- #1
MadMax
- 99
- 0
Using the Einstein summation convention...
Why is
[tex]\mathbf{a}^2 \mathbf{b}^2[/tex]
not the same as
[tex]3 a_i a_j b_j b_i = 3(\mathbf{a} \cdot \mathbf{b})^2[/tex]
given that
[tex]\mathbf{a}^2 = a_i \cdot a_i = a_i a_j \delta_{ij}[/tex]
[tex]\mathbf{b}^2 = b_i \cdot b_i = b_i b_j \delta_{ij}[/tex]
and
[tex]\delta_{ij} \delta_{ji} = 3[/tex]
-> [tex]\mathbf{a}^2 \mathbf{b}^2 = a_i a_j \delta_{ij} b_i b_j \delta_{ij} = 3(\mathbf{a} \cdot \mathbf{b})^2[/tex]
??
Any help would be much appreciated.
Why is
[tex]\mathbf{a}^2 \mathbf{b}^2[/tex]
not the same as
[tex]3 a_i a_j b_j b_i = 3(\mathbf{a} \cdot \mathbf{b})^2[/tex]
given that
[tex]\mathbf{a}^2 = a_i \cdot a_i = a_i a_j \delta_{ij}[/tex]
[tex]\mathbf{b}^2 = b_i \cdot b_i = b_i b_j \delta_{ij}[/tex]
and
[tex]\delta_{ij} \delta_{ji} = 3[/tex]
-> [tex]\mathbf{a}^2 \mathbf{b}^2 = a_i a_j \delta_{ij} b_i b_j \delta_{ij} = 3(\mathbf{a} \cdot \mathbf{b})^2[/tex]
??
Any help would be much appreciated.
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