Concepts in Engineering Mathematics: ECE Webpage ECE298-JA; (Register);
Additional reading:
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: |
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); History Read: Chpt 2, p. 19-33 | |
3 | 4 | M | 1/23 | 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: 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 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 PriNumThm, And how coding theory works: Coding theory, simplified NS-1 Due NS2-sol Homework 2 (NS-2): Prime numbers, GCD, CFA; pdf (1 week) Read: p. 39-50 | |
5 | F | 1/27 | 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: 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 |
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 examples Read: Text p 142, 353-356 NS-2 Due | |
8 | F | 2/3 | 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 | 6 | M | 2/6 | Lecture: Eigen analysis of Pell and Fibonacci matrices Read: Class-notes NS-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 point Read: Class-notes Homework 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, 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 | 2/15 | 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 | 2/17 | 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 | 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) pdf Mobius Transformation (youtube, HiRes), pdf description Mobius composition transformations, as matrices Software: Matlab: ZvizDemo.zip, Matlab Scripts: zviz.zip, python script Read: Class-notes Homework 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 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 | 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. \(u(t)\) |
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. 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: 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) 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 | 3/22 | 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 | 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 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 | 3/29 | 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 | 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 10 Read: 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-notes |
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 handout Read: 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 4 Read: Class-notes |
34 | W | 4/12 | Lecture: Newton's formula for finding roots of polynomials Read: 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 RHP Wiener'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 week | |
39 | 17 | M | 4/24 | Lecture: Short Time Fourier transform (STFT) link, pdf The corner-stone of digital signal processing Read: 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 |
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