varepsilon_{ilm}$, and that if $S_{jk}=S_{kj}$ then $\varepsilon_{ijk}S_{jk}=0$, you will find it much easier to work with the Levi-Civita symbol than any other method. It's definitely worth the effort. It's not a very big effort, since those three things I mentioned are the only basic results you need to know.

4. Sep 4, 2014

Incnis Mrsi

Not sure about A × (B × C), but for A ⋅ (B × C) 3-dimensional Levi-Civita symbol with its 6 non-zero components, if used once as εijkAiBjCk, and (expanded) formula for rank-3 determinant, are the same thing.

As a general remark, not Levi-Civita vs matrix methods makes the difference, but now exactly the algorithm arranges calculation of the cubic polynomial(s) on components for given type of triple product.

Last edited: Sep 4, 2014