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summation of tensors

Radical Edward

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To show that the sum of two tensors is a tensor, I am taking the approach that (here for a vector, or first rank tensor)


[math]V_i' = a_{ij}V_j[/math]


[math]W_i' = a_{ij}W_j[/math]


then summing them to get



[math]V_i' + W_i'= a_{ij}V_j + a_{ij}W_j[/math]


then when you sum and contract that lot you get


[math]V_i' + W_i'= a_{ij}(V_j + W_j)[/math]


which if I am not mistaken fulfils the criteria for a vector, showing that the sum of two vectors is a vector. Now when we get onto higher rank tensors, I am finding this a bit of a pain, since expanding and contracting that lot is tiresome. Is it a general rule that


[math]V_{\alpha\beta\gamma}' + W_{\alpha\beta\gamma}'= a_{\alpha~i}a_{\beta~j}a_{\gamma~k}W_{ijk} + a_{\alpha~i}a_{\beta~j}a_{\gamma~k}W_{ijk}



can be factored to


[math]V_{\alpha\beta\gamma}' + W_{\alpha\beta\gamma}'= a_{\alpha~i}a_{\beta~j}a_{\gamma~k}(V_{ijk}) + W_{ijk}



and am I even taking the right approach.



(I am working through the Mary L Boas book here, chapter ten problems to section 11, q3 and 5.)

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