 ### ECE 493/MATH-487 Daily Schedule Spring 2011

L/WDDateIntegrated Lectures on Mathematical Physics
0/4M1/17 MLK Day; no class
Part I: Complex Variables (10 lectures)
1/4T1/18L1: T25. 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 {$e^{-i\omega T}$}, {$\log(z)$}, {$\sum z^n$}
T26. Singularities (i.e., poles, branch cuts and transformations);Mobius Transformation (youtube, HiRes), pdf description
HW8 Complex Functions and Laplace transforms (Solution)
2R1/20L2: T 27. Differential calculus on {$\mathbb{C}$}
T 28. Cauchy-Riemann Eqs., Analytic functions, Harmonic functions
Read: [21.5] and verify that you can do all the exercises on page 1113.
3T1/25L3: Inverses of Analytic functions;
T 29. Irrotational fields (e.g., velocity potential {$\mathbf{u} = \nabla \phi(x,y,z)$}) [p.~829];
T 28. Discussion on CR conditions: Computing {$\Re f(z)$} and {$\Im f(z)$};
Analytic coloring, via matlab, using zviz.m; Read: [16.10] pp. 826-838;
HW9 Analytic functions; 30. Integration of analytic functions; 33. Cauchy integral formula; Riemann Sheets and Branch cuts; Region of Convergence; inverse Laplace transforms; (Solution)
4R1/28L4: T 30.Integral calculus on {$\mathbb{C}$}
T 31. {$\int z^{n-1} dz$} on the unit circle
Continue discussion of examples of analytic functions, 33. Cauchy integral formula, 37. inverse Laplace transforms, 38. Rational fraction expansions, conservative fields;
Boas' method for computing {$Z(s)$} given {$R(s) \equiv \Re Z(s)$} pdf Δ from R.P. Boas, Invitation to Complex Analysis Random House 1987
5T2/1L5:T 32.Cauchy's theorem; T 33.Cauchy's integral formula [23.5]; Read: [23.3, 23.5];
HW10 (Solution)
6R2/3L6a: Contour integration and Inverse Laplace Transforms
Examples of forward {$\cal L$} and inverse {${\cal L}^{-1}$} Laplace Transform pairs [e.g., {$f(t) \leftrightarrow F(s)$}]
L6b: Special functions and Pole-zero locations (stable/causal, allpass, minimum phase, positive real);
HW11; (Solution)
7T2/8L7: Hilbert Transforms and the Cauchy Integral formula: The difference between the Fourier transform {$2{\tilde u}(t) \leftrightarrow 1+1/j\omega$} and the Laplace {$2u(t) \leftrightarrow 1/s$}
Review of Residues (Examples) and their use in finding solutions to integrals;
T 34. Series: Maclaurin, Taylor, Laurent [24.3]
T 36.Jordan's Lemma
8R2/10L8: Cauer synthesis, Bode plots, Network theory (Brune Positive-real (PR) impedance functions)
Schelkunoff on Impedance (BSTJ, 1938) (djvu(0.6M) Δ, pdf(17M) Δ)
Inverse problems: Tube Area {$A(x)$} given impedance {$Z(s,x=0)$}
9/7T2/15L9: T 37. More on Inverse Transforms: Laplace {${\cal L}^{-1}$} and Fourier {${\cal F}^{-1}$};
The multi-valued {$i^s$}, {$\tanh^{-1}(s) = \frac{1}{2}\ln \left( \frac{1+s}{1-s} \right)$} and: {$\cosh^{-1}(s) = \ln(s \pm \sqrt{s^2 -1} )$}
Analytic continuation
T 35. Cauchy's Residue Theorem [24.5]
Read: [24.2, 24.2] (power series and the ROC);
HW12; (Solution)
10R2/17L10: T 38. 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
0/8T2/22 NO CLASS Optional office hours for review, during class time
0/8T2/22 Exam I Feb 22 Tuesday @ 7-9 PM; Place: 1MEB 135
Part II: Linear (Matrix) Algebra (6 lectures)
1R2/24L1: T 1. Basic definitions, Elementary operations;
T 2. Cramer's Rule, Determininants, Inverse Matrix, Aug Matrix and Gauss Elimination; Vandermonde
Review Exam I;
8.1-2, 10.2;
HW1, (Solution)
2/9T3/1L2: T 3. Solutions to {$Ax=b$} by Gaussian elimination, T 4. Matrix inverse {$x=A^{-1}b$}; Cramer's Rule
3R3/3L3: T 5. The symmetric matrix: Eigenvectors; T 6. Transformations (change of basis);
HW2: Vector space; Schwartz and Triangular inequalities, eigenspaces (Solution)
4/10T3/8L4: T 7. Vector spaces in {$\mathbb{R}^n$}; Innerproduct+Norms; Ortho-normal; Span and Perp ({$\perp$}); Schwartz and Triangular inequalities
5R3/10L5: Gram-Schmidt proceedure; Vector dot-product {$A \cdot B$}, cross-product {$A \times B$}, triple-products {$A \cdot A \times B$}, {$A \times (B \times C)$};
Read: 9.10, 11.4 ; Leykekhman Lecture 9
HW3 Rank-n-Span; Taylor series; Vector products and fields (Solution)
0R3/11-3/12Engineering Open House
6/11T3/15L6: T 5. Asymmetric matrix; T; 8. Optimal approximation and least squares; Singular Value Decomposition
Part III: Vector Calculus (5 lectures)
1R3/17L1: T9. Partial differentiation [Review: 13.1-13.5;]; T 10. Vector fields, Path, volume and surface integrals
HW3-b:;Symmetric and non-symmetric matrices, eigenvectors, Singular value decomposition (Solution)
0/S3/18 Spring Break
0/13M3/28 Instruction Resumes
2T3/29L2: Vector fields: {${\bf R}(x,y,z)$}, Change of variables under integration: Jacobians
HW4: Key vector calculus topics (Solution)
3R3/31L3: Gradient {$\nabla$}, Divergence {$\nabla \cdot$}, Curl {$\nabla \times$}, Scaler (and vector) Laplacian {$\nabla^2$};Vector identies in various coordinate systems
Notes (pdf, djvu)
4/14T4/5L4: Integral and conservation laws: Gauss, Green, Stokes, Divergence
5R4/7L5: Applications of Stokes and Divergence Thms: Maxwell's Equations;
Potentials and Conservative fields;
Review: all 16
0/15T4/12 Exam II Apr 12 Tues @ 7-9 PM Room: 163 Everitt Lab
-T4/12NO Lecture due to Exam I; Class time will be converted to optional Office hours, to review home work solutions and discuss exam
Part IV: Boundary value problems (6 lectures)
Outline: Ch. 17 Fourier Trans.; Ch. 18: Diffusion Eq.; Ch. 19: Wave Eq.; Ch. 20. Laplace's Eq.
1R4/14L1: T 1. PDE: parabolic, hyperbolic, elliptical, discriminant
Read: Chapter 18.3; Look at: Emmy Noether, Noether's Thm. I; Examples of Symmetry in physics
HW6: Separation of variables, BV problems, symmetry (Solution)
2/16T4/19L2: T 21. Special Equations of Physics: Diffusion (Ch. 18); Wave (Ch. 19); Laplace (Ch. 20)
18. Separation of variables; integration by parts
3R4/21L4: T 16. Transmission line theory: Lumped parameter approximations
17. {$2^{nd}$} order PDE: Lecture on: Horns
Read:[17.7, pp.~ 887, 965, 1029, 1070, 1080]
4/17T4/26L3: T 20. Sturm-Liouville BV Theory: Allen out of town; Prof. Levinson to lecture
23. Special functions by Power Series: Bessel, Legendre Polynomials, Riemann Zeta
HW7: Sturm-Liouville, Boundary Value problems, Fourier and Laplace Transforms; (Solution); Hints for problems 3+5 and 4.
5R4/28L5: T 24. Fourier: Integrals, Transforms, Series, DFT
6/18T5/3L6: T Solutions to several geometries for the wave equation (Strum-Liouville cases)
Read: Ch. 20, 5.1-5.3 + Review p.290-1; Study: the solution to HW7
T 40. ODE's with initial condition (vs. Boundary value problems) Di and Gilbert (1993) Δ
Redo HW0:
-W5/4 Instruction Ends
-T5/10 Exam III 7:00-10:00+ PM on HW1-HW11 (Room: EH 106B3)
-/19F5/13 Finals End

 - F 5/13 Backup: Exam III 7:00-10:00+ PM on HW1-HW11 UIUC Final Exam Schedule

L= Lecture #
T= Topic #
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%.