Balakrishnan V. A Miscellany of Mathematical Physics

Bangalore: Indian Academy of Sciences, 2018. — 92 p.This book is based on eight articles that were written for 'Resonance' journal (which objective is targeted primarily at science education for undergraduate students and teachers). Needless to say, the material has been rearranged, revised, and considerably expanded in some directions, so as to have a degree of coherency. The articles themselves grew out of the notion that the mathematical tools and techniques required by the students of physical sciences can, and should, be introduced to them in a more ‘user-friendly’ style than is generally the case. The initial introduction should be heuristic, with adequate motivation. The development of the subject matter should help the student not only to learn the techniques, but also to gain insight and the ability to recognise interconnections. Attention should be paid to the natural unfolding of the subject matter; one thing should lead to another. While correctness can never be sacrificed, formal rigour and exactitude need not be at the forefront. In this book, we take a walking tour — or a casual stroll — through a variety of topics, such as vectors, reciprocal vectors, spherical and hyperspherical coordinates, Cartesian tensors, the true meaning of a vector, linear vector spaces, ket and bra vectors, infinitesimal rotation generators, the finite rotation formula, rotation matrices, the orthogonal group SO(3) and its parameter space, Pauli matrices, the unitary group SU(2) and its parameter space, the Dirac delta function, Green functions, the wave equation, its Fourier transform, and its fundamental solution in spaces of different dimensions. One thing leads to another!What can the answer be?
Oblique axes, reciprocal basis, kets and bras.
Reciprocal basis in two dimensions and other nice things.
Reciprocal basis in d dimensions and other ramifications.
How is a vector rotated?
Rotation matrices and the rotation group.
More fun with the rotation group.
The many avatars of the Dirac delta function.
Signal Propagation: 1. Solution of the Wave Equation.
Signal Propagation: 2. Effect of dimensionality.