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Concepts in Mathematics: ECE Webpage ECE298JA; ECE298JA; UIUC Course Explorer: ECE298JA; Time: 1111:50 MWF; Location: 2013 ECEB (official); Register
 Professor: Jont B. Allen (jontalle@illinois.edu), TA: Sarah Robinson (srrobin2@illinois.edu)
 Syllabus: ECE298JAF15; Flyer; Class notes: NEW: Nov 18 pdf, OLD: pdf; Book (optional): Stillwell 3d edition, google for pdf, Video of Stillwell presentation, pdf, Boas;
 Exams: Exam I; Exam II; Exam III; About the final; Grade statistichs? and distributions
 Office Hours (TA): Mon 121pm & Tues 45pm (Tues 56pm available by appointment) Location: 2137 Beckman (next to ECEB); Allen by apt.
 Extra TA Office Hours Fridays Oct 28, 121pm & Nov 4, 121pm in 2137 Beckman
 General interest: Timeline; luminaries; Math history: site1, site2; Mathematical Notation by topic, alphaorder, use of Functions and their History, Mathematical vignettes
 Tools: MATLAB, Octave, Latex
 This week's schedule
ECE 298JA Schedule (Fall 2016)
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 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. 1524) Homework 1 (NS1): 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. 2429)
 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. 2932)
 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?Publicprivate 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.3233, 7075); A short history of primes, History of PNT NS1 Due Homework 2 (NS2): 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, 7375)
 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. 3435) Homework 3 (NS3): 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 NS2 Due Read: Lec. 7 (p. 36, 7779)
 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. 3637, 7981) 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 {$\sqrt{n}$} NS3 Due Read: Lec 9 (pp. 3738, 8182)
 10

 W
 9/14
 Exam I (In Class): Number Systems

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 Lecture and Assignment




 Part II: Algebraic Equations (12 Lectures)
 11

 F
 9/16
 Lecture: Analytic geometry as physics (Stream 2) The first "algebra" alKhwarizmi (830CE) Polynomials, Analytic functions, {$\infty$} Series: Geometric {$\frac{1}{1z}=\sum_{0}^\infty z^n$}, {$e^z=\sum_{0}^\infty \frac{z^n}{n!}$}; 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. 277278) Read: Lec 12 Homework 4 (AE1): 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; Quadratic EquationSolution of the quadratic (Brahmagupta, 628); Solution of the cubic (c1545),quartic (Tartaglia et al..., 1535), quintic ({$d=5$}) 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 upperdiagional {$U$} Read: Lec 15 Homework 5 (AE2): Nonlinear 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 AE1 Due
 17

 F
 9/30
 Lecture: Introduction to the Riemann sphere (1851); (the extended plane) (JS10, p. 298312) 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;
AE3: pdf Due Oct 10: ABCD method; Colorized mappings; Mobius transformations
 19

 W
 10/5
 Lecture: Fourier Transforms for signals AE2 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. {$u(t)$}
AE3 Due
 22

 W
 10/12
 No class due to Exam II: 710 PM; 2013 ECEB

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 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) Complexanalytic series representations: (1 vs. 2 sided); ROC of {$1/(1s), 1/(1x^2), \ln(1s)$} 1) Series; 2) Residues; 3) polezeros; 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 circularfunction package (Logs, exponentials, sin/cos) Inversion of analytic functions: Example: {$\tan^{1}(z) = \frac{1}{2i}\ln \frac{iz}{i+z}$}, the inverse of Euler's formula (1728) (Stillwell p. 314) Read: Lec 23 Homework 7 (DE1): Series, differentiation, CR conditions, BiHarmonic functions: pdf, Due 10/24/2016
 24
 9 42
 M
 10/17
 Lecture: CauchyRiemann (CR) conditions CauchyRiemann 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 {$\frac{\partial R(\sigma,\omega)}{\partial\omega} = \frac{\partial X(\sigma,\omega)}{\partial\sigma}$} CauchyRiemann 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 1326; 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 Timedomain impedance {$z(t) \leftrightarrow Z(s)$} Read: Lec 25
 26

 F
 10/21
 Lecture: Review session on multivalued 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 (DE2): 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. 3343 Complex Integration; Cauchy's Theorem DE1 due
 28

 W
 10/26
 Lecture: Three complex integration Theorems: Part II 2) Cauchy's Integral Formula: {$\frac{1}{2\pi j} \displaystyle \oint_{{\partial}_{\gamma}} \frac{f(z)}{zz_0}dz = f(z_0) \, U(\gamma) \equiv 0$} 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. 3343 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 (DE3): 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) $} Semicapacitor: {$ u(t)/\sqrt{t} \leftrightarrow \sqrt{\pi/s} $}
Read: Lec 30 DE2 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)$} Read: Lec 32
 33
 12 45
 M
 11/7
 Lecture: Wave equation (DE3) Euler's vs. Riemann's Zeta Function (i.e., poles at the primes); music of primes, Tao Introduction to the Riemann zeta function (Stillwell p. 184) Euler's product formula;
plot of RiemannZeta function showing magnitude and phase separately Inverse Laplace transform of {$\zeta(s) \leftrightarrow \mbox{Zeta}(t)$}
 34

 W
 11/9
 No class due to Exam III: Thursday
 34

 R
 11/10
 Exam III 710 PM; NOTE ROOM CHANGE: 2015ECEB

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 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: {$\psi = F(xct) + G(x+ct)$} Homework 10 (VC1): pdf, Due: Nov 28 Mon (Alt 30 Wed) Read: Class Notes p. 12
 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} $} Read: Class Notes p.310?
 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 38
 38

 F
 11/18
 Lecture: More on the curl and divergence: Stokes' (curl) and Gauss' (divergence) Theorems, Vector Laplacian Homework 11 (VC2): pdf, Due: Dec 7 Wed Read: Lec 39
 
 47
 Sa
 11/19
 Thanksgiving Holiday (11/1911/27)
 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} }$} How a loudspeaker works: {$ \mathbf{F} = \mathbf{J} \times \mathbf{B} $} and EM Reciprocity; Magnetic loop video, citation VC1 due Read: Lec 40
 40

 W
 11/30
 Lecture: The Fundamental theorem of vector calculus: Differential {$\mathbf{F}(x,y,z) = \nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)$} integral forms of ME Incompressable and irrotational fluids and their two critical vector identities Read: Lec 41
 41

 F
 12/2
 Lecture: The lowfrequency quasistatic approximation ({$a < \lambda=c/f$} thus {$f < c/a$}) Brune's Impedance and the quasistatic (QS) approximation ({$a \ll \lambda$}) Kirchhoff's Laws, impedance boundary conditions (integral equations); The telegraph wave equation starting from Maxwell's equations Quantum Mechanics (QM) assumes QS: {$c$}=speed of light; {$v$}=frequency; {$V$}=groupvelocity {$E=h \nu$}, {$p=h/\lambda$}; {$\nu = E/h, \lambda=h/p \rightarrow c = \lambda \nu = E/p$} Electrodynamically vs. classically: {$ c = E/p \gg mV^2/mV = V $}, thus QSs applies to QM Read: Lec 42
 42
 15 49
 M
 12/5
 Lecture: Last day of Class: Review of 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. eigenstates, delay and quasistatics; Read: Lec 43
 43

 W
 12/7
 TBD
VC2 due
 

 R
 12/8
 Reading Day
 

 M
 12/12
 Final Exam Monday Dec 12, 710pm ECEB 2013

