Mathematical Methods In Physics -I

Por: Swayam . en: , ,

Week 1: Concept of Set, Binary composition, Group, Ring, Field, Vector Space, Examples of vector space in Euclidean space (R), Metric Space Week 2: Linearly dependent & independent vectors, Dimensions, Basis, Span, Linear Functional, Dual space, Inner Product, Normed Space, Schwarz inequality, Gram-Schmidt orthonormalization, Completeness Week 3: Linear Operator, Matrix representation, Transformation of axis, Change of Basis, Unitary transformation, Similarity transformation, Eigen value & Eigen vectors, Matrix decomposition Week 4: Elementary Matrices,Rank, Subspace with examples. Diagonalization of matrix, The Cayley-Hamilton theorem, Function, mapping, Function space, Linearly dependent & independent function, Examples, Wronskian, Gram-determinant Week 5: Inner product in function space, Orthogonal functions, Delta function, Completeness, Gram-Schmidt orthogonalization in function space, Legendre polynomials Week 6: Fourier coefficients, Fourier Transform, Examples, Fourier Series, Parseval’s relation, Convolution theorem, Polynomial Space Week 7: Complex numbers, Roots of the complex numbers, Complex variable & Function, Limit and continuity, differentiability of a complex function, Branch Cut and branch point Week 8: Cauchy-Riemann equation, Analytic function, Harmonic conjugate function, Examples, Singularities and their classifications Week 9: Complex integration, Simply and multiply connected regions, Cauchy-Goursat theorem, Cauchy’s integral formula, Examples Week 10: Series & Sequence, Convergence test, Radius of convergence, Taylor’s series, Maclaurin Series, Examples Week 11: Laurent Series, Zeros and poles, Essential singularity, Examples, Residue, Classification of residue, Residue calculations for different orders of poles Week 12: Cauchy’s residue theorem, Application of residue theorem to calculate the definite integrals, Examples