By Nadir Jeevanjee

An advent to Tensors and crew thought for Physicists presents either an intuitive and rigorous method of tensors and teams and their position in theoretical physics and utilized arithmetic. a specific objective is to demystify tensors and supply a unified framework for figuring out them within the context of classical and quantum physics. Connecting the part formalism time-honored in physics calculations with the summary yet extra conceptual formula present in many mathematical texts, the paintings could be a great addition to the literature on tensors and workforce 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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A2n ⎟ ⎜ 1 2 n ⎟ ⎜ ⎟ −1 ⎜ ⎟. 23) , A A=⎜ . = ⎟ . . . . ⎜ ⎜ .. .. .. ⎟ .. .. ⎟ ⎝ ⎠ ⎝ ⎠ An1 An2 An1 . . Ann An2 ... 24) as our notation suggests. 22) under a change of basis. e. ,n you choose). 6 Change of basis matrix for a 2-D rotation As a simple illustration of the formalism, consider the standard basis B in R2 and another basis B obtained by rotating B by an angle θ . This is illustrated in Fig. 2. 14) we have A11 = cos θ, A12 = − sin θ A21 = sin θ, A22 = cos θ. 23) then tells us that A−1 = − sin θ cos θ cos θ sin θ .

37) so matrix multiplication of a vector by the metric matrix gives the corresponding dual vector in the dual basis. Thus, the map L is implemented in coordinates by [η]. Now, we mentioned above that L is invertible; what does L−1 look like in coordinates? Well, by the above, L−1 should be given by matrix multiplication by [η]−1 , the matrix inverse to [η]. 38) or in components f˜μ = ηνμ fν . 39) are probably familiar to you. In physics one usually works with components of vectors, and in relativity the numbers v μ 20 As long as we are talking about ‘standard’ physics notation, you should also be aware that in many texts the indices run from 0 to 3 instead of 1 to 4, and in that case the zeroth coordinate corresponds to time.

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.

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