Mathematical Principles in Science:
Mathematics 5101: Linear Mathematics in Finite Dimensions
Mathematics 5102: Linear Mathematics in Infinite Dimensions
Mathematics 5451: Calculus of Variations and Tensor Calculus

 Autumn Semester: Mathematics 5101 Spring Semester: Mathematics 5102
 Autumn Semester:  Mathematics 5451

Key course topics and texts used (5101, 5102, 5451)

MATH 5101: LIST OF TOPICS:

I.   VECTOR SPACES

Axiomatic properties
Subspaces
Spanning sets
Linear independence
Bases and coordinates
Dimension
Linear functionals and covectors
Dual of a vector space
Bilinear functionals
Metric
Isomorphism between vector space and its dual

II.  LINEAR TRANSFORMATIONS
Null space, range space
Dimension theorem, implicit function theorem for a linear system
Classification of linear transformations
Invertible transformations
Existence and uniqueness of a system of equations
Algebraic operations with linear transformations
The representation theorem
Change of basis, change of representation, and the transition matrix
Invariant subspaces, commuting operators and eigenvectors

III. INNER PRODUCT SPACES

Inner products
Orthogonormal bases
Gram-Schmidt orthogonalization process
Orthogonal matrices
Right and left inverses
Least squares approximation, Bessel's inequality, normal equations
The four fundamental subspaces of a matrix
The Fredholm alternative, uniqueness=existence
Intersection and sum of two vector space

IV.   EIGENVALUES AND EIGENVECTORS ON REAL VECTOR SPACES

Eigenvector basis
Diagonalizing a matrix
Generalized eigenvectors
Phase portrait of a system of linear differential equations
Powers of a matrix
Markov processes

V.   EIGENVALUES AND EIGENVECTORS ON COMPLEX VECTOR SPACES

Adjoint of an operator
Hermetian operators
Spectral theorem
Triangularization via unitary similarity transformation
Diagonalization of normal matrices
Positive definite matrices
Quadratic forms and the generalized eigenvalue problem
Extremization with linear constraints
Rayleigh quotient
Singular value decomposition of a rectangular matrix
Pseudo-inverse of a rectangular matrix

Approximate
Timeline:       I          :10 days
II          :10 days
III+IV  :10 days
V          :12 days

Texts:      (1)  L.W. Johnson, Riess & Arnold: Introduction to Linear Algebra (Chapter 4)
(2)  G. Strang: Linear Algebra and its Applications, 3rd Edition (Selected sections from Chapters 2&3, Chapter 5&6; Appendix A)
(3) Larson and Edwards: Elementary Linear Algebra, 3rd Edition  (Selected sections from Chapter 8)

MATH 5102: LIST OF  TOPICS:

##### Vibrations:

I. INFINITE DIMENSIONAL VECTOR SPACES: EXAMPLES

Sturm-Liouville systems: regular, periodic, and singular
Sturm-Liouville series
##### Signals and Vibrations:

II.  INFINITE DIMENSIONAL VECTOR SPACES: PRINCIPLES

Inner product spaces
Complete metric spaces
Hilbert spaces
Square summable series and square integrable functions
Least squares approximation
Projection theorem
Generalized Fourier coefficients
Bessel's inequality, Parceval's equality and completeness
Unitary transformation between Hilbert spaces

III. FOURIER THEORY

Fourier series
Dirichelet kernel
Fourier's theorem on a finite domain
Sequences leading to the Dirac delta function
Fourier transform representation
Change of basis in Hilbert space:
Orthonormal wave packet representation
Wavelet  representations (if time permits)
##### Signals and Vibrations in Space and Time:
IV.  GREEN'S FUNCTION THEORY: INHOMOGENEOUS DIFFERENTIAL EQUATIONS
Homogeneous sytems
Adjoint systems
Inhomogeneous systems
The concept of a Green's function
Solution via Green's function
Integral equation of a linear system via its Green's function
Classification of integral equations
The Fredholm alternative
Green's function and the resolvent of the operator of a system
Eigenfunctions and eigenvalues via residue calculus
Branches, branch cuts, and Riemann sheets
Singularity structure of the resolvent of a system:
Poles and branch cuts
Effect of boundary conditions and domain size

V. THEORY OF SOLUTIONS TO PARTIAL DIFFERENTIAL EQUATIONS
IN TWO AND THREE DIMENSIONS
Partial differential equations: hyperbolic, parabolic, and elliptic
The Helmholtz equation and its solutions in the Euclidean plane.
Geometry of the space of solutions
Plane waves vs cylinder waves:
Why, and when to use them
Sommerfeld's integral representation
Hankel, Bessel, and Neumann waves
Change of basis in the space of solutions: partial waves
Displaced cylinder waves
The cylindrical addition theorem
Method of steepest descent and stationary phase
Analytic behaviour of cylinder waves
Interior (cavity) and exterior (scattering) boundary value problems
Spherical waves: symmetric and non-symmetric
Cauchy problem and characteristics (if time permits)

Approximate
Timeline:         I+II    :12 days
III+IV  :20 days
V      :10 days

Texts:      (1)  U.H. Gerlach: Linear Mathematics in Infinite Dimensions (Chapter 1,3,2,4,5)
(2)  F.W. Byron  and R.W. Fuller: Mathematics of Classical and Quantum Physics

Possible follow up courses: Math 5451 Math 5756