By Nadir Jeevanjee
An advent to Tensors and workforce conception for Physicists presents either an intuitive and rigorous method of tensors and teams and their position in theoretical physics and utilized arithmetic. a specific goal is to demystify tensors and supply a unified framework for realizing them within the context of classical and quantum physics. Connecting the part formalism customary in physics calculations with the summary yet extra conceptual formula present in many mathematical texts, the paintings should be a great addition to the literature on tensors and staff theory. Advanced undergraduate and graduate scholars in physics and utilized arithmetic will locate readability and perception into the topic during this textbook.
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Extra resources for An Introduction to Tensors and Group Theory for Physicists
From the ‘aside’ above, we know that f (v) is basis-independent, but we also know that f (v) = [f ]TB [v]B . 30): in a new basis B , we have [f ]TB [v]B = A−1 [f ]B T −1 = [f ]B A T T A[v]B A[v]B = [f ]TB [v]B . 31) This makes concrete our claim above that [f ] transforms ‘oppositely’ to [v], so that the basis-independent object f (v) really is invariant under a change of basis. ,n then the components of vectors and their corresponding dual vectors are identical, which is why we were able to ignore the distinction between them for so long.
The octopole moment Q3 has components Qij k which are totally symmetric and which satisfy i Qiij = i Qij i = i Qj ii = 0). We will explain these curious features of the Qr at the end of this chapter. ). 7, and you can verify that these are (0, 2) tensors. ) is just the identity operator. 22. 2 Change of Basis Now we are in a position to derive the usual transformation laws that historically were taken as the definition of a tensor. ,n . 14) j for some numbers Ai . Likewise, there exist numbers Ai (note that here the upper index is primed) such that j ei = Ai ej .
Els 1 r = Aki 1 . . Aki r Al11 . . ls . 22) is the standard tensor transformation law, which is taken as the definition of a tensor in much of the physics literature; here we have derived it as a consequence of our definition of a tensor as a multilinear function on V and V ∗ . 4 below. With the general transformation law in hand, we will now look at specific types of tensors and 2 This is also why we wrote the upper index directly above the lower index, rather than with a horizontal displacement as is customary for tensors.
An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee