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ECE 493/MATH-487 Daily Schedule Spring 2013

L/WDDateIntegrated Lectures on Mathematical Physics
    Part I: Complex Variables (10 lectures)
0/3M1/14 Classes start
1/3T1/15L1: T25. The fundamental Thm of Vector Fields \(\mathbf{F}(x,y,z) = \nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)\)
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 and delay \(e^{-i\omega T}\), \(\log(z)\), \(\sum z^n\)
Read: [Ch. 21.1-21.4]
HW0: Evaluate your present state of knowledge (not graded)
Assignment: CV1 Complex Functions and Laplace transforms
2R1/17L2: T 27. Differential calculus on \(\mathbb{C}\)
compact sets Fr├ęchet and related concepts
T 28. Cauchy-Riemann Eqs., Complex-Analytic functions are harmonic (i.e., irrotational vector fields, where \(\mathbf{A}=0\)) functions
T 34. Series: Maclaurin, Taylor, Laurent [24.3]; Frobenius power series method of solving differential equations [4.2]
Optional: Here is a fun video about B. Riemann.
Read: [21.5] and verify that you can do all the simple exercises on page 1113.
0/4M1/21 MLK Day; no class
3T1/22L3: T 26. Singularities (poles) and Partial fractions (p. 1263-5): \(Z(s) = A + Bs + \sum_{k=1}^K a_k/(s-s_k)\) and
Mobius Transformation (youtube, HiRes), pdf description
T *Inverses of Analytic functions (Riemann Sheets and Branch cuts); Analytic coloring, dial-a-function and doc, Edgar, using zviz.m from www.mathworks.com/company/newsletters/news_notes/clevescorner/summer98.cleve.html
T 28. Discussion on CR conditions: Analytic functions consist of locally-orthogonal pairs of harmonic fields:
i.e. \(\mathbf{u} = \nabla R(\sigma,\omega), \mathbf{w} = \nabla X(\sigma,\omega)\) then \(\mathbf{u} \cdot \mathbf{w} = 0\) (Discussion of physical examples)
T 29. incompressable [p. 839-840]: i.e., \(\nabla \cdot \mathbf{u} =0\) and irrotational [p. 826] \(\nabla \times \mathbf{w} =0\) vector fields
Read: [16.10] pp. 826-838 & 841-843;
Assignment: CV2; Analytic functions; 30. Integration of analytic functions
33. Cauchy integral formula; Riemann Sheets and Branch cuts; Region of Convergence; inverse Laplace transforms
4R1/24L4: 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: Fundamental Theorem of Complex integration
32. Cauchy's Theorem; 37. Inverse Laplace transforms; 38. Rational fraction expansions, conservative fields;
Read: [22.3]
5/5T1/29L5: T 32.Cauchy's theorem;
T 33.Cauchy's integral formula [23.5];
T 35. Cauchy's Residue Theorem [24.5]
Read: [23.3, 23.5];
6R1/31L6a: 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);
Read: pp. 841-843
7/6T2/5L7: Hilbert Transforms and the Cauchy Integral formula: The difference between the Fourier transform:
\( 2{\tilde u}(t) \equiv 1 + sgn(t) \leftrightarrow 2\pi\delta(\omega) + 2/j\omega \) and the Laplace \(2u(t) \leftrightarrow 2/s\)
Review of Residues (Examples) and their use in finding solutions to integrals;
Read: [24.3]
8R2/7L8: 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 \(Z(s,x=0)\)
9/7T2/12L9: 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
Read: [24.2, 24.2] (power series and the ROC);
10R2/14L10: T *38. Rational Impedance (Pade) approximations: \(Z(s)={a+bs+cs^2}/({A+Bs})\)
*Continued fractions: \(Z(s)=s+a/(s+b/(s + c/(s+\cdots)))\) expansions
*Computing the reactance \(X(s) \equiv \Im Z(s)\) given the resistance \(R(s) \equiv \Re Z(s)\)
Boas, R.P., Invitation to Complex Analysis (Boas Ch 4)
*Riemann zeta function: \(\zeta(s) = \sum_{n=1}^\infty 1/n^s\)
*There is also a product form for the Riemann zeta function
*Potpourri of other topics
Read:'' [24.5, Appendix A]
11/8T2/19NO CLASS due to Exam I Optional review and special office hours, of all the material, will be held in the class room 12:30-2PM 441 AH
11/8T2/19 Exam I Feb 19 Tuesday @ 7-9 PM; Place: 241 Altgeld
    Part II: Linear (Matrix) Algebra (6 lectures)
1R2/21LA1: 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;
LA1; (Solution)
2/9T2/26LA2: T 3. Solutions to \(Ax=b\) by Gaussian elimination, T 4. Matrix inverse \(x=A^{-1}b\); Cramer's Rule
Read: 8.3, 10.4 ;
3R2/28LA3:*T5. The symmetric matrix: Eigenvectors; The significance of Reciprocity
*Mechanics of determinates: \(B = P_n P_{n-1} \cdots P_1 A\) with permutation matrix \(P\) such that P1: (i) <- (i)+a(j); P2: (i) <-> (j); P3: (i)<- a(i)
Read: 10.6-10.8, 11.4;
LA2: Vector space; Schwartz and Triangular inequalities, eigenspaces
1/10T3/5 Move this to L1 of Vector Calculus (First lecture of 6 following spring Break)
Vector dot-product \(A \cdot \)B, cross-product \(A \times \)B, triple-products \(A \cdot A \times \)B, \(A \times (B \times C)\)
Gram-Schmidt proceedure
Read:'' 11.4
4R3/7L4: T 7. Vector spaces in \(\mathbb{R}^n\); Innerproduct+Norms; Ortho-normal; Span and Perp (\(\perp\)); Schwartz and Triangular inequalities
* T 6. Transformations (change of basis)
Read: 9.1-9.6, 10.5, 11.1-11.3; Leykekhman Lecture 9
LA3: Rank-n-Span; Taylor series; Vector products and fields
0FS3/8-3/9Engineering Open House
5/11T3/12L5: T 5. Asymmetric matrix; T; 8. Optimal approximation and least squares; Singular Value Decomposition
Read: 9.10, Eigen-analysis and its applications
6R3/14L6: Fourier/Laplace/Hilbert-space lecture: a detailed study of all the Fourier-like transforms
Hilbert space and <bra|c|ket> notation
0/12S3/16 Spring Break
0/13M3/25 Instruction Resumes
    Part III: Vector Calculus (6 lectures)
2/13T3/26L1: T9. Partial differentiation [Review: 13.1-13.4;]; T 10. Vector fields, Path, volume and surface integrals
Read: 15-15.3
VC1: Topics: Rank-n-Span; Taylor series; Vector fields, Gradient Vector field topics (Due 1 week)
3R3/28L2: Vector fields: \({\bf R}(x,y,z)\), Change of variables under integration: Jacobians \(\frac{\partial(x,y,z)}{\partial(u,v,w)}\)
Review 3.5; Read: 13.6,15.4-15.6
4/14T4/2L3: Gradient \(\nabla\), Divergence \(\nabla \cdot\), Curl \(\nabla \times\), Scaler (and vector) Laplacian \(\nabla^2\)
Vector identies in various coordinate systems; Allen's Vector Calculus Summary (partial-pdf, pdf, djvu)
Read: 16.1-16.6
VC2: Key vector calculus topics (Due 1 week)
5R4/4L4: Integral and conservation laws: Gauss, Green, Stokes, Divergence
Read: 16.8-16.10
6/15T4/9L5: Applications of Stokes and Divergence Thms: Maxwell's Equations;
Potentials and Conservative fields;
Review: 16
0/15R4/11 Exam II @ 7-9 PM Room: 163 EVRT Lab (ECE)
-R4/11NO 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.
1/16T4/16L1: T 1. PDE: parabolic, hyperbolic, elliptical, discriminant
Read: Chapter 18.3; Look at: Emmy Noether, Noether's Thm. I; Examples of Symmetry in physics
BV1: Due Apr 25, 2013: Topic: Partial Differential Equations: Separation of variables, BV problems, use of symmetry
2R4/18L2: T 21. Special Equations of Physics: Diffusion (Ch. 18); Wave (Ch. 19); Laplace (Ch. 20)
18. Separation of variables; integration by parts
Read: [20.2-3]
3/17T4/23L4: 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]
4R4/25L3: T 20. Sturm-Liouville BV Theory
23. Special functions by Power Series: Bessel, Legendre Polynomials, Riemann Zeta
Read: 20
BV2: Sturm-Liouville, Boundary Value problems, Fourier and Laplace Transforms; Hints for problems 3+5 and 4.
5T4/30L5: R Solutions to several geometries for the wave equation (Strum-Liouville cases)
WKB solution of the Horn Equation
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)
L6: T 24. Fourier: Integrals, Transforms, Series, DFT
Read: 17.3-17-6
Redo HW0:
-R5/1 Instruction Ends
-F5/2 Reading Day
-/19 R5/9 Exam III 7:00-10:00+ PM, Room: 441AH (UIUC dictate)
-/19F5/10 Finals End

-F5/13 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 (Prof's Discuression), count for 10%.

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