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Concepts in Mathematics: ECE Webpage ECE298-JA; ECE-298JA; UIUC Course Explorer: ECE-298-JA; Time: 1:00-1:50 MWF; Location: 4070 ECEB (official); Register

ECE 298JA Schedule (Fall 2017)

L W D Date Lecture and Assignment

Part I: Number systems (10 Lectures)
1 1
M 8/28 Introduction & Historical Overview; Lecture 0: pdf;

The Pythagorean Theorem & the Three streams:
1) Number systems (Integers, rationals)
2) Geometry
3) {$\infty$} {$\rightarrow$} Set theory {$\rightarrow$} Calculus
Common Math symbols
Matlab tutorial: pdf
Read: Class-notes
Homework 1 (NS-1): Basic Matlab commands: pdf (v. 1.06), Due 9/6 (1 week); help

2 W 8/30 Lecture: Number Systems (Stream 1)
Taxonomy of Numbers, from Primes {$\pi_k$} to Complex {$\mathbb C$}: {$\pi_k \in \mathbb P \subset \mathbb N \subset \mathbb Z \subset \mathbb Z \cup \mathbb F = \mathbb Q \subset \mathbb Q \cup \mathbb I = \mathbb R \subset \mathbb C $}
First use of zero as a number (Brahmagupta defines rules); First use of {$\infty $} (Bhaskara's interpretation)
Floating point numbers IEEE 754 (c1985); History
Read: Class-notes
3 F 9/1 Lecture: The role of physics in Mathematics: Math is a language, designed to do physics
The Fundamental theorems of Mathematics:
1) Arithmetic (i.e., primes), 2) Algebra, 3) Calculus (& Set Theory) and other key concepts:
History review:
BC: Pythagoras; Aristotle;
17C: Mersenne; Galilei, Galileo; Hooke; Boyle; Newton;
18C: Bernoulli, Daniel; Euler; Lagrange; d'Alembert;
19C: Gauss; Laplace; Fourier; Von Helmholtz; Heaviside; Rayleigh;
Read: Class-notes
- 2
M 9/4 Labor Day Holiday -- No class
4 W 9/6 Lecture: Two Prime Number Theorems:
How to identify Primes (Brute force method: Sieve of Eratosthenes)
1) Fundamental Thm of Arith
2) Prime Number Theorem: Statement, Prime number Sieves
Why are integers important?Public-private key systems (internet security) Elliptic curve RSA
Pythagoras and the Beauty of integers: Integers {$\Leftrightarrow$}
1) Physics: The role of Acoustics & Electricity (e.g., light):
2) Eigenmodes: Mathematics in Music and acoustics: Strings, Chinese Bells, chimes;
Read: Class-notes & A short history of primes, History of PNT
NS-1 Due
Homework 2 (NS-2): Prime numbers, GCD, CFA; pdf (v. 1.26) (1 week)
5 F 9/8 Lecture: Euclidean Algorithm for the GCD; Coprimes
Definition of the {$k=\text{gcd}(m,n)$} with examples; Euclidean algorithm
Properties and Derivation of GCD & Coprimes
Algebraic Generalizations of the GCD
Read: Class-notes
6 3
M 9/11 Lecture: Continued Fraction algorithm (Euclid & Gauss, JS10, p. 47)

The Rational Approximations of irrational {$\sqrt{2} \approx 17/12\pm 0.25%)$} and transcendental {$(\pi \approx 22/7)$} numbers
Matlab's {$rat()$} function
Read: Class-notes
Homework 3 (NS-3): Pythagorean triplets, Pell's equation, Fibonacci sequence; pdf (v.1.24), (1 week)

7 W 9/13 Lecture: Pythagorean triplets {$[a, b, c] \in {\mathbb N}$} such that {$c^2=a^2+b^2$}
Euclid's formula, Properties & examples
Read: Class-notes
NS-2 Due
8 F 9/15 Lecture: Pell's Equation: Lenstra (2002) pdf; General solution; Brahmagupta's solution by Pell's Eq
Fibonacci Series
Geometry & irrational numbers {$\sqrt{n}$}; History of {$\mathbb R$}
Read: Class-notes
9 4
M 9/18 Lecture: Eigen analysis of Pell and Fibonacci matrices
Read: Class-notes
NS-3 Due
10 W 9/20 Exam I (In Class): Number Systems
L W D Date Lecture and Assignment
Part II: Algebraic Equations (12 Lectures)
11 F 9/22 Lecture: Analytic geometry as physics (Stream 2)
The first "algebra" al-Khwarizmi (830CE)
Polynomials, Analytic functions, {$\infty$} Series: Geometric {$\frac{1}{1-z}=\sum_{0}^\infty z^n$}, {$e^z=\sum_{0}^\infty \frac{z^n}{n!}$}; Taylor series; ROC; expansion point
Read: Class-notes
Homework 4 (AE-1, V. 1.29): Polynomials & Analytic functions and their inverse, Convolution, Newton's method (pdf, 1 week)
12 5
M 9/25 Lecture: Polynomial root classification by convolution
Fundamental Thm of Algebra (pdf) &
Summarize Lec 11: Series representations of analytic functions, ROC
Historical notes on complex numbers: Solution of the quadratic (Brahmagupta, 628), cubic (c1545), quartic (Tartaglia et al..., 1535),
quintic cannot be solved (Abel, 1826)
Read: Class-notes
13 W 9/27 Lecture: Residue expansions of rational functions {$F(s) = \frac{P_m(s)}{P_n(s)}$}
Utility in Engineering applications
Read: Class-notes
14 F 9/29 Lecture: Analytic Geometry; Scalar and vector products of two vectors
Read: Class-notes
AE-1 due
Homework 5 (AE-2): Linear systems of equations; Gaussian elimination; ABCD method; (pdf Due 1 week)
15 6
M 10/2 Lecture: Gaussian elimination (intersection); Pivot matrices {$(\Pi_n)$}: {$U = \Pi_n^N P_n A$} gives upper-diagional {$U$}
Read: Class-notes
16 W 10/4 Lecture: Transmission matrix method (composition of polynomials)
Read: Class-notes
17 F 10/6 Allen out of town Nathan Bleier (TA) lecture
Lecture: Visualizing complex valued functions Colorized plots of rational functions vs. the Riemann sphere (1851); (the extended plane)pdf
Mobius Transformation (youtube, HiRes), pdf description
Mobius composition transformations, as matrices
Software: Matlab: zviz.zip, python
Read: Class-notes
AE-3: Complex algebra; visualizing complex functions; Mobius transformations; (pdf due 1 week)
18 7
M 10/9

Lecture topic 2: Fourier Transforms of signals
Fourier Transform (wikipedia), Notes on the Fourier series and transform from ECE 310
(including tables of transforms and derivations of transform properties)
Read: Class-notes

19 W 10/11 Lecture: Fourier Transforms of systems
AE-2 Due
Read: Class-notes;
20 F 10/13 Lecture: Laplace transforms continued
The importance of Causality
Cauchy Riemann role in the acceptance of complex functions:
Convolution of the step function: {$u(t) \leftrightarrow 1/s$} vs. {$2\tilde{u}(t) \equiv 1+ \mbox{sgn}(t) \leftrightarrow 2\pi \delta(\omega) + 2/j\omega$}

Read: Class-notes; Laplace Transform, Types of Fourier transforms

21 8
M 10/16 Lecture: The nine postulates of Systems (aka, Network) pdf
The important role of the Laplace transform re impedance: {$z(t) \leftrightarrow Z(s)$}
A.E. Kennelly introduces complex impedance, 1893 pdf
Fundamental limits of the Fourier re the Laplace Transform: {$\tilde{u}(t)$} vs. {$u(t)$}

AE-3 Due

22 W 10/18 Optional Class Review for Exam II (No official class): 7-10 PM; 3013 ECEB
L W D Date Lecture and Assignment
Part III: Scaler Differential Equations (12 Lectures)
23 F 10/20 Lecture: Integration in the complex plane: FTC vs. FTCC
Analytic vs complex analytic functions and Taylor formula
Calculus of the complex {$s=\sigma+j\omega$} plane: {$dF(s)/ds$}, {$\int F(s) ds$} (Boas, see page 8)
The convergent analytic power series: Region of convergence (ROC)
Complex-analytic series representations: (1 vs. 2 sided); ROC of {$1/(1-s), 1/(1-x^2), -\ln(1-s)$}
1) Series; 2) Residues; 3) pole-zeros; 4) Continued fractions; 5) Analytic properties
History: The amazing Bernoulli family; Fluid mechanics; airplane wings; natural logarithms
Beginnings of modern mathematics: Euler and Bernoulli, Euler's standard circular-function package (Logs, exp, sin/cos);
D'Angelo {$e^z$} lecture lecture
Inversion of analytic functions: Example: {$\tan^{-1}(z) = \frac{1}{2i}\ln \frac{i-z}{i+z}$}, the inverse of Euler's formula (1728) (Stillwell p. 314)
Read: Class-notes
Homework 7 (DE-1): Series, differentiation, CR conditions, Bi-Harmonic functions: pdf, Due 10/24/2017
24 9
M 10/23 Lecture: Cauchy-Riemann (CR) conditions
Cauchy-Riemann conditions and differentiation wrt {$s$}: {$Z^\prime(s) \equiv \frac{dZ(s)}{ds} = \frac{dZ(s)}{d\sigma} = \frac{dZ(s)}{dj\omega}$}
Differentiation independent of direction in {$s$} plane: {$Z(s)$} obeys CR conditions:
{$\frac{\partial R(\sigma,\omega)}{\partial\sigma} = \frac{\partial X(\sigma,\omega)}{\partial\omega}$} and {$\frac{\partial R(\sigma,\omega)}{\partial\omega} = -\frac{\partial X(\sigma,\omega)}{\partial\sigma}$}
Cauchy-Riemann conditions require that Real and Imag parts of {$Z(s) = R(\sigma,\omega) + j X(\sigma,\omega)$} obey Laplace's Equation:
{$\nabla^2 R=0$}, namely: {$\frac{\partial^2R(\sigma,\omega)}{\partial^2\sigma} + \frac{\partial^2 R(\sigma,\omega)}{\partial^2 \omega} =0 $}
{$\nabla^2 X=0$}, namely: {$\frac{\partial^2 X(\sigma,\omega)}{\partial^2\sigma} + \frac{\partial^2 X(\sigma,\omega)}{\partial^2 \omega} =0$},
Biharmonic grid (zviz.m)
Discussion of the solution of Laplace's equation given boundary conditions (conservative vector fields)
Read: Class-notes & Boas pages 13-26; Derivatives; Convergence and Power series
25 W 10/25 Lecture: Complex analytic functions and Brune impedance
Complex impedance functions {$Z(s)$}, {$\Re Z(\sigma>0) \ge 0$}, Simple poles and zeros & 9 Postulates
Time-domain impedance {$z(t) \leftrightarrow Z(s)$}
Read: Class-notes
26 F 10/27 Lecture: Review session on multi-valued functions and complex integration
Riemann sheets, colorized plots, branch cuts, Review of Fundamental Theorems of complex analytic functions.
Laplace's equation and its role in Engineering Physics. What is the difference between a mass and an inductor?
nonlinear elements; Examples of systems and the Nine postulates of systems.

Homework 8 (DE-2): Inverse Laplace Transforms; Residue integration: pdf, Due 10/31/2017

27 10
M 10/30 Lecture: Three complex integration Theorems: Part I
1) Cauchy's Integral Theorem: {$\oint f(z) dz =0$} (Boas p. 45) vs. 2D Green's Thm (p. 49); Stokes (Thm, Bio)
Read: Class-notes & Boas p. 33-43 Complex Integration; Cauchy's Theorem
DE-1 due
28 W 11/1 Lecture: Three complex integration Theorems: Part II
2) Cauchy's Integral Formula: {$\frac{1}{2\pi j} \displaystyle \oint_{{\partial}_{\gamma}} \frac{f(z)}{z-z_0}dz = f(z_0) \, U(\gamma) \equiv 0$} if {$z_0 \notin \gamma^\circ$}
3) Cauchy's Residue Theorem
Example by brute force integration: {$\oint_{|s|=1} \frac{ds}{s}= 2\pi j$}

Read: Class-notes & Boas p. 33-43 Complex Integration; Cauchy's Theorem

29 F 11/3 Lecture: The Inverse Laplace Transform (ILT); poles and the Residue expansion: The case for causality {$t\le0$}
Cauchy's Residue theorem {$\Leftrightarrow$} 2D Green's Thm (in {$\mathbb C$})
Homework 9 (DE-3): pdf, Due 11/7/2017
Read: Class-notes
30 11
M 11/6 Lecture: Inverse Laplace Transform: Use of the Residue theorem {$t>0$}
Case for causality: Closing the contour: ROC as a function of {$e^{st}$}.
Examples: {$F(s)=1 \leftrightarrow \delta(t)$} and {$u(t) \leftrightarrow 1/s$}
Case of RC impedance {$ z(t) = R\delta(t)+u(t)/C \leftrightarrow R+1/sC $}
RC admittance {$ y(t) = e^{-t}u(t) \leftrightarrow 1/(s+1) $}
Semi-capacitor: {$ u(t)/\sqrt{t} \leftrightarrow \sqrt{\pi/s} $}

Read: Class-notes
DE-2 Due

31 W 11/8 Lecture: General properties of Laplace Transforms:
Modulation, Translation, Convolution, periodic functions, etc. (png)
Table of common LT pairs (png)
Read: Class-notes
32 F 11/10 Lecture: General properties of Impedance (Z) and Transmission (ABCD) functions:
Impedance {$Z(s) = V(s)/I(s) \rightarrow $} Generalized impedance and interesting story Raoul Bott
Minimum phase impedance {$\rightarrow$} Simple poles & zeros in LHP ({$\sigma \le 0$})
Transfer {$H(s)=V_2/V_1, I_2/I_1 \rightarrow $} Allpass: {$|e^{-\jmath\phi(\omega)}|=1 \rightarrow$} poles in LHP, zeros in RHP
Wiener's factorization theorem: {$H(s) = M(s)A(s)$} with factors Minimum phase {$M(s)$} & Allpass {$A(s)$}
Read: Class-notes
33 12
M 11/13 Lecture: Riemann Zeta function {$\zeta(s)=\sum \frac{1}{n^s}$}
Euler's vs. Riemann's Zeta Function (i.e., poles at the primes), music of primes, Analytic continuation, Tao
Introduction to the Riemann zeta function (Stillwell p. 184) Euler's product formula;
Inverse Laplace transform of {$\zeta(s) \leftrightarrow \mbox{Zeta}(t)$} & Sieve of Eratosthenes (Lec 4)
plot of Riemann-Zeta function showing magnitude and phase
DE-3 Due
34 W 11/15 Lecture: Time-domain Riemann-Zeta function

35 R 11/16 Exam III 7-10 PM; ?ROOM? ECEB
35 F 11/17 No class due to Thursday Exam III:
- 47 Sa 11/19 Thanksgiving Holiday (11/18-11/26)
L W D Date Lecture and Assignment
Part IV: Vector (Partial) Differential Equations (9 Lectures)
36 13
M 11/27 Lecture: Scaler wave equation {$\nabla^2 p = \frac{1}{c^2} \ddot{p}$} with {$c=\sqrt{ \eta P_o/\rho_o }$}
Newton's formula: {$c=\sqrt{P_o/\rho_o}$} with an error of {$\sqrt{1.4}$}
What Newton missed: Adiabatic compression {$PV^\eta=$} const with {$\eta = \frac{c_p}{c_v} = \frac{dof+2}{dof}=\frac{7}{5}$}
d'Alembert solution: {$\psi = F(x-ct) + G(x+ct)$}
Homework 10 (VC-1): pdf, Due: Nov 28 Mon (Alt 30 Wed)
Read: Class-notes
37 W 11/29 Lecture: The Webster Horn Equation {$ \frac{1}{A(x)}\frac{\partial}{\partial x}A(x)\frac{\partial}{\partial x}{\cal P}(x,\omega) = \frac{s^2}{c^2}{\cal P}(x,\omega) $}
Dot and cross product of vectors: {$ \mathbf{A} \!\cdot\! \mathbf{B}, \mathbf{A} \!\times\! \mathbf{B} $} vs. {$ \nabla \phi, \nabla\!\cdot\!\mathbf{B}, \nabla \!\times\! \mathbf{B} $} Curl examples
Read: Class Notes
38 F 12/1 Lecture: Gradient, divergence, curl, scalar Laplacian and Vector Laplacian
Gradient {$\nabla p(x,y,z)$}, divergence {$\nabla \cdot \mathbf{D}$} and Curl {$\nabla \times \mathbf{A}(x,y,z)$}, Scalar Laplacian {$\nabla^2 \phi$}, Vector Laplacian {$\nabla^2 \mathbf{E}$}
Read: Class-notes
39 14
M 12/4 Lecture: More on the curl and divergence: Stokes' (curl) and Gauss' (divergence) Theorems, Vector Laplacian
Homework 11 (VC-2): pdf, Due: Dec 7 Wed
Read: Class-notes
40 W 12/6 Lecture: J.C. Maxwell unifies Electricity and Magnetism (1861); Basic definitions: {$ \mathbf{E}, \mathbf{H}, \mathbf{B}, \mathbf{D} $};
O. Heaviside's (1884) vector form of Maxwell's Eqs.: {$\nabla \times \mathbf{E} = - \dot{\mathbf{B}} $}, {$\nabla \times \mathbf{H} = \dot{ \mathbf{D} }$}
Differential and integral forms of Maxwell's Eqs.
How a loudspeaker works: {$ \mathbf{F} = \mathbf{J} \times \mathbf{B} $} and EM Reciprocity; Magnetic loop video, citation
VC-1 due
Read: Class-notes
41 F 12/8 Lecture: The Fundamental theorem of vector calculus: {$\mathbf{F}(x,y,z) = -\nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)$},
Definitions of Incompressable and irrotational fluids depend on two null-vector identities:
DoC: {$\nabla\cdot\nabla\times(\text{vector})=0$} & CoG: {$\nabla\times\nabla(\text{scalar}) =0$}.
Definition of the Conservative vector fields.
Read: Class-notes
42 15
M 12/11 Lecture: The low-frequency quasi-static approximation: i.e., {$a < \lambda=c/f$} or {$f < c/a$}) are used for:
Brune's Impedance ({$a \ll \lambda$}), Kirchhoff's Laws, the telegraph wave equation starting from Maxwell's equations.
Impedance boundary conditions and generalized impedance:
{$Z(s)\equiv \frac{\cal P}{\cal V} = r_0 \frac{1+\Gamma(s)}{1-\Gamma(s)}$} where {$ \Gamma(s) \equiv {\cal P}_-/{\cal P}_+ $} and {$r_0 = {\cal P_+}/{\cal V_+}$}, with {${\cal P}= {\cal P}_+ +{\cal P}_-$} and {${\cal V}= {\cal V}_+ -{\cal V}_-$}.

Read: Class-notes

43 W 12/13 Lecture: Review The Fundamental Thms of Mathematics & their applications Theorems of Mathematics;
Fundamental Thms of Mathematics (Ch. 9)
QM: Normal modes vs. eigen-states, delay vs. quasi-statics;
The Hydrogen atom is an exponential horn: it is a waveguide with radial normal modes (eigen-states),
occupied with electrons (EM energy), which escapes (i.e., radiates) as photons (free particles). This explains {$E=h\nu$}.

Read: Class-notes

- R 12/14 Reading Day

- M 12/18 Final Exam Monday Dec 18, 7-10pm ECEB 2013

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