Author: CHRISTINE ADELLE L. RICO
Here is another method of verifying a vector formula using the Levi-Civita symbol. Levi-Civita symbol is a tensor of rank three and is defined by +1 if the indices are in even permutation of , -1 if the indices are in odd permutation, and 0 if any two indices are the same.
Prove that .
Using the Levi-Civita symbol, we rewrite the cross products,
Since each term is only scalar, they can be rearranged such that,
where is summed over.
Note that is only nonzero if are all different from . There are two possibilities of its implications, either,
Consider the case of and , which gives for any value of . If and , . Therefore,
We can now write the proof so that the th component is,
Adelle is currently pursuing her MS Physics degree at the Mindanao State University- Iligan Institute of Technology in Iligan City.