Tensor Mechanics Problem

Joang

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Joined
May 3, 2024
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1
Please, colleagues, I would greatly appreciate your assistance with the following problem.


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Or any criteria that may help solve it
Thank you
 
This is way too vague. What are aαβa_{\alpha\beta}, xαx^\alpha and yβy^\beta ? For example, in xαx^\alpha is α\alpha a superscript or is it xx to the α\alpha-th power ? Please provide a fuller description.
Thank you.
 
Here is what I think your post means: you are given a bilinear form in Einstein notation. In my preferred notation it looks like this: S(x,y)=xTAy=i,jxiaijyjS(\mathbf x, \mathbf y) = \mathbf x^T \mathbf A \mathbf y = \sum_{i,j} x_i a_{ij} y_j and condition S(x,x)=i,jxiaijxj=0S(\mathbf x, \mathbf x) = \sum_{i,j} x_i a_{ij} x_j = 0. Then you need to show that aij=ajia_{ij} = - a_{ji}, i.e., that A\mathbf A is "skew-symmetric".
Can you see that this equivalent to equation S(x,y)=S(y,x)S(\mathbf x, \mathbf y) = - S(\mathbf y, \mathbf x)? The latter can be proven by looking at S(u+v,u+v)S(\mathbf u+\mathbf v , \mathbf u+\mathbf v).
 
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