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ECE298JA-S23EngMath

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

Additional reading:

ECE 298JA Schedule (Spring 2023)

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) Geometry
3) \(\infty\) \(\rightarrow\) Set theory \(\rightarrow\) Calculus
Common Math symbols
Matlab tutorial: pdf
Read: 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); 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 NS1-sol pdf
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
Eigen-analysis of a matrix
Read: Class-notes p. 62-68
Homework 3 (NS-3): Pythagorean triplets, Pell's equation, Fibonacci sequence; pdf, (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 examples
Read: Text p 142, 353-356
NS-2 Due NS2-sol pdf
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 pdf
 
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: 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. \(u(t)\)

AE-3 Due

22 W 3/8 Optional Class Review for Exam II (In class)
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: 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: 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): 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
DE-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 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: Vector (Partial) Differential Equations (11 Lectures)
33 15 M 4/10 Lecture: Gradient, divergence, curl, scalar Laplacian and Vector Laplacian
Gradient \(\nabla p(x,y,z)\), 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 4
Read: Class-notes
34 W 4/12 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 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 (\(\sigma \le 0\))
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: 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
37 W 4/19 Lecture: Four examples of horns: 1) Uniform, 2) 2D parabolic, 3) 3D conical, 4) Exponential
Read: Class Notes
38 F 4/21 Lecture: More on the curl and divergence: Stokes' (curl) and Gauss' (divergence) Theorems, Vector Laplacian
Dot 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 examples
Homework 11 (VC-2): pdf, Due: Dec 13
VC-1 due
Read: Class-notes
39 17 M 4/24 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
40 W 4/26 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, citation
Read: Class-notes
41 F 4/28 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

42 18 M 5/1 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 due

W 5/3 Review/Q&A
R 5/4 Reading Day

19 TBD Final Exam Dates & Times

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