Concepts in Mathematics: ECE Webpage ECE298-JA; ECE-298JA; UIUC Course Explorer: ECE-298-JA;
L | W | D | Date | Lecture and Assignment |
Part I: Number systems (12 Lectures) | ||||
1 | 1 35 | M | 8/24 | ''Introduction & Historical Overview; Lecture 0: pdf;
Three streams: |
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; | |
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 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) | |
- | 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 | |
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 Polynomials (p. 87) and the first "algebra" (al-jabr) Bombelli (1572) first uses complex numbers (p. 277-278) Composition of polynomial equations Read: |
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 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. 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 | |
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 \(x^2-Ny^2=1\) with \(x,y\in \mathbb Z\) |
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 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 [\(z, b, c, d\)] \(\equiv (z-b)(c-d)/(z-d)(c-b)\) 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 42 | 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) 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: 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 44 | 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: |
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 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 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 | |
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 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 (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 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 Derivation of Sterling's formula Inverse Laplace transform of \(\zeta(s) \leftrightarrow \mbox{Zeta}(t)\) |
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) Read: | |
Thur | 11/19 | Review session for exam: Bring your questions Room 2013, 1-2 PM | ||
- | F | 11/20 | Exam III: (NO CLASS) | |
- | 47 49 | Sa Su | - | Thanksgiving Holiday (11/21-11/29) |
L | W | D | Date | Lecture and Assignment |
Part IV: Vector (Partial) Differential Equations (5 Lectures) | ||||
37 | 14 49 | 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 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 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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