An Introduction to Tensors and Group Theory for Physicists by Nadir Jeevanjee

By Nadir Jeevanjee

An advent to Tensors and team idea for Physicists presents either an intuitive and rigorous method of tensors and teams and their function in theoretical physics and utilized arithmetic. a specific target 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 commonplace 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 team 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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We will also meet several bases for this space which you should find familiar. 47) where n is arbitrary. Verify that P (R) is a (real) vector space. Then show that P (R) is infinite-dimensional by showing that, for any finite set S ⊂ P (R), there is a polynomial that is not in Span S. Exhibit a simple infinite basis for P (R). d (b) Compute the matrix corresponding to the operator dx ∈ L(P (R)) with respect to the basis you found in part (a). 8 Problems 37 where W (x) is a nonnegative weight function.

20) are linearly independent. 7. Now consider the element f j of V ∗ which eats a vector in Rn or Cn and spits out the j th j component; clearly f j (ei ) = δi so the f j are just the dual vectors ej described above. Similarly, for Mn (R) or Mn (C) consider the dual vector f ij defined by f ij (A) = Aij ; these vectors are clearly dual to the Eij and thus form the corresponding dual basis. While the f ij may seem a little unnatural or artificial, you should note that there is one linear functional on Mn (R) and Mn (C) which is familiar: the trace functional, denoted Tr and defined by 14 If V is infinite-dimensional then this may not work as the sum required may be infinite, and as mentioned before care must be taken in defining infinite linear combinations.

This corresponds to lowering the second index, and we write the components of T˜ as Tij , omitting the tilde since the fact that we lowered the second index implies that we precomposed with L. This is in accord with the conventions in relativity, where given a vector v ∈ R4 we write vμ for the components of v˜ when we should really write v˜μ . From this point on, if we have a non-degenerate bilinear form on a vector space then we permit ourselves to raise and lower indices at will and without comment.

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