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

By Nadir Jeevanjee

ISBN-10: 0817647147

ISBN-13: 9780817647148

An advent to Tensors and staff conception for Physicists offers either an intuitive and rigorous method of tensors and teams and their position in theoretical physics and utilized arithmetic. a selected goal 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 widespread in physics calculations with the summary yet extra conceptual formula present in many mathematical texts, the paintings can 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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Extra resources for An Introduction to Tensors and Group Theory for Physicists

Example text

20) 26 2 Vector Spaces where δji is the usual Kronecker delta. If V is finite-dimensional with dimension n, it is then easy to check (by evaluating both sides on basis vectors) that we can write14 n f= fi e i i=1 so that the fi really are the components of f . Since f was arbitrary, this means that the ei span V ∗ . ,n is actually a basis for V ∗ . We sometimes say that the ei are dual to the ei . Note that we have shown that V and V ∗ always have the same dimension. We can use the dual basis {ei } ≡ B ∗ to write f in components, ⎛ ⎞ f1 ⎜ f2 ⎟ ⎜ ⎟ [f ]B∗ = ⎜ .

28) Aij Notice that the components of v transform with the whereas the basis vectors j transform with the Ai , so the components of a vector obey the law opposite (‘contra’) to the basis vectors. This is the origin of the term ‘contravariant’. Note also that the components of a dual vector transform in the same way as the basis vectors, hence the term ‘covariant’. It makes sense that the basis vectors and the components of a vector should transform oppositely; v exists independently of any basis for V and should not change under a change of basis, so if the ei change one way the v i should change oppositely.

And vn ⎜ ⎜ T (v) B = ⎜ ⎜ ⎝ n i 1⎞ i=1 v Ti n i 2⎟ i=1 v Ti ⎟ .. ⎟, ⎟ ⎠ n i n i=1 v Ti which looks suspiciously like matrix multiplication. In fact, we can define the matrix of T in the basis B, denoted [T ]B , by the matrix equation T (v) B = [T ]B [v]B where the product on the right-hand side is given by the usual matrix multiplication. Comparison of the last two equations then shows that ⎛ 1 ⎞ T1 T 2 1 . . Tn 1 ⎜T 2 T 2 ... 15) .. .. ⎟ . ⎜ .. ⎝ . . ⎠ T1 n T2 n . . Tn n Thus, we really can use the components of T to represent it as a matrix, and once we do so the action of T becomes just matrix multiplication by [T ]B !

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


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