Levi-Civita Tensor: Index Interchange Identity

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The identity involving the Levi-Civita tensor, ε_{ijk} a_j b_k = -ε_{ijk} a_k b_j, is confirmed to hold true. This is demonstrated by recognizing that ε_{ijk} a_j b_k can be rewritten as -ε_{ikj} a_j b_k. By relabeling the dummy indices j and k, the original identity is obtained. This confirms the validity of the index interchange identity within the context of the Levi-Civita tensor. The discussion emphasizes the importance of understanding index manipulation in tensor calculus.
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Does the following identity hold?:

<br /> \epsilon_{ijk} a_j b_k = -\epsilon_{ijk} a_k b_j<br />
 
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Yes.
 


The answer is yes; clearly we have:
<br /> \epsilon_{ijk} a_j b_k = -\epsilon_{ikj} a_j b_k<br />

from which we can obtain your identity by relabeling the dummy indices j and k.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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