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Concepts in Mathematics: ECE Webpage ECE298JA; ECE298JA; UIUC Course Explorer: ECE298JA;
 UIUC Calendars: Fall 2015; Academic
 Professor: Jont B. Allen (jontalle@illinois.edu), TA: Sarah Robinson (srrobin2@illinois.edu)
 Time: 1111:50 MWF; Location: 2013 ECEB (official)
 Syllabus: ECE298JAF15;Flyer; Book: Stillwell 3d edition, google for pdf, Video of Stillwell presentation, pdf, Boas
 Exams: Exam I; Exam II; About the final;
 Office Hours (TA): Week of 12/7 and Finals week: Tuesday 12/8, 12pm, Tuesday 12/15 24pm, and Wednesday 12/16 11am1pm (or by appointment) Location: 2137 Beckman; Allen by apt.
 Mathematical Notation, use of Functions and their History
 General interest: Timeline; luminaries; Math history: site1, site2
 Tools: Matlab, Latex
 This week's schedule
ECE 298JA Schedule (Fall 2015)
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 Date
 Lecture and Assignment




 Part I: Number systems (12 Lectures)
 1
 1 35
 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. 5367, 56 Homework 1 (NS1): 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 36
 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 Coprime integers have no gcd's: {$m \perp n$} satisfy gcd{$(m,n)$}=1 Prime number Theorem Statement, History, Bertrand's conjecture, Prime number Sieves Definition of Pell's Equation: {$m^2  N n^2 = 1$} Introduction to the Riemann zeta function (p. 184) {$\zeta(s)$};
Relation to (primes & coprimes, {$\phi(n)$}) 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 Read: 1.2, 1.3 HW1 Due Homework 2 (NS2): Histogram of Primes; Pythagorean triplets; gcd(m,n): pdf Δ, 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
Read: 3.3, 3.4 plot Δ of RiemannZeta function showing magnitude and phase separately
 
 3
37
 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 (NS3): 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 38
 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 39
 M
 9/21
 Lecture: Introduction to analytic geometry Composition of polynomial equations Read:
 13

 W
 9/23
 Exam I (In Class): Number Systems (Text Chapters 15)

L
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 Date
 Lecture and Assignment




 Part II: Algebraic Equations(11 Lectures)
 14

 F
 9/25
 Lecture Stream 2: Ch. 6: Polynomials (p. 87) and the first "algebra" (aljabr) Geometry + Algebra {$\Rightarrow$} Analytic Geometry: From Euclid to Descartes+Newton Bombelli (1572) first uses complex numbers (p. 277278) Newton (1667) labels complex cubic roots as "impossible" (p. 112 (3rd Ed.)); Newton's "irrational" power series Read: Sect. [5.56.3] (p. 7895)
 15
 6 40
 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. 9596) (Tartaglia et al..., 1535) Quintic ($d=5$}) cannot be solved (Abel, 1826) Homework 4 (AE1): Linear systems of equations; Gaussian elimination; Matrix permutations; determinants, pdf Due 10/5 Read: Ch 6, p. 95108 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. 104119 (2nd Ed.), p. 109125 (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 upperdiagional {$N\times N$} matrix
ABCD Matrix composition; Commuting vs. Noncommuting operators Read: Ch. 7, p. 104119 (2nd Ed.), p. 109125 (3rd Ed.) Analytic Geometry
 18
 7 41
 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 {$x^2Ny^2=1$} with {$x,y\in \mathbb Z$} HW4 Due Homework 5 (AE2): 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. 279292 (2nd Ed.), p. 298312 (3rd Ed.)) Mobius Transformation (youtube, HiRes), pdf description Understanding {$\infty$} by closing the complex plane; Composition of line and sphere pdf Read:
 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 [{$z, b, c, d$}] {$\equiv (zb)(cd)/(zd)(cb)$} to a Mobius transformation 3D representations of 2D systems; Perspective (3D) drawing. Read: p. 111119 (2nd Ed.), p. 118125 (3rd Ed.)Bezout's Thm
 20'
 8 42
 M
 10/12
 Lecture: Fourier Transforms for signals vs. Laplace transforms for systems; HW5 Due Homework 6 (AE3): 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, crossratio) (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)
Read:
 23
 9 43
 M
 10/19
 Lecture: General discussion and review of Exam II Read: HW6 Due
 24

 W
 10/21
 No class due to Exam II: 710 PM; 3013 ECEB

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 Lecture and Assignment




 Part III: 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 44
 M
 10/26
 Lecture: CauchyRiemann 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) Complexanalytic series representations: (1 vs. 2 sided); ROC of {$1/(1s), 1/(1x^2)$} 1) Series; 2) residue; 3) polezero; 4) continued fraction
Homework 7 (DE1): Series, differentiation, CR conditions, BiHarmonic 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 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) Detailed discussion of the solution of Laplace's equation in 2 dimensions given the boundary values.
Read: Boas pages 1326; Derivatives; Convergence and Power series
 29
 11 45
 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 2733
 30

 W
 11/4
 Lecture: Three complex integration Theorems: 1) Cauchy's Thm: {$\oint f(z) dz =0$} (Boas p. 45) vs. 2D Green's Thm (p. 49); Stokes (Thm, Bio) 2) Cauchy's Integral Thm: {$\frac{1}{2\pi j} \displaystyle \oint_{{\partial}_{\gamma}} \frac{f(z)}{zz_0}dz = f(z_0) \, U(\gamma) \equiv 0$} if {$z_0 \notin \gamma^\circ$} 3) Cauchy's Residue Theorem Examples: Brute force integration of {$\oint_{s=1} \frac{ds}{s}= 2\pi j$}
Homework 8 (DE2): Inverse Laplace Transforms; Residue integration: pdf v1.0, pdf v1.11, Due 11/11/2015 Read: Boas p. 3343 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 319322; Boas 49
 32
 12 46
 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 {$u(t) \leftrightarrow 1/s$}
Read:
 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 (DE3): Version 1.13 pdf, Due 11/18/2015 Read:
 34

 F
 11/13
 Lecture: General properties of Laplace Transforms: modulation, translation, convolution, periodic functions, etc. png Table of common LT pairs png Read:
 35
 13 47
 M
 11/16
 Lecture: Analytic functions: Euler's vs Riemann's Zeta Function (i.e., poles at the primes); music of primes, Tao 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
Read:
 36

 W
 11/18
 Lecture: Multivalued functions (and their many manyvalued inverses!); branch cuts The extended complex plane (regularization at {$\infty$}) (p. 280) Complex analytic functions of Genus 1 (p. 343) Read:


 Thur
 11/19
 Review session for exam: Bring your questions Room 2013, 12 PM
 

 F
 11/20
 Exam III: (NO CLASS)
 
 47 49
 Sa Su
 
 Thanksgiving Holiday (11/2111/29)

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 Date
 Lecture and Assignment




 Part IV: Vector calculus & Partial Differential Equations (5 Lectures)
 37
 14 49
 M
 11/30
 Lecture: Partial differential equations of Physics; A realworld example where the branchcut 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 (VC1): : 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 Read:
 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 50
 M
 12/7
 Lecture: The lowfrequency quasistatic approximation ({$a < \lambda=c/f$} thus {$f < c/a$}) Brune's Impedance and the quasistatic 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. eigenstates, delay and quasistatics; Lecture Notes pdf
 

 R
 12/10
 Reading Day HW10 (VC1) due
 

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

