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ECE298JA-F16

Concepts in Mathematics: ECE Webpage ECE298-JA; ECE-298JA; UIUC Course Explorer: ECE-298-JA; Time: 11-11:50 MWF; Location: 2013 ECEB (official); Register

ECE 298JA Schedule (Fall 2016)

L W D Date Lecture and Assignment

Part I: Number systems (10 Lectures)
1 1
34
M 8/22 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: Lec. 1 (pp. 15-24)
Homework 1 (NS-1): Basic Matlab commands: pdf, Due 8/29 (1 week)

2 W 8/24 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 $} (Bhaskara's interpretation)
Floating point numbers IEEE 754 (c1985); History
Read: Lec. 2 (pp. 24-29)
3 F 8/26 Lecture: The role of physics in Mathematics: Math is a language, designed to do physics
The Fundamental theorems of Mathematics:
1) Arithmetic (i.e., primes), 2) Algebra, 3) Calculus (& Set Theory) and other key concepts:
History review:
BC: Pythagoras; Aristotle;
17C: Mersenne; Galilei, Galileo; Hooke; Boyle; Newton;
18C: Bernoulli, Daniel; Euler; Lagrange; d'Alembert;
19C: Gauss; Laplace; Fourier; Von Helmholtz; Heaviside; Rayleigh;
Read: Lec. 3 (pp. 29-32)
4 2
35
M 8/29 Lecture: Two Prime Number Theorems:
How to identify Primes (Brute force method: Sieve of Eratosthenes)
1) Fundamental Thm of Arith
2) Prime Number Theorem: Statement, Prime number Sieves
Why are integers important?Public-private key systems (internet security) Elliptic curve RSA
Pythagoras and the Beauty of integers: Integers {$\Leftrightarrow$}
1) Physics: The role of Acoustics & Electricity (e.g., light):
2) Eigenmodes: Mathematics in Music and acoustics: Strings, Chinese Bells, chimes;
Read: Lec 4 (pp.32-33, 70-75); A short history of primes, History of PNT
NS-1 Due
Homework 2 (NS-2): Prime numbers, GCD, CFA; pdf, Due 9/7
5 W 8/31 Lecture: Euclidean Algorithm for the GCD; Coprimes
Definition of the {$k=\text{gcd}(m,n)$} with examples; Euclidean algorithm
Properties and Derivation of GCD & Coprimes
Algebraic Generalizations of the GCD
Read: Lec. 5 (pp. 33, 73-75)
6 F 9/2 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: Lec 6 (pp. 34-35)
Homework 3 (NS-3): Pythagorean triplets, Pell's equation, Fibonacci sequence; pdf, Due Mon 9/12

- 3

36

M 9/5 Labor Day Holiday -- No class
7 W 9/7 Lecture: Pythagorean triplets {$[a, b, c] \in {\mathbb N}$} such that {$c^2=a^2+b^2$}
Examples of PTs & Euclid's formula
Properties, examples, History
NS-2 Due
Read: Lec. 7 (p. 36, 77-79)
8 F 9/9 Lecture: Pell's Equation: General solution; Brahmagupta's solution by composition
Chord and tangent solution (Diophantus {$\approx$}250CE) methods
Read: Lec. 8 (pp. 36-37, 79-81) History of {$\mathbb R$}
Optional: GCD Algorithm - Stillwell sections 3.3 & 5.3
9 4
37
M 9/12 Lecture: Fibbonacci Series
Geometry & irrational numbers {$\sqrt{n}$}
NS-3 Due
Read: Lec 9 (pp. 37-38, 81-82)

10 W 9/14 Exam I (In Class): Number Systems
L W D Date Lecture and Assignment
Part II: Algebraic Equations (12 Lectures)
11 F 9/16 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: Lec 11
12 5
38
M 9/19 Lecture: Complex analytic functions; Physical equations in several variables
Summarize Lec 11:Detailed review of series representations of analytic functions: Poles, residues, ROC, etc.
Geometry + Algebra {$\Rightarrow$} Analytic Geometry: From Euclid to Descartes+Newton
Newton (1667) labels complex cubic roots as "impossible." Bombelli (1572) first uses complex numbers (JS10: p. 277-278)
Read: Lec 12
Homework 4 (AE-1): Polynomials & Analytic functions and their inverse, Convolution, Newton's method (pdf, Due 9/28)
13 W 9/21 Lecture: Root classification for polynomials by convolution; Residue expansion
Chinese discover Gaussian elimination (Jiuzhang suanshu) (JS10: p. 89) Gaussian elimination in one & two variables
Solution of the quadratic (Brahmagupta, 628), cubic (c1545), quartic (Tartaglia et al..., 1535), quintic cannot be solved (Abel, 1826)
Composition of polynomial equations (Bezout's Thm)
Read: Lec 13
14 F 9/23 Lecture: Analytic Geometry (Fermat 1629; Descartes 1637)
Descartes' insight: Composition of two polynomials of degrees: ({$m$}, {$n$} {$\rightarrow$} one of degree {$n\cdot m$})
Composition, elimination vs. intersection of polynomials: What is the difference?
Detailed comparison of Euclid's Geometry (300BCE) and Algebra (830CE)
Computing and interpreting the roots of the characteristic polynomial (CP)
Linear equations are Hyperplanes in {$N$} dimensional space; 2 planes compose a line, 3 planes compose to a point
Vectors, Complex planes & lines, Dot and cross products of vectors
Read: Lec 14
15 6
39
M 9/26 Lecture: Gaussian elimination (intersection); Pivot matrices {$(\Pi_n)$}: {$U = \Pi_n^N P_n A$} gives upper-diagional {$U$}
Read: Lec 15
Homework 5 (AE-2): Non-linear and linear systems of equations; Gaussian elimination; pdf Due 10/5
16 W 9/28 Lecture: Composition of polynomials, ABCD matrix method
ABCD Composition relations of transmission lines
Read Lec 16
AE-1 Due
17 F 9/30 Lecture: Introduction to the Riemann sphere (1851); (the extended plane) (JS10, p. 298-312)
Mobius Transformation (youtube, HiRes), pdf description
Understanding {$\infty$} by closing the complex plane;
Chords on the sphere pdf
Mobius transformations in matrix format
Read: Lec. 17
18 7
40
M 10/3 Lecture: Fundamental Thm of Algebra (pdf) & Colorized plots

Software: Matlab: zviz.zip, python
Bezout's Thm: Mathpages, Wikipediaby Example
Exponential {$e^z$} D'Angelo lecture
3D representations of 2D systems; Perspective (3D) drawing.
Read: Lec 18; AE-3: pdf Due Oct 10: ABCD method; Colorized mappings; Mobius transformations

19 W 10/5 Lecture: Fourier Transforms for signals
AE-2 Due
Read: Lec 19; Fourier Transform (wikipedia), Notes on the Fourier series and transform from ECE 310
(including tables of transforms and derivations of transform properties)
20 F 10/7 Lecture: Laplace transforms for systems
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: Lec 20; Laplace Transform,Table of transforms

21 8
41
M 10/10 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 10/12 No class due to Exam II: 7-10 PM; 2013 ECEB
L W D Date Lecture and Assignment
Part III: Scaler Differential Equations (12 Lectures)
23 F 10/14 Lecture: Integration in the complex plane: FTC vs. FTCC
Analytic vs complex analytic functions and Taylor formula
Calculus of the complex {$s$} plane ({$s=\sigma+j\omega$}): {$dF(s)/ds$}, {$\int F(s) ds$} (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, exponentials, sin/cos)
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: Lec 23
Homework 7 (DE-1): Series, differentiation, CR conditions, Bi-Harmonic functions: pdf, Due 10/24/2016
24 9
42
M 10/17 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$},
Biharmonic grid (zviz.m)
Discussion of the solution of Laplace's equation given boundary conditions (conservative vector fields)
Read: Lec 24 & Boas pages 13-26; Derivatives; Convergence and Power series
25 W 10/19 Lecture: Complex analytic functions and Brune impedance
Complex impedance functions {$Z(s)$}, {$\Re Z(\sigma>0) \ge 0$}, Simple poles and zeros & 9 Postulates
Time-domain impedance {$z(t) \leftrightarrow Z(s)$}
Read: Lec 25
26 F 10/21 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 10/31/2016

27 M 10/24 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: Lec 27 & Boas p. 33-43 Complex Integration; Cauchy's Theorem
DE-1 due
28 W 10/26 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: Lec 28 & Boas p. 33-43 Complex Integration; Cauchy's Theorem

29 F 10/28 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 11/7/2016
Read: Lec 29
30 11
44
M 10/31 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: Lec 30
DE-2 Due

31 W 11/2 Lecture: General properties of Laplace Transforms:
Modulation, Translation, Convolution, periodic functions, etc. (png)
Table of common LT pairs (png)
Read: Lec 31
32 F 11/4 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)$}
Read: Lec 32
33 12
45
M 11/7 Lecture: Riemann Zeta function {$\zeta(s)=\sum \frac{1}{n^s}$}
Euler's vs. Riemann's Zeta Function (i.e., poles at the primes), music of primes, Analytic continuation, Tao
Introduction to the Riemann zeta function (Stillwell p. 184) Euler's product formula;
plot of Riemann-Zeta function showing magnitude and phase separately
Inverse Laplace transform of {$\zeta(s) \leftrightarrow \mbox{Zeta}(t)$}
DE-3 Due
34 W 11/9 No class due to Exam III: Thursday

34 R 11/10 Exam III 7-10 PM; NOTE ROOM CHANGE: 2015ECEB
L W D Date Lecture and Assignment
Part IV: Vector (Partial) Differential Equations (9 Lectures)
35 F 11/11 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 p. 1-2
36 13
46
M 11/14 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) $}
Dot and cross product of vectors (repeat of Lec 14): {$ \mathbf{A} \!\cdot\! \mathbf{B}, \mathbf{A} \!\times\! \mathbf{B} $} vs. {$ \nabla \phi, \nabla\!\cdot\!\mathbf{B}, \nabla \!\times\! \mathbf{B} $} Curl examples
Read: Class Notes p.3-10?
37 W 11/16 Lecture: Gradient, divergence, curl, scalar Laplacian and Vector Laplacian
Gradient {$\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: Lec 37
38 F 11/18 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: Lec 38
- 47 Sa 11/19 Thanksgiving Holiday (11/19-11/27)

39 14
48
M 11/28 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} $} and EM Reciprocity; Magnetic loop video, citation
VC-1 due
Read: Lec 39
40 W 11/30 Lecture: The Fundamental theorem of vector calculus: {$\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$}.
Definition of the Conservative vector fields.
Read: Lec 40
41 F 12/2 Lecture: The low-frequency quasi-static approximation: i.e., {$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: Lec 41

42 15
49
M 12/5 Lecture: Review The Fundamental Thms of Mathematics & their applications Theorems of Mathematics;
Fundamental Thms of Mathematics (Ch. 9)
Normal modes vs. eigen-states, delay and 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: Lec 42

43 W 12/7 Last day of Class: The ``quantum'' in QM refers to normal modes.
Quantum mechanics is quasi-static, which assumes no delay. {$\Rightarrow$}
Let {$c$}=speed of light; {$v$}=frequency; {$V$}=group-velocity, then

{$E=h \nu$}, {$p=h/\lambda$} {$\rightarrow$} {$\nu = E/h, \lambda=h/p \rightarrow c = \lambda \nu = E/p$} (pdf)
Electro-dynamically vs. classically: {$ c = E/p \gg mV^2/mV = V $}, thus QSs applies to QM VC-2 due

- R 12/8 Reading Day

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

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