A Course in Mathematical Physics, Vol. 1: Classical by Walter E Thirring

y(x). Minimal conditions are that 'rix, Yi,j =f. 5J). Yi,j E Coo and Det 5. 13; 2). 7.

The mr+s-dimensional vector space of these tensors is denoted by 1'q~(M). 5) On IRn with the basis {el' e2"'" en}, each eil ® ei2 ® ... , ... , V:leil ® e;, ® ... ® ei) == (eit ® ei2 ® ... , ... , v:) == (vi leit)(v! lei,) ... (v: lei). If we let (i1o i 2 , ••• , ir) run through all r-tuples of indices, we obtain a basis for the vector space of tensors of degree r. Using this basis we can identify the space of tensors of degree r with IRnr. Every such tensor can be written as t="cit ..... ire. e.

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