 L W D Date Lecture and Assignment Part I: Number systems (12 Lectures) 1 135 M 8/24 ''Introduction & Historical Overview; Lecture 0: pdf; Three streams:1) Number systems (Integers, rationals); 2) Geometry; 3) $$\infty$$ $$\rightarrow$$ Set theoryCommon Math symbolsMatlab tutorial: pdfRead: Stillwell Ch. 1, 1.7 2 W 8/26 Lecture: Number SystemsFirst 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, CalculusRead: 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;17C: Mersenne, Marin; Galilei, Galileo; Hooke, Robert; Boyle, Robert; Newton, Sir Issac;18C: Bernoulli, Daniel; Euler; Lagrange; d'Alembert;19C: Gauss; Laplace; Fourier; Helmholtz; Heaviside; Strutt, William; Rayleigh, Lord; Bell, AGRead: Sections 1.1, 1.2, 4.5, 5.7 4 236 M 8/31 Lecture: Brief tutorial on Prime Numbers $$\pi_k$$; Fundamental Thm of ArithDefinition 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$$(m,n)$$=1Prime number Theorem Statement, History, Bertrand's conjecture, Prime number SievesDefinition of Pell's Equation: $$m^2 - N n^2 = 1$$Read: 1.2, 1.3HW1 DueHomework 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 $$\Leftrightarrow$$ PhysicsLecture: Euclid's Algorithm: The GCD (p. 41, 66)Properties and Derivation of GCDWhy integers are important for internet security? Elliptic curve DSA Introduction to the Riemann zeta function (p. 184) $$\zeta(s)$$; Relation to (primes & co-primes)plot of Riemann-Zeta function showing magnitude and phase separatelyRead: 3.3, 3.4 - 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 $$(\pi \approx 22/7)$$ numbersAlgebraic Generalizations of GCDreal $$\mathbb{R}$$ vs. complex $$\mathbb{C}$$ numbers, vectors, matricesRead: 3.6, 5.3, History of $$\mathbb R$$HW2 Due Homework 3 (NS-3): Continued fractions [rats()]; Pell's Eq.; Solutions {m,n,1} of am+bn=1 via GCD(a,b), pdf Due 9/18GCD Algorithm - Stillwell sections 3.3 & 5.3 8 F 9/11 Lecture: Euclid: Ruler and Compass constructions: Conic sectionsComplex numbers (Bombelli, 1575, p. 259) and the Radius of convergence (ROC)Read: 2.3, 2.4; 4.2, 4.3 9 438 M 9/14 Lecture: Pell's Equation: General solution (p. 72); Brahmagupta's solution by compositionPell equation solver,historyPart 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 IHW3 Due 12 539 M 9/21 Lecture: Introduction to analytic geometryPolynomials (p. 87) and the first "algebra" (al-jabr)Bombelli (1572) first uses complex numbers (p. 277-278)Composition of polynomial equationsRead: 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+NewtonNewton (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) 15 640 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 10/5Read: 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 pointComplex planes & linesRead: 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 Read: Ch. 7, p. 104-119 (2nd Ed.), p. 109-125 (3rd Ed.) Analytic Geometry 18 741 M 10/5 Lecture: Review of Composition of polynomials, ABCD matrix method, convolution of sequencesGaussian elimination; Permutation and Pivot matrices; Formula for Pell Triplets (solutions to $$x^2-Ny^2=1$$ with $$x,y\in \mathbb Z$$HW4 Due Homework 5 (AE-2): Linear and nonlinear systems of equations; Gaussian elimination; Matrix permutations; Convolution pdf Due 10/12 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 pdfRead: 20 F 10/9 Lecture: Fundamental Thm of Algebra Colorized plots, (pdf & Matlab code zviz.m, OLD), pythonBezout's Thm: Mathpages, Wikipedia by ExampleMore on the ABCD relations for transmission linesMobius transformations in matrix formatInvariance of the cross ratio [$$z, b, c, d$$] $$\equiv (z-b)(c-d)/(z-d)(c-b)$$ to a Mobius transformation3D representations of 2D systems; Perspective (3D) drawing.Read: p. 111-119 (2nd Ed.), p. 118-125 (3rd Ed.)Bezout's Thm 20' 842 M 10/12 Lecture: Fourier Transforms for signals vs. Laplace transforms for systems;HW5 Due Homework 6 (AE-3): ABCD method; Colorized mappings; Mobius transformationsFund. Thm Alg vs Bezout's Thm; Fourier vs Laplace Transforms pdf Due 10/19Read: 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 signalsCauchy 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 causalConvolution of the step functionMobius'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: 23 943 M 10/19 Lecture: General discussion and review of Exam IIRead: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 1044 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: $$\frac{\partial R(\sigma,\omega)}{\partial\sigma} = \frac{\partial X(\sigma,\omega)}{\partial\omega}$$ and $$\frac{\partial R(\sigma,\omega)}{\partial\omega} = -\frac{\partial R(\sigma,\omega)}{\partial\sigma}$$Read: Chap 1 Boas (Handout) 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 conditionsBasic equations of mathematical Physics: Wave equation, diffusion equation, Laplace's EquationCauchy-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 1145 M 11/2 Lecture: Integration in the complex plane: Basic definitions1) 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 poleRead: 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 TheoremExamples: 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 11/11/2015Read: Boas p. 33-43 Complex Integration; Cauchy's Theorm 31 F 11/6 Lecture: The Inverse Laplace Transform (ILT); poles and the Residue expansion: The case for causalityFundamental Thm of Complex Calculus: $$F(s) = F(a) + \int_a^s f(\zeta) d\zeta \Rightarrow f(s) = dF/ds$$ is independent of the pathCauchy'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 1246 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 11/18/2015Read: 34 F 11/13 Lecture: General properties of Laplace Transforms: modulation, translation, convolution, periodic functions, etc. pngTable of common LT pairs pngRead: 35 1347 M 11/16 Lecture: Analytic functions: Euler's vs Riemann's Zeta Function (i.e., poles at the primes); music of primes, TaoDerivation of Sterling's formula Inverse Laplace transform of $$\zeta(s) \leftrightarrow \mbox{Zeta}(t)$$Analytic continuation (continued)Why is the convergence of a series/integral important?The role of Sets; Why closing a set important (the fear of $$\infty$$)? p. 56 Read: 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 questionsRoom 2013, 1-2 PM - F 11/20 Exam III: (NO CLASS) - 4749 SaSu - Thanksgiving Holiday (11/21-11/29)  L W D Date Lecture and Assignment Part IV: Vector (Partial) Differential Equations (5 Lectures) 37 1449 M 11/30 Lecture: Partial differential equations of Physics; A real-world example where the branch-cut placement is criticalThe 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 formsHomework 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, citationRead: 39 F 12/4 Lecture: More on curl and divergence; Stokes' (curl) and Gauss' (divergence) TheoremsThe telegraph wave equation starting from Maxwell's equationsJ.C. Maxwell unifies Electricity and Magnetism (1861); O. Heaviside's vector form of MEs (1884)Lecture 39 Notes pdf 40 1550 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 DayHW10 (VC-1) due - Tr 12/17 Final Exam 1:30 PM -- 4:30 PM ECEB 2013