This would be the first of a two-part series on Mathematical Methods in Physics. The aim here is to provide a solid mathematical foundation for the budding Physicist eager to climb the ladder of self-learning. This course is designed to cater to an undergraduate student who excitedly embarks on a study of Physics, but is obstructed by the mathematics which appears so forbidding. The approach we will follow is one of showing many examples, and weaving the theory around examples. When possible, we will show how Mathematica can be used to offer extra insight. INTENDED AUDIENCE :NonePREREQUISITES : We try to make this self-contained. However, some familiarity with the basic mathematical tools used in say, the first courses on Mechanics, Electromagnetism, would help.INDUSTRIES SUPPORT :Quantitative Finance and Scientific consulting companies
Week 1:Linear Algebra 1 : vectors, linear vector spaces, inner product, C-S inequality, linear independence, row-reduction Week 2:Linear Algebra 2 : Matrices, determinants, span, basis, orthonormal basis, subspaces, linear operators. Week 3: Linear Algebra 3: Direct sum, eigenvalues and eigenvectors, unitary, Hermitian, normal operators, transformations, defective matrices, diagonalization. Week 4: Fourier Series and Transforms: periodic functions, series expansion, Fourier coefficients, Completeness relation, Fourier transforms. Week 5: Ordinary Differential Equations 1: Introduction, Separable variables, orthogonal trajectories, linear first-order ODEs, Wronksian, exact ODEs, auxiliary equation. Week 6: Ordinary Differential Equations 2: Inhomogeneous second order ODEs, method of undetermined coefficients, vibrations in mechanical systems, forced vibrations, resonance, linear superposition. Week 7: Ordinary Differential Equations 3: Laplace transforms, Solving ODEs using Laplace transforms, Dirac Delta function. Week 8: Ordinary Differential Equations 4: Green's function method, power series method, Frobenius method.