How to Show Pij = Pji When Epsilon ijk Pkj = 0?

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Homework Help Overview

The discussion revolves around demonstrating the equality \( P_{ij} = P_{ji} \) under the condition that \( \epsilon_{ijk} P_{kj} = 0 \). The subject area pertains to tensor analysis and properties of antisymmetric tensors.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of the given condition and consider expanding the left-hand side of the equation. There is a discussion about whether numerical substitution is necessary or if the proof can be conducted entirely in index notation.

Discussion Status

Some participants have provided specific cases to illustrate the reasoning, particularly focusing on the scenario where \( i \neq j \). There is an ongoing exploration of different values for \( i \) to validate the equality, but no consensus has been reached yet.

Contextual Notes

Participants are considering the implications of the antisymmetry in the context of the indices and the specific cases where \( i \) and \( j \) are equal versus when they are not. The discussion is framed within the constraints of the problem statement without additional assumptions.

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if [itex]\epsilon_{ijk} P_{kj}=0[/itex] how do we show [itex]P_{ij}=P_{ji}[/itex]?
 
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Expand out the LHS for each i.
 


do you mean just substitute in numbers and see what happens because that works - i was just wondering if i could do it without explicitly using numbers (i.e. all of it in index notation).
 


For i = j, clearly Aij = Aji, so you only need to check the cases in which i ≠ j.

For example, setting i = 1 in εijkAkj = 0, you get ε123A32 + ε132A23 = A32 - A23 = 0 ⇒ A32 = A23.

You just need to do the same with i = 2 and i = 3.
 

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