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Homework Statement
Evaluate the following sums, implied according to the Einstein Summation Convention.
[tex]\begin{array}{l}
\delta _{ii} = \\
\varepsilon _{12j} \delta _{j3} = \\
\varepsilon _{12k} \delta _{1k} = \\
\varepsilon _{1jj} = \\
\end{array}[/tex]
The Attempt at a Solution
[tex]
\begin{array}{l}
\delta _{ii} = \delta _{11} + \delta _{12} + \delta _{13} + \delta _{21} + \delta _{22} + \delta _{23} + \delta _{31} + \delta _{32} + \delta _{33} = 1 + 0 + 0 + 0 + 1 + 0 + 0 + 0 + 1 =3 \\
\varepsilon _{12j} \delta _{j3} = \left( {\varepsilon _{121} + \varepsilon _{122} + \varepsilon _{123} } \right)\left( {\delta _{13} + \delta _{23} + \delta _{33} } \right) = \left( {0 + 0 + 1} \right)\left( {0 + 0 + 1} \right) = \left( 1 \right)\left( 1 \right) = 1 \\
\varepsilon _{12k} \delta _{1k} = \left( {\varepsilon _{121} + \varepsilon _{122} + \varepsilon _{123} } \right)\left( {\delta _{11} + \delta _{12} + \delta _{13} } \right) = \left( {0 + 0 + 1} \right)\left( {1 + 0 + 0} \right) = \left( 1 \right)\left( 1 \right) = 1 \\
\varepsilon _{1jj} = \left( {\varepsilon _{111} + \varepsilon _{122} + \varepsilon _{133} } \right) = 0 + 0 + 0 = 0\\
\end{array}
[/tex]
Am I doing these right? Thanks!