"Mathematical
Principles in Science"
(Math 5101, 5202)
In Math 5101 ("Linear Mathematics in Finite
Dimensions") you will learn, among
others
The mathematics of the
states of a
linear system (vector space theory)
The mathematics of
reference frames
and their transformations (bases and the representation
theorem)
The mathematics of linear
systems (the
four fundamental subspaces of a matrix)
The mathematics of
covectors and metrics
(linear functionals and dual vector space theory)
The mathematics of least
squares approximations
(subspaces and inner products)
The mathematics of stable
and unstable
systems (phase portrait of coupled system of differential
equations)
The mathematics of
vibrating systems
(eigenvalue and eigenvector theory)
The mathematics of spectra
(normal
matrices and the spectral theorem)
The mathematics of normal
modes and
of linear systems with quadratic and linear constraints
(Rayleigh quotient and its
geometry)
The mathematics of
rectangular matrices
(singular value decomposition theory)
In Math 5102 ("Linear Mathematics in Infinite Dimensions") you
will learn,
among
others
The mathematics of bounded
linear systems
(Sturm-Liouville theory)
The mathematics of
infinite dimensional
inner product spaces (Hilbert space theory)
The mathematics of a
ticking clock
and a phase-locked laser (Fourier theory)
The mathematics of signal
processing
(wavelet and orthonormal wave packet analysis)
The mathematics of the
unit impulse
response of a linear system (Green's function theory)
The mathematics of
representing symmetries
of the Euclidean plane by cylinder waves and plane waves
(special function
theory)
The mathematics of the
rotationally
and translationally symmetric waves in the Euclidean plane
(Hankel and Bessel function
theory)
The mathematics of high
frequency waves
(the method of steepest descent and stationary phase)
The mathematics of
translating cylindrical
waves (the cylindrical addition theorem)
The mathematics of wave
propagation
inside and outside a cylindrical pipe
(interior and exterior
boundary
value problems)
The mathematics of
potentials and waves
on a spherical background (algebraic method for solving
linear partial
differential equations)