PHYS 5401 MATHEMATICAL METHODS OF PHYSICS I

TARGET AUDIENCES:

• This is a required course for all entering, degree-seeking graduate students in Physics. The course is also helpful for graduate students in signal processing, telecommunications and other areas of Electrical Engineering.
• This course is recommended for all students who intend to take PHYS 5403, Numerical Methods in Physics.
• Others who can benefit from this course include employees of local high-technology companies who need an introduction to modern mathematics in order to work more effectively with chaotic systems, wavelets or fractals.

CONCEPTS/TOOLS TO BE ACQUIRED IN THIS COURSE:

GENERAL CONCEPTS AND SKILLS:

• An acquaintance with linear algebra and functional analysis which is sufficient to prepare students for an understanding of modern theoretical and computational methods
• Preparation for reading the modern literature in applied mathematics

SPECIFIC CONCEPTS AND TOOLS:

1. Sets and mappings
   • Basic definitions
     • Sets
     • Mappings
     • Axiom of choice
     • Cartesian products
     • Equivalence and equivalence classes
   • Union, intersection and complement
   • Infinite sets
     • Basic properties
     • Induction and recursion, with examples
     • Countable sets; the rational numbers
     • Countable unions and intersections
     • Uncountable sets; the real numbers
   • Ordered and partially ordered sets
     • Partial orderings
     • Orderings; upper and lower bounds
2. Groups, rings and fields
   • Groups
     • Axioms
       • Finite and infinite groups
       • Abelian and non-Abelian groups
       • Consequences of associativity
       • Right identity and right inverses
     • The two-element group
       • Physical realizations of the two-element group
       • Generators of a group
     • Subgroups
       • Definition of a subgroup
       • Cosets
       • Lagrange's theorem
       • The three-element group
     • Cyclic groups
       • Definition of a cyclic group
       • One-dimensional crystal lattices
       • Cyclic groups of non-prime order
       • The four-element groups
       • Complex n-th roots of unity
       • The integers modulo n
       • Difference equations and n-cycles
     • Dihedral groups
   • Group homomorphisms
     • Definitions and basic properties
       • Group homomorphisms
       • Group representations
       • Group isomorphisms
     • Normal subgroups
       • The kernel of a homomorphism
       • Quotient groups
     • Classes of conjugate elements
       • Conjugacy
       • Classes in SO(3,R)
   • The symmetric groups
     • Permutations
       • Definitions
       • Groups of permutations
     • Cayley's theorem
     • Cyclic permutations
       • Transpositions and m-cycles
       • Expression of a permutation as a product of disjoint m-cycles
   • Rings and integral domains
     • Axioms
       • Rings
       • Integral domains
     • Basic properties of rings
       • Multiplicative properties
       • The characteristic of a ring
     • The rational numbers
     • Ring homomorphisms
       • Definition and examples
       • Ideals
   • Fields
     • Axioms and examples
     • Galois fields
       • Fermat's theorem
       • Construction of GF(4)
       • The order of a Galois field
3. Vector spaces
   • Motivation for vector-space axioms
   • Basic definitions and examples
     • Axioms for a vector space
     • Vector subspaces
       • Translations and affine subspaces
     • Comments on the vector-space axioms
       • The zero vector versus the zero scalar
       • Finite versus infinite linear combinations
       • The number field
       • Distance
       • Why are real numbers necessary?
       • Modules over rings
       • Notation
     • Selected realizations of the vector-space axioms
       • Any field
       • Spaces of column vectors
       • Matrix spaces
       • Sequence spaces
       • Spaces of square-summable sequences
       • Spaces of continuously differentiable functions
       • Spaces of square-integrable functions
       • Sturm-Liouville solution spaces
   • Linear independence and linear dependence
     • Definitions and basic results
     • Examples of linear independence
       • The space of complex numbers
       • Spaces of column vectors
       • Matrix spaces
       • Sequence spaces
       • Spaces of differentiable functions
       • Sturm-Liouville solution spaces
   • Bases and dimension
     • The dimension of a vector space
       • Infinite bases
       • Finite-dimensional vector spaces
       • Summation convention
       • Extension of a linearly independent list to a basis
       • R-modules
     • Vector-space isomorphisms
       • Definitions
       • Products of vector isomorphisms
       • Physical examples of vector isomorphisms
       • Characterization of a vector space by its dimension
     • Selected realizations of vector-space bases
       • The complex numbers
       • Spaces of column vectors
       • Matrix spaces
       • Sequence spaces
       • Spaces of continuous functions
       • Spaces of square-integrable functions
       • Sturm-Liouville solution spaces
     • Gaussian elimination and linear dependence
       • Gaussian elimination applied to rows
       • Gaussian elimination applied to columns
   • Affine transformations
     • The affine group of a vector space
     • Coordinate transformations
       • Active transformations
       • Passive transformations
       • Relation between active and passive transformations
   • Complementary subspaces
     • Vector complements and direct sums
       • Complementary subspaces
       • Direct sums of vector spaces
     • Examples of direct sums of vector spaces
       • The space of complex numbers
       • Spaces of column vectors
       • Matrix spaces
4. Linear mappings I
   • Linear mappings and their matrices
     • Basic properties
       • Axioms for a linear mapping
       • The product of two linear mappings
       • Vector spaces of linear mappings
       • The domain and range of a linear mapping
     • The matrix of a linear mapping
       • The matrix elements of a linear mapping
       • The column rank of a linear mapping
       • The matrix of a linear mapping
       • The matrix-vector product
       • Visualization of linear mappings
       • Computation of the column rank
       • Computation of a basis of the range
       • The matrix of the product of two linear mappings
     • Other examples of linear mappings
       • The complex numbers
       • l2 spaces
       • Spaces of differentiable functions
       • Discretization and interpolation mappings
   • Non-singular linear mappings
     • Definitions and basic properties
     • Matrix formulation of a change of basis
     • Permutation matrices
     • The general linear group of a vector space
   • Singular linear mappings
     • Singularity and linear dependence
     • Visualization of singular linear mappings
     • The null space of a linear mapping
     • Other examples of singular linear mappings
       • l2 spaces
       • Null space of a differential operator
   • The trace and determinant
     • The trace of a linear mapping
     • Determinants
       • Definition
       • Basic properties of determinants
       • The determinant of a singular matrix
       • The determinant of a matrix product
       • Determinants and the handedness of coordinate systems
       • The Laplace expansion of a determinant
       • A determinantal formula for the matrix inverse
       • Vandermonde determinants
   • Structural properties
     • The quotient space V/null[A]
     • Isomorphism of the range to a complement of the null space
     • The rank-nullity theorem
     • Right inverses of a linear mapping
     • Examples of right inverses
       • Finite-dimensional vector spaces
       • Solutions of differential equations
   • Solution of linear equations
     • Basic facts about linear equations
       • Classification of linear-equation problems
     • Matrix formulation of Gaussian elimination
     • The LU decomposition
     • Bases of the range and null space
5. Functionals on vector spaces
   • Linear functionals
     • Dual spaces
       • Motivation for studying functionals
       • Definitions
       • The range and null space of a linear functional
     • Coordinate functionals
       • Definitions
       • Computation of coordinate functionals on Fn
       • Isomorphism of V* to V
       • Coordinate functionals in two-dimensional Euclidean space
       • Coordinate functionals and the reciprocal lattice
     • Isomorphism of V to V**
     • The annihilator of a subspace
     • Other realizations of dual spaces
       • The dual space of C
       • The dual of Fn
     • Polynomial interpolation
       • Formulation of the problem
       • Lagrange interpolation
     • Boundary and initial conditions for differential equations
   • Inner-product spaces
     • Definitions and examples
       • Notation for inner products
       • The space of complex numbers
       • Spaces of column vectors
       • Finite, discrete quantum systems
       • Matrix spaces
     • The metric tensor
       • Definitions
       • Properties of the Gram matrix
       • Positive-definite inner products
       • Computation of inner products on finite-dimensional spaces
       • Sequence spaces
       • Cn and L2 spaces
       • Spaces of sampled functions
     • Indefinite inner products
       • Minkowski space
     • Orthogonality
       • Definition
       • Mutually orthogonal sets
   • The geometry of inner-product spaces
     • Pythagoras's theorem
       • The generalized Pythagorean theorem
       • The Cauchy-Schwarz-Bunyakovsky inequality
       • Other geometrical properties of Euclidean and unitary spaces
     • Orthonormal bases
       • Definitions and examples}44}
       • Contravariant components relative to an orthonormal basis}45}
       • The Gram-Schmidt construction for Euclidean and unitary spaces
     • Orthogonal polynomials I
       • Definition
       • Three-term recurrence relations
       • Chebyshev polynomials
   • Projection methods
     • Projection of a vector onto a subspace
       • Definition
       • Orthogonal projectors
     • The orthogonal complement
       • Computation of a basis of the orthogonal complement
       • The orthogonal complement of the orthogonal complement
     • Least-squares approximation
       • Motivation
       • Abstract formulation
       • Approximation by finite Fourier sums
       • Chebyshev approximations
     • Volume of an m-parallelepiped
   • Vector and matrix norms
     • Vector norms
     • The norm of a linear mapping
     • Matrix norms
       • Matrix p-norms
       • The Frobenius norm
     • Norm of a linear functional on C0([a,b]; R)
   • Properties of the metric tensor
     • The covariant components of a vector
     • The inner-product mapping
       • Definition and examples
       • Mapping from ``kets'' to ``bras'' in quantum mechanics
       • The image of a basis under the inner-product mapping
     • The inverse inner-product mapping
       • Definition
       • The reciprocal basis
       • Transformation from covariant to contravariant components
       • Inner product in the dual space
       • Relation of the orthogonal complement to the annihilator
6. Linear mappings II
   • Dyads
     • Definition of a dyad
     • Dyadic expansions
     • Resolutions of the identity mapping
   • The transpose and adjoint of a linear mapping
     • The transpose
     • The adjoint
       • Definitions
     • Other realizations of the adjoint
       • l2 spaces
       • Spaces of differentiable functions
       • The adjoint of a Sturm-Liouville differential operator
     • Properties of the adjoint
       • Linearity of the Hermitian adjoint
       • Relation of the transpose to the Hermitian adjoint
       • Dyadic expansion of the Hermitian adjoint
       • The range and null space of the Hermitian adjoint
     • Hermitian and self-adjoint mappings
       • Definition of a Hermitian mapping
       • Hermitian and self-adjoint operators
       • Projection operators
     • Isometric and unitary mappings
       • Isometric operators in infinite-dimensional spaces
       • Isometric mappings in finite-dimensional spaces
       • Unitary mappings in finite-dimensional spaces
       • Unitary groups
   • Eigenvalues and eigenvectors
     • The secular equation
     • Diagonalization of Hermitian matrices
       • Decomposition into orthogonal eigensubspaces
       • Matrix version of the orthogonal decomposition
       • Hermitian matrices and inner products
       • The principal-axis theorem
       • Spectral decomposition of a self-adjoint linear mapping
     • Normal linear mappings
       • Non-normal linear mappings
   • The singular-value decomposition
     • Derivation of the SVD
     • Matrix version of the SVD
     • The fundamental subspaces
     • The inverse and pseudo-inverse in the SVD
   • Linear equations II
     • Numerical vs. analytical methods
     • Diagonal dominance
     • Condition number of the linear-equation problem
     • Accuracy of Gaussian elimination
     • Relatives of the LU decomposition
       • The Cholesky decomposition
   • Selected applications of linear equations
     • The linear least-squares problem
     • Linear difference equations
       • Vandermonde determinants
       • Degenerate characteristic roots
     • Tridiagonal systems
       • Solution of a tridiagonal system of linear equations
       • Eigenvalues of certain tridiagonal matrices