Concepts in Mathematics: ECE Webpage ECE298-JA; ECE-298JA; UIUC Course Explorer: ECE-298-JA; Time: 1:00-1:50 MWF; Location: 4070 ECEB (official); Register
L | W | D | Date | Lecture and Assignment |
Part I: Number systems (10 Lectures) | ||||
1 | 1 35 | M | 8/28 | Introduction & Historical Overview; Lecture 0: pdf;
The Pythagorean Theorem & the Three streams: |
2 | W | 8/30 | 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: Class-notes | |
3 | F | 9/1 | 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: Class-notes | |
- | 2 36 | M | 9/4 | Labor Day Holiday -- No class |
4 | W | 9/6 | 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: Class-notes & A short history of primes, History of PNT NS-1 Due Homework 2 (NS-2): Prime numbers, GCD, CFA; pdf (v. 1.26) (1 week) | |
5 | F | 9/8 | 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: Class-notes | |
6 | 3 37 | M | 9/11 | 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 |
7 | W | 9/13 | Lecture: Pythagorean triplets \([a, b, c] \in {\mathbb N}\) such that \(c^2=a^2+b^2\) Euclid's formula, Properties & examples Read: Class-notes NS-2 Due | |
8 | F | 9/15 | Lecture: Pell's Equation: Lenstra (2002) pdf; General solution; Brahmagupta's solution by Pell's Eq Fibonacci Series Geometry & irrational numbers \(\sqrt{n}\); History of \(\mathbb R\) Read: Class-notes | |
9 | 4 38 | M | 9/18 | Lecture: Eigen analysis of Pell and Fibonacci matrices Read: Class-notes NS-3 Due |
10 | W | 9/20 | Exam I (In Class): Number Systems |
L | W | D | Date | Lecture and Assignment |
Part II: Algebraic Equations (12 Lectures) | ||||
11 | F | 9/22 | 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: Class-notes Homework 4 (AE-1, V. 1.29): Polynomials & Analytic functions and their inverse, Convolution, Newton's method (pdf, 1 week) | |
12 | 5 39 | M | 9/25 | Lecture: Polynomial root classification by convolution; Fundamental Thm of Algebra (pdf) & Summarize Lec 11: Series representations of analytic functions, ROC Historical notes on complex numbers: Solution of the quadratic (Brahmagupta, 628), cubic (c1545), quartic (Tartaglia et al..., 1535), quintic cannot be solved (Abel, 1826) and much much more Read: Class-notes |
13 | W | 9/27 | Lecture: Residue expansions of rational functions Impedance \(Z(s) = \frac{P_m(s)}{P_n(s)}\) and its utility in Engineering applications Read: Class-notes | |
14 | F | 9/29 | Lecture: Analytic Geometry; Scalar and vector products of two vectors Read: Class-notes AE-1 due Homework 5 (AE-2): Linear systems of equations; Gaussian elimination; ABCD method; (pdf Due 1 week) | |
15 | 6 40 | M | 10/2 | Lecture: Gaussian elimination (intersection); Pivot matrices \((\Pi_n)\): \(U = \Pi_n^N P_n A\) gives upper-diagional \(U\) Read: Class-notes |
16 | W | 10/4 | Lecture: Transmission matrix method (composition of polynomials) Read: Class-notes | |
17 | F | 10/6 | Allen out of town Nathan Bleier (TA) lecture Lecture: The Riemann sphere (1851); (the extended plane) pdf Mobius Transformation (youtube, HiRes), pdf description Mobius composition transformations, as matrices Software: Matlab: zviz.zip, python Read: Class-notes AE-3: Complex algebra; visualizing complex functions; Mobius transformations; (pdf due 1 week) | |
18 | 7 41 | M | 10/9 | Lecture: Visualizing complex valued functions Colorized plots of rational functions
Read: Class-notes |
19 | W | 10/11 | Lecture: Fourier Transforms (signals) Fourier Transform (wikipedia), Notes on the Fourier series and transform from ECE 310 (including tables of transforms and derivations of transform properties) AE-2 Due Read: Class-notes; | |
20 | F | 10/13 | Lecture: Laplace transforms (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: Class-notes; Laplace Transform, Types of Fourier transforms | |
21 | 8 42 | M | 10/16 | Lecture: The 10 postulates of Systems (aka, Networks) 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 | 10/18 | Optional Class Review for Exam II (No official class): 7-10 PM; 3015 ECEB |
L | W | D | Date | Lecture and Assignment |
Part III: Scaler Differential Equations (10 Lectures) | ||||
23 | F | 10/20 | Lecture: Integration in the complex plane: FTC vs. FTCC Analytic vs complex analytic functions and Taylor formula Calculus of the complex \(s=\sigma+j\omega\) plane: \(dF(s)/ds\), \(\int F(s) ds\) (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, exp, sin/cos); D'Angelo \(e^z\) & \(\log(z)\) Math 446 lecture 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: Class-notes Homework 7 (DE-1): Series, differentiation, CR conditions, Bi-Harmonic functions: pdf, Due Oct 30 | |
24 | 9 43 | M | 10/23 | 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: \(Z(s)\) results in 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: \(\frac{\partial^2R(\sigma,\omega)}{\partial \sigma^2} + \frac{\partial^2 R(\sigma,\omega)}{\partial \omega^2} =0 \) and \(\nabla^2 X=0\), namely: \(\frac{\partial^2 X(\sigma,\omega)}{\partial \sigma^2} + \frac{\partial^2 X(\sigma,\omega)}{\partial \omega^2} =0\), Biharmonic grid (zviz.m) Discussion: Laplace's equation means conservative vector fields: (1, 2) Read: Class-notes & Boas pages 13-26; Derivatives; Convergence and Power series |
25 | W | 10/25 | 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: Class-notes | |
26 | F | 10/27 | Lecture: Time out: Come with questions: Review session on: multi-valued functions, 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. Impedance. What is the difference between a mass and an inductor? Nonlinear elements; Examples of systems and the 10 postulates of systems. Homework 8 (DE-2): Inverse Laplace Transforms; Residue integration: pdf, Due Nov 6 | |
27 | 10 44 | M | 10/30 | 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: Class-notes & Boas p. 33-43 Complex Integration; Cauchy's Theorem DE-1 due |
28 | W | 11/1 | 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: Class-notes & Boas p. 33-43 Complex Integration; Cauchy's Theorem | |
29 | F | 11/3 | Lecture: The Inverse Laplace Transform (ILT); poles and the Residue expansion: The case for causality \(t<0\) Cauchy's Residue theorem \(\Leftrightarrow\) 2D Green's Thm (in \(\mathbb C\)) Homework 9 (DE-3): pdf, Due Nov 10 Read: Class-notes | |
30 | 11 45 | M | 11/6 | 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: Class-notes |
31 | W | 11/8 | Lecture: General properties of Laplace Transforms: Modulation, Translation, Convolution, periodic functions, etc. (png) Table of common LT pairs (png) Sol to DE-3 handout Read: Class-notes | |
32 | F | 11/10 | Lecture: Review of Laplace Transforms, Integral theorems, etc Exam III 7-10 PM; 2013 ECEB DE-3 Due |
L | W | D | Date | Lecture and Assignment |
Part IV: Vector (Partial) Differential Equations (11 Lectures) | ||||
33 | 12 46 | M | 11/13 | 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 \(\nabla^2 \mathbf{E}\) Homework 10 (VC-1): pdf, Due: Dec 4 Read: Class-notes |
34 | W | 11/15 | 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: \(\psi = F(x-ct) + G(x+ct)\) Read: Class-notes | |
35 | F | 11/17 | No class because of Friday (Nov 10) Exam III: | |
- | 47 | Sa | 11/19 | Thanksgiving Holiday (11/18-11/26) |
36 | 13 48 | M | 11/27 | 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: Class-notes |
37 | W | 11/29 | Lecture: The Webster Horn Equation \( \frac{1}{A(x)}\frac{\partial}{\partial x}A(x)\frac{\partial}{\partial x}{\cal P}(x,\omega) = \frac{s^2}{c^2}{\cal P}(x,\omega) \) Read: Class Notes | |
38 | F | 12/1 | Lecture: Four examples of horns: 1) Uniform, 2) 2D parabolic, 3) 3D conical, 4) Exponential Read: Class Notes | |
39 | 14 49 | M | 12/4 | Lecture: More on the curl and divergence: Stokes' (curl) and Gauss' (divergence) Theorems, Vector Laplacian Dot and cross product of vectors: \( \mathbf{A} \!\cdot\! \mathbf{B}, \mathbf{A} \!\times\! \mathbf{B} \) vs. \( \nabla \phi, \nabla\!\cdot\!\mathbf{B}, \nabla \!\times\! \mathbf{B} \); some Curl examples Homework 11 (VC-2): pdf, Due: Dec 13 VC-1 due Read: Class-notes |
40 | W | 12/6 | 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: Class-notes | |
41 | F | 12/8 | 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 Read: Class-notes | |
42 | 15 50 | M | 12/11 | 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: Class-notes |
43 | W | 12/13 | Lecture: Review The Fundamental Thms of Mathematics & their applications Theorems of Mathematics; Fundamental Thms of Mathematics (Ch. 9) QM: Normal modes vs. eigen-states, delay vs. 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 \(E=h\nu\). VC-2 due | |
- | R | 12/14 | Reading Day | |
- | M | 12/18 | Final Exam Monday Dec 18, 7-10pm 3269 (3d floor tower) Beckman Inst |
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