 ### ECE 493/MATH-487 {$\sqrt{Daily Schedule}$} (Spring 2009)

L/WDDateIntegrated Lectures on Mathematical Physics
0/4M1/19 MLK Day; no class
Part I: Linear (Matrix) Algebra (5 lectures)
1/4T1/20Lecture 1: Topic 1. Basic definitions, 2. Elementary operations;
Assignment: Flag terminology you don't understand in Class;
2R1/22Lecture 2: 3. Solutions to {$Ax=b$}, 4. Matrix inverse {$x=A^{-1}b$};
3/5T1/27Lecture 3: 5. Matrix Algebra; 6. Transformations;
HW1 2. Cramer's Rule, Determininants, Inverse Matrix, Aug Matrix and Gauss Elimination; Vandermonde (Solution) Scores: 5x100 6x90 2x85 2x65
4R1/29Lecture 4: 7. Vector spaces in {$\mathbb{R}^n$};
5/6T2/3Lecture 5: 5. Eigenvalues & vectors; 8. Optimal approximation and least squares;
HW2: Vector space; Schwartz and Triangular inequalities, eigenspaces (Solution)
Part II: Vector Calculus (5 lectures)
6R2/5Lecture 6: 9. Partial differentiation [Review: 13.1-13.5;]; 10. Vector fields, Path, volume and surface integrals
7/7T2/10Lecture 7: Vector fields: {${\bf R}(x,y,z)$}, vector dot-product {$A \cdot B$}, cross-product {$A \times B$}, triple-products {$A \cdot A \times B$}, {$A \times (B \times C)$}; Change of variables under integration: Jacobians
HW3: Rank-n-Span; Taylor series; Vector products and fields (Solution)
8R2/12Lecture 8:Gradient {$\nabla$}, Divergence {$\nabla \cdot$}, Curl {$\nabla \times$}, Laplacian {$\nabla^2$};
9/8T2/17Lecture 9: Integral and conservation laws: Gauss, Green, Stokes, Divergence
Lecture 9 Notes
HW4: Key vector calculus topics (Solution)
10R2/19Lecture 10: Potentials and Conservative fields;
Review: all 16 (not 16.7)
T24 ExamI Feb 24 Tues @ 7-9 PM
Exam I grade distribution: [81,93 95, 102 109 103 105, 110 110 115 118, 122, 131, 143]
-/9T2/24NO Lecture due to Exam I; Class time will be converted to optional Office hours, to review home work solutions and discuss exam
HW5: Not assigned
Part III: Boundary value problems (6 lectures)
11R2/26Lecture 11:15. PDE: parabolic, hyperbolic, elliptical, discriminant
Read: 18-19 Emmy Noether, Noether's Thm. I; Examples of Symmetry in physics
12/10T3/3Lecture 12:21. Special Equations of Physics: Wave, Laplace, Diffusion;
18.~Separation of variables; integration by parts
HW6: Separation of variables, BV problems, symmetry (Solution Ver-1.4)
13R3/5Guest Lecture Prof. Levinson Lecture 13:20.Sturm-Liouville BV Theory;
23. Special functions by Power Series: Bessel, Legendre Polynomials, Riemann Zeta
14/11T3/10Lecture 14: 16. Transmission line theory: Lumped parameter approximations,
17. {$2^{nd}$} order PDE from a pair of first order ODEs as unit-cell -> 0
Read:[17.7, pp.~ 887, 965, 1029, 1070, 1080]
HW7-v1.2: Sturm-Liouville, Boundary Value problems, Fourier and Laplace Transforms; (Solution-3/18/09); Hints for problems 3+5 and 4.
15R3/12Lecture 15:24.Fourier: Integrals, Transforms, Series, DFT
16/12T3/17Lecture 16:25.Laplace and z Transforms
19.The vector space {$\mathbb{C}$}
Read: 5.1-1.3+Review p.290-1; Study: the solution to HW7
R3/19No Lecture: due to Exam II
A3/21 Spring Break Starts
M3/30 Instruction Resumes
Part IV: Complex Variables (10 lectures)
17/14T3/31Lecture 17:The frequency domain: Complex {$Z(s) = R(s)+iX(s)$} as a function of complex frequency {$s=\sigma+i\omega$}; e.g., {$Z,s \in \mathbb{C}$}), phasors, Phasers and delay {$e^{-i\omega T}$}, {$\log(z)$}, {$\sum z^n$}
27.Singularities (i.e., poles, branch cuts and transformations);Mobius Transformation (youtube, HiRes), complex color-coding
HW8-v1.01 Complex Functions and Laplace transforms (Solution)
T3/31 Exam II March 31 Tues @ 7-9 PM 441 Altgeld Hall
A: 114-98; A-: 88 89 86; B+ 77 ;C+ 57
[114 111 109 108 105 103 99 98; 88 89 86; 77; 57] out of a maximum of 137 points
18R4/2Lecture 18:28.Differential calculus on {$\mathbb{C}$}
29. Cauchy-Riemann Eqs., Analytic functions, Harmonic functions
19/15T4/7Lecture 19: Inverses of Analytic functions;
30. Irrotational fields (e.g., velocity potential {$\mathbf{u} = \nabla \phi(x,y,z)$}) [p.~829];
Discussion on CR conditions: Computing {$\Re f(z)$} and {$\Im f(z)$};
Analytic coloring; Read: [16.10] pp. 826-838;
HW9 Analytic functions; Integration of analytic functions; Cauchy integral formula; Riemann Sheets and Branch cuts; Region of Convergence; inverse Laplace transforms; (Solution)
20R4/9Lecture 20:31.Integral calculus on {$\mathbb{C}$}
32. {$\int z^{n-1} dz$} on the unit circle (Hardy spaces {${\cal H}^2$})
Continue discussion of examples of analytic functions,, Cauchy integral formula, inverse Laplace transforms, Rational fraction expansions, conservative fields;
21/16T4/14Lecture 21: Special functions and Pole-zero locations (stable/causal, allpass, minimum phase, positive real); 33.Cauchy's theorem
34.Cauchy's integral formula [23.5]
36.Cauchy's Residue Theorem [24.5]
HW10 (ver 1.2) (Solution)
22R4/16Lecture 22: Hilbert Transforms and the Cauchy Integral formula; Review of Residues (Examples) and their use in finding solutions to integrals; Topic 35.Series: Maclaurin, Taylor, Laurent [24.3]
Topic 37.Jordan's Lemma
23/17T4/21Lecture 23: Topic 38. More on Inverse Transforms: Laplace {${\cal L}^{-1}$} and Fourier {${\cal F}^{-1}$};
The multi-valued {$i^s$} and {$\tanh^{-1}(log(s))$};
24R4/23Lecture 24: 39. Rational Impedance (Pade) approximations: {$Z(s)={a+bs+cs^2}/({A+Bs})$}
Partial fraction: {$Z(s) = \sum a_i/(s-s_i)$} and
Continued fractions: {$Z(s)=s+a/(s+b/(s + c/(s+\cdots)))$} expansions
25T4/28Lecture 25: Guest Lecture: Prof. J. d'Angelo; Contour integration and Fourier Transforms
HW11; (Solution)
26R4/30Lecture 26: Inverse problems;
Cauer synthesis, Bode plots, Network theory (Brune Positive-real (PR) impedance functions) Schelkunoff Impedance (1940?) Δ;
40.~ODE's with initial condition (vs. Boundary value problems) Di and Gilbert (1993) Δ
27/19T5/5Last class; Review for Final
Redo HW0:
-W5/6 Instruction Ends
-F5/8 Exam III 7:00-10:00+ PM on HW6-HW11:
The Exam III score Distribution is trimodal: [74 76 76 77; 92 92 92 92 92; 104 105 106] out of 130.
The top three top scores are JB, LM, VC. Nice job! Final scores to follow.
-/20F5/15 Finals End

L= Lecture #
W=week of the year, starting from Jan 1
D=day: T is Tue, W Wed, R Thur, S Sat, etc.
The somewhat random-ordered numbers in front of many (not all) topics, are the topic numbers defined in the 2009 Syllabus Δ:
ECE-493 is divided into 4 basic sections (I-IV), divided into 40 topics, delivered as 24=4*6 lectures. There are two mid-term exams and one final. There are 12 homework assignments, with a HW0 that does not count toward your final grade. Each exam (I, II and Final) will count as 30% of your final grade, while the Assignments (HW1-12) plus class participation, count for 10%.

#### Assignments:

All homework assigned on Tuesday will be due in class the following Tuesday. In 2009 there was no HW5 due to resource-constraints.

#### Exams

There are 3 exams total:
Exam I following Sections I, II, Exam II following Section III, Exam III following Section IV
The final is similar to the two midterms, and is only on the final 10 lectures on complex variables.

 Not proofed beyond here