Topics covered: Mathematical Methods 2008

• Lecture 1 (Aug 4):
  • Introduction to the course and course contents
• Lecture 2 (Aug 6):
  • Constructing differential equations, solving and plotting them: cascade radioactive decay, uniform acceleration with drag, predator-prey problem
  • Mathematica demonstration
• Lecture 3 (Aug 11):
  • Real numbers, complex numbers
  • Vector space, inner product space, norm, angle
  • Vector as a quantity that transforms like coordinates
  • Vectors in 2 dimensions: transformation properties, explicit transformation (rotation) matrix
  • constructing invariants by combining vectors: length, dot product, area, introduction to epsilon(ij)
  • A general transformation matrix "a" in any dimensions, orthogonality of [a], i.e. [a^T a = I] for orthogonal coordinates
• Lecture 4 (Aug 13):
  • Vectors in three dimensions: different conventions for rotation matrices, Euler angles, mixing angles
  • Combining vectors to form invariants: length, dot product, scalar triple product. Cross product as a vector
  • 4-dim Euclidean vectors: 4-volume, 3-area using [epsilon(ijkl)]
  • 4-dim Lorentz vectors: Lorentz transformations, [x, p, J, A, k] as Lorentz vectors, [p.p, x.k] as Lorentz invariants
  • Tensors and their ranks, [delta(ij)] as a rank-2 tensor, [epsilon(ij)] as a rank-2 tensor only in 2 dimensions
  • Rank vs. dimensions in tensors, examples
  • Multiplication of tensors, contraction to get lower rank tensors
  • Tensorial nature of electromagnetic field tensor [F(mu,nu)] leading to physical relations between [E] and [B] in different frames
• Lecture 5 (Aug 18) :
  • Differential and integral operators: [del/delx_i] and [dx_i] as vectors, combinations with other vectors and tensors to form area element, volume element etc.
  • Grad, Div and Curl: physical interpretation, Gauss's theorem, Stokes' theorem
  • Curvilinear orthogonal coordinates, Grad, Div, Curl and Laplacian
• Assignment 1 given (Aug 19)
• Lecture 6 (Aug 20) :
  • Linear operators on vector spaces: linearity, addition, product, examples with [A = d/dx] and [B = x d/dx]
  • linear independence, dimension, basis, orthogonality, orthonormality
  • Inverse: left and right. If both exist, they are unique and identical. If one exists and is unique, then the other is identical to the first. Finite dimensions: if an inverse exists, it is unique and both left and right inverse.
  • Dual vector space, inner product, adjoint operator, |x><y| and |y><x| as operators and adjoints of each other
  • Special operators: Hermitian, unitary, antilinear, projection, normal
• Lecture 7 (Aug 25) :
  • Eigenvectors as special vectors that do not change directions under an operation, generalization to eigenfunctions: examples for the operators [d/dx], [ integral(0,x) dx/x ], in general [ integral(0,x) dx' f(x') ]
  • Normal operators: [ A|x> = lambda |x>] implies [ A^dagger |x> = lambda^* |x>], eigenvectors corresponding to different eigenvalues are orthogonal.
  • Similarity transformation [ A -> S A S^(-1)]: eigenvalues unaltered, equivalent to changing to a basis of linearly independent (not necessarily orthogonal) vectors. S^(-1) as reciprocal lattice in three dimensions
  • Matrices as representations of finite dimensional vector spaces, representaions for vectors, operator, operator product implying matrix multiplication rule
  • Examples of matrix representation for space of quadratic polynomials, action of differential / integral operators and their products
  • Matrices for discretizing a differential equation
• Lecture 8 (Aug 27) :
  • Some more special matrices: idempotent [A^n = A], nilpotent [A^n = 0]
  • Upper triangular matrices [U] : diagonal elements as eigenvalues, determinant as their product
  • [U] with nonzero diagonal elements form a group, unit [U] matrices also form a group.
  • Strictly upper triangular matrices: all eigenvalues zero, nilpotent
  • lower triangular matrices [L]: results similar to upper triangular
  • Simplification of [A x = b] if [A] is [L] or [U]: no need of solving simultaneous equations, simple substitution enough
  • Gaussian elimination as left multiplication with unit [L] matrices. "Any" matrix may be written as [A=LU] not uniquely, but as [A=LDU] uniquely, where [D] is a diagonal matrix. Caveat in Gaussian elimination: pivot
  • Positive definite matrices: [x^T A x > 0] for all [x]. All eigenvalues positive, [x^T A y] as a valid inner product, [A_ij = x_i^T x_j], principle subdeterminant criterion for recognizing positive definiteness
  • Applications of positive definite matrices: density matrix, variance matrix, least square fitting
• Lecture 9 (Sep 1) :
  • Least equare fitting using LU decomposition
  • Cholesky decomposition [A = L L^T] for positive definite symmetric matrices
  • QR decomposition for [m x n] matrices: explicit construction, solving the least square problem with QR decomposition
  • Eigenvalues and eigenvector results: each matrix has at least one eigenvalue, distinct eigenvalues imply linearly independent eigenvectors, if all eigenvalues are distinct then eigenvectors span the whole space.
  • If eigenvectors span the whole vector space, similarity transformation that diagonalizes the matrix can be constructed with S having its columns as the eigenvectors
  • If S is a unitary matrix, the original matrix A has to be normal
  • Numerical examples of eigenvectors not spanning the vector space
  • Similarity transform preserves the characteistic equation
• Lecture 10 (Sep 8) :
  • Rank-nullity theorem (only statement and examples, no proof)
  • Caley-Hamilton theorem: algebraic proof using adjugate matrices, implications
  • Generalized eigenvectors, constructive proof of Jordan normal form [P^(-1) M P = J] for matrices with degenerate eigenvalues
• Assignment 2 given (Sep 8)
• Lecture 11 (Sep 10) :
  • Ironing out the proof of Jordan normal form
  • Characteristic and minimal polynomial, connection with Jordan form
  • Application of Jordan form: exponentiation
  • Application of Caley-Hamilton theorem: solving differential equations
• Lecture 12 (Sep 15) :
  • Infinite series: definition of convergence
  • Convergence tests: Cauchy root test, Cauchy (D'Alembert) ratio test, Cauchy (Maclauren) integral test
  • Kummer's test and Gauss's test with proofs
  • Absolute convergence
  • Alternating series and conditional convergence
  • Example of improvement of convergence of [ln(1+x)] through multiplication by a polynomial
• Lecture 13 (Sep 29) :
  • Convergence of series of functions: uniform convergence, example of non-uniformly convergent (but absolutely convergent) series
  • Taylor series, remainder term, generalization to any dimensions, Taylor series as "generated" translations.
  • Power series: radius of convergence, uniform and absolute convergence within this radius
• Lecture 14 (Sep 30) :
  • Problems from the first two assignments
• Lecture 15 (Oct 1) :
  • Clarification of QR decomposition for least square fits
  • Problems on infinite series
• Midterm (Oct 2)
• Lecture 16 (Oct 6) :
  • Uniform convergence: Weierstrass's M-test
  • Taylor series: the remainder term goes to zero as [1/n!]
  • An infinitely diffrentiable real function need not be equal to its Taylor series expansion, example [Exp(-1/x)]. "Real analytic" functions as the ones that are equal to their Taylor series.
  • Asymptotic series: concept, motivation and definition
  • Examples of asymptotic series: Exponential integral function [Ei(x)], Incomplete Gamma function [I(x,p)], Sterlings's formula for factorial. Demonstration on Mathematica.
• Lecture 17 (Oct 8) :
  • The Kobayashi-Maskawa problem in terms of a unitary matrix
  • Infinite product: [(product) (1+a_i)], convergence in terms of an infinite series [(sum) a_i]
  • Constructing a power series from the zeroes of a function: some motivation about complex analytic functions
  • Fibonacci series: recurrence relation, n^th term
  • A second order recurrence relation, "companion" matrix method to obtain the n^th term, use of Jordan form when the companion matrix has degenerate eigenvalues
  • Generalization to a recurrence relation of arbitrary order.
• Assignment 3 given (Oct 13)
• Lecture 18 (Oct 13) :
  • Complex analysis: motivation and preview, complex functions [f= u+ iv]
  • Cauchy-Riemann conditions [u_x = v_y, u_y = - v_x] as necessary ones for differentiability at a point
  • CR conditions in terms of [z] and [z*]
  • Sufficient conditions for differentiability at a point
  • "Analytic" functions: differentiable in a whole neighbourhood, however small,
  • algebraic operations on analytic functions, composition of two analytic functions
  • Generalization of real fuctions to complex ones: polynomials, exponential, trigonometric, logarithmic [log and its principle value, Log], exponent [z^w = exp(w log z)]. Choice of domain to make functions single-valued
• Lecture 19 (Oct 15) :
  • Definitions of arc, simple arc, smooth/regular arc, contour, simple contour, inside/outside, clockwise/anticlockwise
  • Definition of a contour inegral [intC] in terms of real integrals
  • Darboux inequality for putting upper bounds on contour integrals
  • Cauchy's integral theorem: [f] analytic and [f'] continuous inside a region R enclosed by a simple closed contour C implies [intC f(z) dz =0]. Proof through Stokes' theorem (vector analysis) and Green's theorem (real functions)
  • Generalization to Cauchy-Goursat theorem: [f'] need not be continuous everywhere inside R.
  • Extension of Cauchy-Goursat theorem to non-simple closed contours, multiply connected regions
  • Cauchy's integral formula: [f(z) = 1/(2 pi i) IntC f(w) dw /(w-z)]
• Lecture 20 (Oct 20) :
  • Derivatives of analytic functions through contour integrals: [f^{n} (z) = n! /(2 pi i) IntC f(w) dw /(w-z)^(n+1)]. All derivatives of an analytic functions are analytic
  • Taylor series: if f is analytic everywhere inside R, then [f(z) = Sum(n=0,infinity) a_n (z-z_0)^n], where [a_n = 1/(2 pi i) IntC f(w) dw /(w-z)^(n+1)]
  • Examples of Taylor seies, radius of convergence
  • Laurent series: if z0 is a singular point inside R, then [f(z) = Sum(n=0,infinity) a_n (z-z_0)^n + Sum(n=1,infinity) b_n (z-z0)^(-n)], where [b_n = 1/(2 pi i) IntC f(w) dw /(w-z)^(-n+1)].
  • Alterative form for Laurent series: [f(z) = Sum(n=-infinity,infinity) c_n (z-z_0)^n], where [c_n = 1/(2 pi i) IntC f(w) dw /(w-z)^(n+1)].
• Lecture 21 (Oct 22) :
  • Convergence of taylor series: if [Sum(n=0,infinity) a_n z^n] is convergent at z=z0], then it is (a) absolutely convergent for [|z| < |z0| = r_c] (b) convergent for [|z| < |z0|], (c) uniformly convergent for [|z| < |z1| < r_c]
  • Generalization of the above results to a Laurent series inside an annulus (not shown explicitly)
  • Uniform convergence of a series implies differentiation and integration can be carried out term by term.
  • Uniqueness of Laurent series, [intC dz/(z-z0)^(n+1) = 2 pi i delta_{n0}]
• Lecture 22 (Oct 27) :
  • Nature of singularities: removable singularity (all [b_n] in Laurent series vanish), poles of finite order m (finite number of [b_n] are nonzero), essential singularity (infinite [b_n] nonzero)
  • Example of Laurent series, poles and their order, zeroes and their order
  • Residue [= b_1], Residue theorem [intC f(z) dz = 2 pi i b_1]
  • Calculating residues: examples. When f(z) in the form [phi(z)/(z-z0)^m], residue is [b_1 = phi^{m-1}(z0)/(m-1)! ].
  • Procedure when [f(z) = P(z)/Q(z)] and Q(z) has a zero of order m. When [Q(z)] has a simple zero, residue = [P(z0)/Q'(z0)]. Examples with [f(z)] of this form
• Lecture 23 (Nov 3) :
  • Branch points and branch cuts
  • Contour integrals and residue theorem for evaluating real integrals
  • Integrals of the form [Int(0,2 pi) f(sin x, cos x) dx]
  • Integrals of the form [Int(-infinity, infinity) f(x) dx], where [f(z)] goes to zero faster than [1/z]
  • Integrals of the form [Int(-infinity,infinity) e^(i a z) f(z) dz] when [|f(z)| -> 0] uniformly over a large semicircle
• Lecture 24 (Nov 5) :
  • Presription when a simple pole is on the contour
  • Avoiding the poles: limiting procedures that give different answers for different physical situations
  • Integral with a branch cut
  • Defining a single-values function for multiple branch points
  • Asymptotic value of real integrals, Striling expansion
  • Steepest descent method for asymptotic limit of complex integrals
• Lecture 25 (Nov 10) :
  • Explanation of bypassing singularities
  • Steepest descent method: clarification
  • Analytic continuation
  • Conformal mapping: preservation of angles between curves
  • 2-d electrostatics: generalizing a real potential to complex, using analyticity properties
• Assignment 4 given (Nov 10)
• Lecture 26 (Nov 12) :
  • Common terminology for differential equations: first order, second order, linear, homogeneous
  • First order ODE: exact, integrating factors to bring an ODE to exact form
  • PDE: separation of variables, streamline method (method of characteristics) for converting a linear PDE to a ODE along a curve
  • Second order linear homogeneous ODE: series solution, Fuchs theorem, finding at least one series solution
  • Wronskian for linear independence of functions
• Lecture 27 (Nov 15) :
  • Wronskian for finding the second solution of a homogeneous ODE,
  • Showing that there can be maximum two solutions to a second order linear ODE
• Lecture 28 (Nov 17) :
  • Nature of solutions of a ODE [y'' + p(z) y' + q(z)=0] through the behaviour of the quantities [I] and [Q]
  • Effect of a pole of [p(z)] of order more than one and a pole of [q(z)] of order more than two
  • Classification of points: ordinary, regular singular, irregular singular
  • Singularity at infinity
  • Differential equations with the same nature of singularities are connected: example with Hypergeometric and Legendre
• Lecture 29 (Nov 19) :
  • Linear algebra of ODE: boundary conditions + self-adjoint operator (Sturm-Liouville) leading to Hermiticity
  • Bringing a linear second order operator to self-adjoint form
  • Definition of eigenvalue and eigenfunction of a S-L operator, eigenvalues real, eigenfunctions orthogonal (with inner product defined with respect to a weighting function)
  • Gram-Schmidt orthonormalization for a set of functions, completeness of a basis
  • Application of a basis expansion for solving inhomogeneous ODEs
• Lecture 30 (Nov 21) :
  • Solving inhomogeneous ODEs by expansion in orthonormal basis
  • Green's function: definition, representation in the orthonormal basis
• Assignment 5 given (Nov 21)