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Concepts in Mathematics: ECE Webpage ECE298-JA; ECE-298JA; UIUC Course Explorer: ECE-298-JA;

ECE 298JA Schedule (Fall 2015)

L W D Date Lecture and Assignment

Part I: Number systems (12 Lectures)
1 1
M 8/24 ''Introduction & Historical Overview; Lecture 0: pdf;

Three streams:
1) Number systems (Integers, rationals); 2) Geometry; 3) {$\infty$} {$\rightarrow$} Set theory
Common Math symbols
Matlab tutorial: pdf
Read: Stillwell Ch. 1, 1.7

2 W 8/26 Lecture: Number Systems
First use of zero as a number (Brahmagupta defines rules)
First use of {$\infty$} (Bhaskara's interpretation)
Taxonomy of Numbers: {$\pi_k \in \mathbb P \subset \mathbb N \equiv Z^+ \subset \mathbb Z \subset \mathbb Q \subset \mathbb J \subset \mathbb R \subset \mathbb C$}
Three fundamental theorems: Arithmetic, Algebra, Calculus
Read: Ch. 4.5, 5.1, 5.7 p. 53-67, 56
Homework 1 (NS-1): Basic Matlab commands: pdf, Due 9/2
3 F 8/28 Lecture: Aristotle, Pythagoras and the beauty of integers; Why are integers important?
Eigenmodes: Mathematics in Music and acoustics: Strings, Chinese Bells, chimes;
History of acoustics:

BC: Pythagoras; Aristotle;
17C: Mersenne, Marin; Galilei, Galileo; Hooke, Robert; Boyle, Robert; Newton, Sir Issac;
18C: Bernoulli, Daniel; Euler; Lagrange; d'Alembert;
19C: Gauss; Laplace; Fourier; Helmholtz; Heaviside; Strutt, William; Rayleigh, Lord; Bell, AG
Read: Sections 1.1, 1.2, 4.5, 5.7

4 2
M 8/31 Lecture: Brief tutorial on Prime Numbers {$\pi_k$}; Fundamental Thm of Arith
Definition of Pythagorean triplets with examples; Euclid's formula
Definition of the gcd{$(m,n)$} with examples; Euclid's algorithm
Read: 1.4, 1.5, 5.3; Short history of primes
5 W 9/2 Lecture: Pythagorean triplets (p. 43) [{$a, b, c$}] such that {$c^2=a^2+b^2$}
Properties, examples, History
Coprime integers ({$m \perp n$}) have no gcd: gcd{$(m,n)$}=1
Prime number Theorem Statement, History, Bertrand's conjecture, Prime number Sieves
Definition of Pell's Equation: {$m^2 - N n^2 = 1$}
Read: 1.2, 1.3
HW1 Due
Homework 2 (NS-2): Histogram of Primes; Pythagorean triplets; gcd(m,n): pdf, Due 9/9
6 F 9/4 Lecture: Greek Number Theory; Why are integers so important to the Greeks? (Eudoxus, Archimedes) (p. 57)
Integers {$\Leftrightarrow$} Physics
Lecture: Euclid's Algorithm: The GCD (p. 41, 66)
Properties and Derivation of GCD
Why integers are important for internet security? Elliptic curve DSA

Introduction to the Riemann zeta function (p. 184) {$\zeta(s)$}; Relation to (primes & co-primes)
plot of Riemann-Zeta function showing magnitude and phase separately
Read: 3.3, 3.4

- 3


M 9/7 Labor Day Holiday -- No class
7 W 9/9 Lecture: Continued Fraction algorithm (Euclid & Gauss, p. 47)

The Rational Approximations of irrational {$(\sqrt{2} \approx 17/12\pm 0.25%)$} and transcendental {$(\pi \approx 22/7)$} numbers
Algebraic Generalizations of GCD
real {$\mathbb{R}$} vs. complex {$\mathbb{C}$} numbers, vectors, matrices
Read: 3.6, 5.3, History of {$\mathbb R$}
HW2 Due
Homework 3 (NS-3): Continued fractions [rats()]; Pell's Eq.; Solutions {m,n,1} of am+bn=1 via GCD(a,b), pdf Due 9/18
GCD Algorithm - Stillwell sections 3.3 & 5.3

8 F 9/11 Lecture: Euclid: Ruler and Compass constructions: Conic sections
Complex numbers (Bombelli, 1575, p. 259) and the Radius of convergence (ROC)
Read: 2.3, 2.4; 4.2, 4.3
9 4
M 9/14 Lecture: Pell's Equation: General solution (p. 72); Brahmagupta's solution by composition
Pell equation solver,history
Part I (Number systems) Notes (pdf), Compressed 3x3 format(pdf)
Read: 5.3, 5.4
10 W 9/16 Lecture: Pythagorean geometry: Euclidean Lengths

Read: 1.6, 6.3

11 F 9/18 Review for Exam I
HW3 Due
12 5
M 9/21 Lecture: Introduction to analytic geometry
Polynomials (p. 87) and the first "algebra" (al-jabr)
Bombelli (1572) first uses complex numbers (p. 277-278)
Composition of polynomial equations
13 W 9/23 Exam I (In Class): Number Systems (Text Chapters 1-5)
L W D Date Lecture and Assignment
Part II: Algebraic Equations(11 Lectures)
14 F 9/25 Lecture Stream 2: Ch. 6: Geometry + Algebra {$\Rightarrow$} Analytic Geometry: From Euclid to Descartes+Newton
Newton (1667) labels complex cubic roots as "impossible" (p. 112 (3rd Ed.)); Newton's "irrational" power series
Read: Sect. [5.5-6.3] (p. 78-95)
15 6
M 9/28 Lecture: Root classification for polynomials of Degree {$d =$} 1, 2, 3, 4 (p.102)
Chinese discover Gaussian elimination (Jiuzhang suanshu) (p. 89)
Solution of the quadratic (Brahmagupta, 628); Solution of the cubic (c1545) (p. 95-96) (Tartaglia et al..., 1535)
Quintic ($d=5$}) cannot be solved (Abel, 1826)
Homework 4 (AE-1): Linear systems of equations; Gaussian elimination; Matrix permutations; determinants, pdf Due 10/5
Read: Ch 6, p. 95-108 Cubic, Quatric, Quintic; Descartes' Thm p. 103
16 W 9/30 Lecture: First Analytic Geometry (Fermat 1629; Descartes 1637) (p. 118)
Descartes' insight: Composition of two polynomials of degrees ($m$}, {$n$} {$\rightarrow$} one of degree {$n\cdot m$})
Composition vs. intersection of polynomials: What is the difference?
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
Complex planes & lines
Read: Ch. 7, p. 104-119 (2nd Ed.), p. 109-125 (3rd Ed.)
17 F 10/2 Lecture: Composition and the Mobius (aka bilinear) transformation {$\Rightarrow$} Ratios of polynomials (aka: Poles & zeros)
Projection operations and Gaussian Elimination: {$\Pi_n^N P_n A$} gives upper-diagional {$N\times N$} matrix

ABCD Matrix composition; Commuting vs. Noncommuting operators
Read: Ch. 7, p. 104-119 (2nd Ed.), p. 109-125 (3rd Ed.) Analytic Geometry

18 7
M 10/5 Lecture: Review of Composition of polynomials, ABCD matrix method, convolution of sequences
Gaussian elimination; Permutation and Pivot matrices;

Formula for Pell Triplets (solutions to {$x^2-Ny^2=1$} with {$x,y\in \mathbb Z$}
HW4 Due
Homework 5 (AE-2): Linear and nonlinear systems of equations; Gaussian elimination; Matrix permutations; Convolution pdf Due 10/12

19 W 10/7 Lecture:Introduction to the Riemann sphere (1851); (the extended plane) (p. 279-292 (2nd Ed.), p. 298-312 (3rd Ed.))
Mobius Transformation (youtube, HiRes), pdf description
Understanding {$\infty$} by closing the complex plane;
Composition of line and sphere pdf
20 F 10/9 Lecture: Fundamental Thm of Algebra Colorized plots, (pdf & Matlab code zviz.m, OLD), python
Bezout's Thm: Mathpages, Wikipedia by Example
More on the ABCD relations for transmission lines
Mobius transformations in matrix format
Invariance of the cross ratio [{$z, b, c, d$}] {$\equiv (z-b)(c-d)/(z-d)(c-b)$} to a Mobius transformation
3D representations of 2D systems; Perspective (3D) drawing.
Read: p. 111-119 (2nd Ed.), p. 118-125 (3rd Ed.)Bezout's Thm
20' 8
M 10/12 Lecture: Fourier Transforms for signals vs. Laplace transforms for systems;
HW5 Due
Homework 6 (AE-3): ABCD method; Colorized mappings; Mobius transformations
Fund. Thm Alg vs Bezout's Thm; Fourier vs Laplace Transforms pdf Due 10/19
Read: Fourier Transform (wikipedia), Notes on the Fourier series and transform from ECE 310
(including tables of transforms and derivations of transform properties)
21 W 10/14 Lecture: Laplace transforms for systems & Fourier Transforms (Hilbert space) for signals
Cauchy Riemann role in the acceptance of complex functions:
The importance of Causality; Why {$2\tilde{u}(t) \equiv 1+ \mbox{sgn}(t) \leftrightarrow 2\pi \delta(\omega) + 1/j\omega$} is not causal
Convolution of the step function
Mobius's Homogeneous Coordinates (1827) (Genus, cross-ratio) (p. 134 (2nd Ed.), p. 147 (3rd Ed.));
Read: Laplace Transform,Table of transforms
22 F 10/16 Lecture: The 6 postulates of System (aka, Network) Theory pdf
The important role of the Laplace transform re impedance: {$z(t) \leftrightarrow Z(s)$}
Heaviside & Maxwell's Eqs. 1880, p. 402 (2nd Ed.), p. 436 (3rd Ed.); A.E. Kennelly introduces complex impedance, 1893;
Fundamental limits of the Fourier re the Laplace Transform: {$\tilde{u}(t)$} vs. {$u(t)$}
Calculus of the complex {$s$} plane ($s=\sigma+j\omega$}): {$dF(s)/ds$}, {$\int F(s) ds$} (Boas, see page 8)


23 9
M 10/19 Lecture: General discussion and review of Exam II
HW6 Due
24 W 10/21 No class due to Exam II: 7-10 PM; 3013 ECEB
L W D Date Lecture and Assignment
Part III: Scaler Differential Equations (10 Lectures)
25 F 10/23 Lecture: The amazing Bernoulli family; Fluid mechanics; airplane wings; natural logarithms

Read: Chapter 1 of Boas (handout)

26 10
M 10/26 Lecture: Cauchy-Riemann conditions follow from 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 {$\frac{\partial R(\sigma,\omega)}{\partial\omega} = -\frac{\partial R(\sigma,\omega)}{\partial\sigma}$}
Read: Chap 1 Boas (Handout)

27 W 10/28 Lecture: Infinite power Series and analytic function theory (p 171) as an {$\infty$} degree extension of the polynomial;
The convergent analytic power series: Region of convergence (ROC)
Complex-analytic series representations: (1 vs. 2 sided); ROC of {$1/(1-s), 1/(1-x^2)$}
1) Series; 2) residue; 3) pole-zero; 4) continued fraction

Homework 7 (DE-1): Series, differentiation, CR conditions, Bi-Harmonic functions: Ver 1.11 pdf, Due 11/4/2015

28 F 10/30 Lecture: Integration in the complex plane: Laplace's equation and the CR conditions
Basic equations of mathematical Physics: Wave equation, diffusion equation, Laplace's Equation
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)
Detailed discussion of the solution of Laplace's equation in 2 dimensions given the boundary values.

Read: Boas pages 13-26; Derivatives; Convergence and Power series

29 11
M 11/2 Lecture: Integration in the complex plane: Basic definitions
1) Fundamental Thm of complex calculus (FTCC): {$\int_a^z f(\zeta) d\zeta = F(z)-F(a)$}
2) Differentiation {$f(z) = dF(z)/dz$} independent of path (follows from FTCC)
3) ROC along path of integration, close to a pole
Read: Boas pages 27-33
30 W 11/4 Lecture: Three complex integration Theorems:
1) Cauchy's Integral Theorem: {$\oint f(z) dz =0$} (Boas p. 45) vs. 2D Green's Thm (p. 49); Stokes (Thm, Bio)
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
Examples: Brute force integration of {$\oint_{|s|=1} \frac{ds}{s}= 2\pi j$}

Homework 8 (DE-2): Inverse Laplace Transforms; Residue integration: pdf v1.0, pdf v1.11, Due 11/11/2015
Read: Boas p. 33-43 Complex Integration; Cauchy's Theorm

31 F 11/6 Lecture: The Inverse Laplace Transform (ILT); poles and the Residue expansion: The case for causality
Fundamental Thm of Complex Calculus: {$F(s) = F(a) + \int_a^s f(\zeta) d\zeta \Rightarrow f(s) = dF/ds$} is independent of the path
Cauchy's Residue theorem {$\Leftrightarrow$} 2D Green's Thm (in {$\mathbb C$})
Example: {$e^{-t}u(t) \leftrightarrow \frac{1}{s+1}$}
Read: Stillwell 319-322; Boas 49
32 12
M 11/9 Lecture: Introduction to the inverse Laplace Transform: Use of the Residue theorem.
ROC as a function of {$e^{st}$}. Cases of {$F(s)=1 \leftrightarrow \delta(t)$} and {$u(t) \leftrightarrow 1/s$}


33 W 11/11 Lecture: Detailed examples of the inverse Laplace Transform: Role of {$\Re\{st\}$}; Closing the contour as {$s\rightarrow \infty$}

Homework 9 (DE-3): Version 1.13 pdf, Due 11/18/2015

34 F 11/13 Lecture: General properties of Laplace Transforms: modulation, translation, convolution, periodic functions, etc. png
Table of common LT pairs png

35 13
M 11/16 Lecture: Analytic functions: Euler's vs Riemann's Zeta Function (i.e., poles at the primes); music of primes, Tao
Derivation of Sterling's formula

Inverse Laplace transform of {$\zeta(s) \leftrightarrow \mbox{Zeta}(t)$}
Analytic continuation (continued)
Why is the convergence of a series/integral important?
The role of Sets; Why closing a set important (the fear of {$\infty$})? p. 56

36 W 11/18 Lecture: Multi-valued functions (and their many many-valued inverses!); branch cuts
The extended complex plane (regularization at {$\infty$}) (p. 280)
Complex analytic functions of Genus 1 (p. 343)
Thur 11/19 Review session for exam: Bring your questions
Room 2013, 1-2 PM
- F 11/20 Exam III: (NO CLASS)
- 47
- Thanksgiving Holiday (11/21-11/29)
L W D Date Lecture and Assignment
Part IV: Vector (Partial) Differential Equations (5 Lectures)
37 14
M 11/30 Lecture: Partial differential equations of Physics; A real-world example where the branch-cut placement is critical
The Fundamental theorem of vector calculus: Differential {$\mathbf{F}(x,y,z) = -\nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)$} and integral forms
Homework 10 (VC-1): : pdf, Due Wed 12/9/15
38 W 12/2 Lecture: Scaler (acoustics) and vector (Maxwell's EM) wave Equations: Basic definitions
$E, H, B, D$} definitions; Maxwell's Eqs: {$\nabla \times E = - \dot{B}$}; {$\nabla \times H = \dot{D}$}
Gradient {$\nabla p(x,y,z)$}, divergence {$\nabla \cdot D$} and Curl {$\nabla \times \mathbf{A}(x,y,z)$}
How a loudspeaker works: {$F = J \times B$} and EM Reciprocity; Magnetic loop video, citation
39 F 12/4 Lecture: More on curl and divergence; Stokes' (curl) and Gauss' (divergence) Theorems
The telegraph wave equation starting from Maxwell's equations
J.C. Maxwell unifies Electricity and Magnetism (1861); O. Heaviside's vector form of MEs (1884)
Lecture 39 Notes pdf
40 15
M 12/7 Lecture: The low-frequency quasi-static approximation ({$a < \lambda=c/f$} thus {$f < c/a$})
Brune's Impedance and the quasi-static approximation ({$a << \lambda$})
Impedance boundary conditions (integral equations);
Quantum Mechanics assumes very long wavelengths: {$E=h \nu$}, {$p=h/\lambda$}; {$\nu = E/h, \lambda=h/p$} thus
{$\lambda \nu = E/p =mv^2/mv = v < c$}
41 W 12/9 Lecture: Closure on Numbers, Algebra, Differential Equations and Vector Calculus.
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;
Lecture Notes pdf
- R 12/10 Reading Day
HW10 (VC-1) due

- Tr 12/17 Final Exam 1:30 PM -- 4:30 PM ECEB 2013

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