 L W D Date Lecture and Assignment Part I: Number systems (10 Lectures) 1 135 M 8/28 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: Class-notesHomework 1 (NS-1): Basic Matlab commands: pdf (v. 1.06), Due 9/6 (1 week); help 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); HistoryRead: Class-notes 3 F 9/1 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: Class-notes - 236 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 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 PNT NS-1 DueHomework 2 (NS-2): Prime numbers, GCD, CFA; pdf (v. 1.26) (1 week) 5 F 9/8 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: Class-notes 6 337 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 $$rat()$$ functionRead: Class-notesHomework 3 (NS-3): Pythagorean triplets, Pell's equation, Fibonacci sequence; pdf (v.1.24), (1 week) 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 & examplesRead: Class-notesNS-2 Due 8 F 9/15 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 9 438 M 9/18 Lecture: Eigen analysis of Pell and Fibonacci matricesRead: Class-notesNS-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 pointRead: Class-notesHomework 4 (AE-1, V. 1.29): Polynomials & Analytic functions and their inverse, Convolution, Newton's method (pdf, 1 week) 12 539 M 9/25 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 9/27 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 9/29 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) 15 640 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) pdfMobius Transformation (youtube, HiRes), pdf description Mobius composition transformations, as matricesSoftware: Matlab: zviz.zip, pythonRead: Class-notesAE-3: Complex algebra; visualizing complex functions; Mobius transformations; (pdf due 1 week) 18 741 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 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 21 842 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. 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: pdf, Due Oct 30 24 943 M 10/23 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 10/25 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 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 1044 M 10/30 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 11/1 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 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 10Read: Class-notes 30 1145 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-notesDE-2 Due 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 handoutRead: Class-notes 32 F 11/10 Lecture: Review of Laplace Transforms, Integral theorems, etc Exam III 7-10 PM; 2013 ECEBDE-3 Due
 L W D Date Lecture and Assignment Part IV: Vector (Partial) Differential Equations (11 Lectures) 33 1246 M 11/13 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 $$\nabla^2 \mathbf{E}$$Homework 10 (VC-1): pdf, Due: Dec 4Read: 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 1348 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 RHPWiener'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) ExponentialRead: Class Notes 39 1449 M 12/4 Lecture: More on the curl and divergence: Stokes' (curl) and Gauss' (divergence) Theorems, Vector LaplacianDot 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 examplesHomework 11 (VC-2): pdf, Due: Dec 13VC-1 dueRead: 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, citationRead: Class-notes 42 1550 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 dueRead: Class-notes - R 12/14 Reading Day - M 12/18 Final Exam Monday Dec 18, 7-10pm 3269 (3d floor tower) Beckman Inst