 L W D Date Lecture and Assignment Part I: Number systems (10 Lectures) 1 134 M 8/22 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: Lec. 1 (pp. 15-24)Homework 1 (NS-1): 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); HistoryRead: Lec. 2 (pp. 24-29) 3 F 8/26 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: Lec. 3 (pp. 29-32) 4 235 M 8/29 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: Lec 4 (pp.32-33, 70-75); A short history of primes, History of PNT NS-1 DueHomework 2 (NS-2): Prime numbers, GCD, CFA; pdf, Due 9/7 5 W 8/31 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: Lec. 5 (pp. 33, 73-75) 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 {$rat()$} functionRead: Lec 6 (pp. 34-35)Homework 3 (NS-3): 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$}Examples of PTs & Euclid's formula Properties, examples, History NS-2 DueRead: Lec. 7 (p. 36, 77-79) 8 F 9/9 Lecture: Pell's Equation: General solution; Brahmagupta's solution by compositionChord and tangent solution (Diophantus {$\approx$}250CE) methodsRead: Lec. 8 (pp. 36-37, 79-81) History of {$\mathbb R$}Optional: GCD Algorithm - Stillwell sections 3.3 & 5.3 9 437 M 9/12 Lecture: Fibbonacci SeriesGeometry & irrational numbers {$\sqrt{n}$}NS-3 DueRead: Lec 9 (pp. 37-38, 81-82) 10 W 9/14 Exam I (In Class): Number Systems  L W D Date Lecture and Assignment Part II: Algebraic Equations (12 Lectures) 11 F 9/16 Lecture: Analytic geometry as physics (Stream 2)The first "algebra" al-Khwarizmi (830CE)Polynomials, Analytic functions, {$\infty$} Series: Geometric {$\frac{1}{1-z}=\sum_{0}^\infty z^n$}, {$e^z=\sum_{0}^\infty \frac{z^n}{n!}$}; Taylor series; ROC; expansion pointRead: Lec 11 12 538 M 9/19 Lecture: Complex analytic functions; Physical equations in several variablesSummarize Lec 11:Detailed review of series representations of analytic functions: Poles, residues, ROC, etc.Geometry + Algebra {$\Rightarrow$} Analytic Geometry: From Euclid to Descartes+NewtonNewton (1667) labels complex cubic roots as "impossible." Bombelli (1572) first uses complex numbers (JS10: p. 277-278)Read: Lec 12Homework 4 (AE-1): Polynomials & Analytic functions and their inverse, Convolution, Newton's method (pdf, Due 9/28) 13 W 9/21 Lecture: Root classification for polynomials by convolution; Residue expansion Chinese discover Gaussian elimination (Jiuzhang suanshu) (JS10: p. 89) Gaussian elimination in one & two variablesSolution of the quadratic (Brahmagupta, 628), cubic (c1545), quartic (Tartaglia et al..., 1535), quintic cannot be solved (Abel, 1826)Composition of polynomial equations (Bezout's Thm)Read: Lec 13 14 F 9/23 Lecture: Analytic Geometry (Fermat 1629; Descartes 1637)Descartes' insight: Composition of two polynomials of degrees: ({$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$} dimensional space; 2 planes compose a line, 3 planes compose to a pointVectors, Complex planes & lines, Dot and cross products of vectorsRead: Lec 14 15 639 M 9/26 Lecture: Gaussian elimination (intersection); Pivot matrices {$(\Pi_n)$}: {$U = \Pi_n^N P_n A$} gives upper-diagional {$U$}Read: Lec 15Homework 5 (AE-2): Non-linear and linear systems of equations; Gaussian elimination; pdf Due 10/5 16 W 9/28 Lecture: Composition of polynomials, ABCD matrix methodABCD Composition relations of transmission linesRead Lec 16AE-1 Due 17 F 9/30 Lecture: Introduction to the Riemann sphere (1851); (the extended plane) (JS10, p. 298-312)Mobius Transformation (youtube, HiRes), pdf description Understanding {$\infty$} by closing the complex plane;Chords on the sphere pdfMobius transformations in matrix formatRead: Lec. 17 18 740 M 10/3 Lecture: Fundamental Thm of Algebra (pdf) & Colorized plots Software: Matlab: zviz.zip, pythonBezout's Thm: Mathpages, Wikipediaby ExampleExponential {$e^z$} D'Angelo lecture 3D representations of 2D systems; Perspective (3D) drawing.Read: Lec 18; AE-3: pdf Due Oct 10: ABCD method; Colorized mappings; Mobius transformations 19 W 10/5 Lecture: Fourier Transforms for signalsAE-2 Due Read: Lec 19; Fourier Transform (wikipedia), Notes on the Fourier series and transform from ECE 310(including tables of transforms and derivations of transform properties) 20 F 10/7 Lecture: Laplace transforms for systemsThe importance of CausalityCauchy Riemann role in the acceptance of complex functions:Convolution of the step function: {$u(t) \leftrightarrow 1/s$} vs. {$2\tilde{u}(t) \equiv 1+ \mbox{sgn}(t) \leftrightarrow 2\pi \delta(\omega) + 2/j\omega$} Read: Lec 20; Laplace Transform,Table of transforms 21 841 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. {$u(t)$} AE-3 Due 22 W 10/12 No class due to Exam II: 7-10 PM; 2013 ECEB  L W D Date Lecture and Assignment Part III: Scaler Differential Equations (12 Lectures) 23 F 10/14 Lecture: Integration in the complex plane: FTC vs. FTCCAnalytic vs complex analytic functions and Taylor formulaCalculus 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)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, exponentials, sin/cos)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: Lec 23 Homework 7 (DE-1): Series, differentiation, CR conditions, Bi-Harmonic functions: pdf, Due 10/24/2016 24 942 M 10/17 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: {$Z(s)$} obeys CR conditions: {$\frac{\partial R(\sigma,\omega)}{\partial\sigma} = \frac{\partial X(\sigma,\omega)}{\partial\omega}$} and {$\frac{\partial R(\sigma,\omega)}{\partial\omega} = -\frac{\partial X(\sigma,\omega)}{\partial\sigma}$}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$},Biharmonic grid (zviz.m)Discussion of the solution of Laplace's equation given boundary conditions (conservative vector fields)Read: Lec 24 & Boas pages 13-26; Derivatives; Convergence and Power series 25 W 10/19 Lecture: Complex analytic functions and Brune impedanceComplex impedance functions {$Z(s)$}, {$\Re Z(\sigma>0) \ge 0$}, Simple poles and zeros & 9 PostulatesTime-domain impedance {$z(t) \leftrightarrow Z(s)$}Read: Lec 25 26 F 10/21 Lecture: Review session on multi-valued functions and complex integrationRiemann sheets, colorized plots, branch cuts, Review of Fundamental Theorems of complex analytic functions.Laplace's equation and its role in Engineering Physics. What is the difference between a mass and an inductor?nonlinear elements; Examples of systems and the Nine postulates of systems. Homework 8 (DE-2): Inverse Laplace Transforms; Residue integration: pdf, Due 10/31/2016 27 M 10/24 Lecture: Three complex integration Theorems: Part I1) Cauchy's Integral Theorem: {$\oint f(z) dz =0$} (Boas p. 45) vs. 2D Green's Thm (p. 49); Stokes (Thm, Bio)Read: Lec 27 & Boas p. 33-43 Complex Integration; Cauchy's TheoremDE-1 due 28 W 10/26 Lecture: Three complex integration Theorems: Part II2) Cauchy's Integral Formula: {$\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$}3) Cauchy's Residue TheoremExample by brute force integration: {$\oint_{|s|=1} \frac{ds}{s}= 2\pi j$} Read: Lec 28 & Boas p. 33-43 Complex Integration; Cauchy's Theorem 29 F 10/28 Lecture: The Inverse Laplace Transform (ILT); poles and the Residue expansion: The case for causality {$t\le0$}Cauchy's Residue theorem {$\Leftrightarrow$} 2D Green's Thm (in {$\mathbb C$})Homework 9 (DE-3): pdf, Due 11/7/2016Read: Lec 29 30 1144 M 10/31 Lecture: Inverse Laplace Transform: Use of the Residue theorem {$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) $}Semi-capacitor: {$ u(t)/\sqrt{t} \leftrightarrow \sqrt{\pi/s} $} Read: Lec 30DE-2 Due 31 W 11/2 Lecture: General properties of Laplace Transforms:Modulation, Translation, Convolution, periodic functions, etc. (png)Table of common LT pairs (png)Read: Lec 31 32 F 11/4 Lecture: General properties of Impedance (Z) and Transmission (ABCD) functions:Impedance {$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$} poles in LHP, zeros in RHPWiener's factorization theorem: {$H(s) = M(s)A(s)$} with factors Minimum phase {$M(s)$} & Allpass {$A(s)$}Read: Lec 32 33 1245 M 11/7 Lecture: Riemann Zeta function {$\zeta(s)=\sum \frac{1}{n^s}$}Euler's vs. Riemann's Zeta Function (i.e., poles at the primes), music of primes, Analytic continuation, TaoIntroduction to the Riemann zeta function (Stillwell p. 184) Euler's product formula; plot of Riemann-Zeta function showing magnitude and phase separatelyInverse Laplace transform of {$\zeta(s) \leftrightarrow \mbox{Zeta}(t)$}DE-3 Due 34 W 11/9 No class due to Exam III: Thursday 34 R 11/10 Exam III 7-10 PM; NOTE ROOM CHANGE: 2015ECEB  L W D Date Lecture and Assignment Part IV: Vector (Partial) Differential Equations (9 Lectures) 35 F 11/11 Lecture: Scaler wave equation {$\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(x-ct) + G(x+ct)$}Homework 10 (VC-1): pdf, Due: Nov 28 Mon (Alt 30 Wed) Read: Class Notes p. 1-2 36 1346 M 11/14 Lecture: The Webster Horn Equation {$ \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} $} Curl examplesRead: Class Notes p.3-10? 37 W 11/16 Lecture: Gradient, divergence, curl, scalar Laplacian and Vector LaplacianGradient {$\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}$}Read: Lec 37 38 F 11/18 Lecture: More on the curl and divergence: Stokes' (curl) and Gauss' (divergence) Theorems, Vector LaplacianHomework 11 (VC-2): pdf, Due: Dec 7 WedRead: Lec 38 - 47 Sa 11/19 Thanksgiving Holiday (11/19-11/27) 39 1448 M 11/28 Lecture: J.C. Maxwell unifies Electricity and Magnetism (1861); Basic definitions: {$ \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} }$}Differential and integral forms of Maxwell's Eqs.How a loudspeaker works: {$ \mathbf{F} = \mathbf{J} \times \mathbf{B} $} and EM Reciprocity; Magnetic loop video, citationVC-1 dueRead: Lec 39 40 W 11/30 Lecture: The Fundamental theorem of vector calculus: {$\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$}. Definition of the Conservative vector fields.Read: Lec 40 41 F 12/2 Lecture: The low-frequency quasi-static approximation: i.e., {$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}_-$}. Read: Lec 41 42 1549 M 12/5 Lecture: Review The Fundamental Thms of Mathematics & their applications Theorems of Mathematics; Fundamental Thms of Mathematics (Ch. 9)Normal modes vs. eigen-states, delay and quasi-statics;The Hydrogen atom is an exponential horn: it is a waveguide with radial normal modes (eigen-states),occupied with electrons (EM energy), which escapes (i.e., radiates) as photons (free particles). This explains {$E=h\nu$}. Read: Lec 42 43 W 12/7 Last day of Class: The quantum'' in QM refers to normal modes.Quantum mechanics is quasi-static, which assumes no delay. {$\Rightarrow$} Let {$c$}=speed of light; {$v$}=frequency; {$V$}=group-velocity, then {$E=h \nu$}, {$p=h/\lambda$} {$\rightarrow$} {$\nu = E/h, \lambda=h/p \rightarrow c = \lambda \nu = E/p$} (pdf)Electro-dynamically vs. classically: {$ c = E/p \gg mV^2/mV = V \$}, thus QSs applies to QM VC-2 due - R 12/8 Reading Day - M 12/12 Final Exam Monday Dec 12, 7-10pm ECEB 2013