## ECE298JA-S23EngMath

Concepts in Engineering Mathematics: ECE Webpage ECE298-JA; (Register);

 L W D Date Lecture and Assignment Part I: Number systems (10 Lectures) 1 3 W 1/18 Introduction & Historical Overview; Lecture 0: pdf; The Pythagorean Theorem & the Three streams:1) Number systems (Integers, rationals)2) Geometry3) $$\infty$$ $$\rightarrow$$ Set theory $$\rightarrow$$ CalculusCommon Math symbolsMatlab tutorial: pdfRead: Chapter 1 (p. 1-17)Homework 1 (NS-1): Basic Matlab commands: pdfs, Due 9/6 (1 week); NS1.m: Matlab script 2 F 1/20 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); HistoryRead: Chpt 2, p. 19-33 3 4 M 1/23 Lecture: The role of physics in Mathematics: Math is a language, designed to do physicsThe 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: Ch 2, p. 33-39 4 W 1/25 Lecture: Two Prime Number Theorems:How to identify Primes (Brute force method: Sieve of Eratosthenes)1) Fundamental Thm of Arith2) Prime Number Theorem: Statement, Prime number SievesWhy are integers important?Public-private key systems (internet security) Elliptic curve RSAPythagoras 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 PriNumThm, And how coding theory works: Coding theory, simplifiedNS-1 Due NS2-solHomework 2 (NS-2): Prime numbers, GCD, CFA; pdf (1 week)Read: p. 39-50 5 F 1/27 Lecture: Euclidean Algorithm for the GCD; CoprimesDefinition of the $$k=\text{gcd}(m,n)$$ with examples; Euclidean algorithmProperties and Derivation of GCD & CoprimesAlgebraic Generalizations of the GCDRead: p. 56-62: Pyth-Triplets; Pell and Fibonacci & their Eigen Matrix 6 5 M 1/30 Lecture: Continued Fraction algorithm (Euclid & Gauss, Stewart 2010, 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 Eigen-analysis of a matrixRead: Class-notes p. 62-68 NS3-solHomework 3 (NS-3): Pythagorean triplets, Pell's equation, Fibonacci sequence; pdf, (Due 1 week) 7 W 2/1 Lecture: Pythagorean triplets $$[a, b, c] \in {\mathbb N}$$ such that $$c^2=a^2+b^2$$Euclid's formula, Properties & Euclid's formula; Rydberg formula uses Euclid's formula; motr examplesRead: Text p 142, 353-356NS-2 Due 8 F 2/3 Lecture: Pell's Equation: Lenstra (2002) pdf; General solution; Brahmagupta's solution by Pell's EqFibonacci SeriesGeometry & irrational numbers $$\sqrt{n}$$; History of $$\mathbb R$$Read: Class-notes 9 6 M 2/6 Lecture: Eigen analysis of Pell and Fibonacci matricesRead: Class-notesNS-3 Due NS3-sol   10 W 2/8 Exam I (In Class): Number Systems
 L W D Date Lecture and Assignment Part II: Algebraic Equations (12 Lectures) 11 F 2/10 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 pointRead: Class-notesHomework 4 (AE-1): Polynomials & Analytic functions and their inverse, Convolution, Newton's method (pdf, 1 week) 12 7 M 2/13 Lecture: Polynomial root classification by convolution; Fundamental Thm of Algebra (pdf) & Summarize Lec 11: Series representations of analytic functions, ROCHistorical 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 moreRead: Class-notes 13 W 2/15 Lecture: Residue expansions of rational functionsImpedance $$Z(s) = \frac{P_m(s)}{P_n(s)}$$ and its utility in Engineering applicationsRead: Class-notes 14 F 2/17 Lecture: Analytic Geometry; Scalar and vector products of two vectorsRead: Class-notesAE-1 dueHomework 5 (AE-2): Linear systems of equations; Gaussian elimination; ABCD method; (pdf Due 1 week) 15 8 M 2/20 Lecture: Gaussian elimination (intersection); Pivot matrices $$(\Pi_n)$$: $$U = \Pi_n^N P_n A$$ gives upper-diagional $$U$$Read: Class-notes 16 W 2/22 Lecture: Transmission matrix method (composition of polynomials)Read: Class-notes 17 F 2/24 Lecture: The Riemann sphere (1851); (the extended plane) pdfMobius Transformation (youtube, HiRes), pdf description Mobius composition transformations, as matricesSoftware: Matlab: ZvizDemo.zip, Matlab Scripts: zviz.zip, python scriptRead: Class-notesHomework 6 (AE-3): Complex algebra; visualizing complex functions; Mobius transformations; (pdf due 1 week) 18 9 M 2/27 Lecture: Visualizing complex valued functions Colorized plots of rational functions Read: Class-notes 19 W 3/1 Lecture: Fourier Transforms (signals) Fourier Transform (wikipedia), Notes on the Fourier series & transform from ECE 310 (tables of transforms & derivations of transform properties)AE-2 Due Read: Class-notes; 20 F 3/3 Lecture: Laplace transforms (systems); The importance of CausalityCauchy 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 10 M 3/6 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 3/8 EXAM II CANCELLED due to student illness
 L W D Date Lecture and Assignment Part III: Scaler Differential Equations (10 Lectures) 23 F 3/10 Lecture: Integration in the complex plane: FTC vs. FTCCAnalytic vs complex analytic functions and Taylor formulaCalculus 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 lectureInversion 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: Sol pdf, Due Oct 30 pdf, Due Oct 30 Spring Break) - 11 Sa [Spring Break (3/11-3/19)] 24 12 M 3/20 Lecture: Cauchy-Riemann (CR) conditionsCauchy-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 3/22 Lecture: Complex analytic functions and Brune impedanceComplex impedance functions $$Z(s)$$, $$\Re Z(\sigma>0) \ge 0$$, Simple poles and zeros & 9 PostulatesTime-domain impedance $$z(t) \leftrightarrow Z(s)$$Read: Class-notes 26 F 3/24 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: SOL pdf, pdf, Due Nov 6 27 13 M 2/27 Lecture: Three complex integration Theorems: Part I1) 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 TheoremDE-1 due 28 W 3/29 Lecture: Three complex integration Theorems: Part II2) 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 3/31 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): SOL pdf, pdf, Due Nov 10Read: Class-notes 30 14 M 4/3 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-notesDE-2 Due 31 W 4/5 Lecture: General properties of Laplace Transforms:Modulation, Translation, Convolution, periodic functions, etc. (png)Table of common LT pairs (png)Sol to DE-3 handoutRead: Class-notes 32 F 4/7 Lecture: Review of Laplace Transforms, Integral theorems, etc Exam III (In Class) DE-3 Due
 L W D Date Lecture and Assignment Part IV: Signal processing (11 Lectures) 33 15 M 4/10 Lecture: Scaler wave equation $$\nabla^2 p = \frac{1}{c^2} \ddot{p}$$ with\ $$c=\sqrt{ \eta P_o/\rho_o }$$d'Alembert solution: $$\psi = F(x-ct) + G(x+ct)$$(:cell:) Short discussion on the utility of the $$Continued fraction Algorithm$$ (CFA)Homework 10 (VC-1): pdf, Due: one week 4Read: Class-notes 34 W 4/12 Lecture: Newton's formula for finding roots of polynomialsRead: Class-notes 35 F 4/14 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 ($$\sigma \le 0$$) More on causal functions;Transfer $$H(s)=V_2/V_1, I_2/I_1 \rightarrow$$ Allpass: $$|e^{-\jmath\phi(\omega)}|=1 \rightarrow$$ poles in LHP, zeros in RHPWiener's factorization theorem: $$H(s) = M(s)A(s)$$ with factors Minimum phase $$M(s)$$ & Allpass $$A(s)$$Read: Class-notes 36 16 M 4/17 Lecture: Nyquist Sampling Theorm: The corner stone of Digital signal processing Read: Class Notes 37 W 4/19 Lecture: The Wedge product of two vectors:Read: Class Notes 38 F 4/21 Lecture: The Wedgie definition and its utility Homework 11 (VC-2): pdf, Due: one weekVC-1 dueRead: Class-notes 39 17 M 4/24 Lecture: Short Time Fourier transform (STFT) link, pdf The corner-stone of digital signal processingRead: Class-notes 40 W 4/26 Lecture: More on the STFT: Examples of how it works Basic definitions: video,Read: Class-notes 41 F 4/28 Lecture: Train-mission problem: solved full analysis Read: Class-notes 42 18 M 5/1 Lecture: Review '' Theorems of Mathematics; Fundamental Thms of Mathematics (Ch. 9)VC-2 due W 5/3 Review/Q&A R 5/4 Reading Day 19 TBD Final Exam Dates & Times