### 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 $$e^{-i\omega T}$$, $$\log(z)$$, $$\sum z^n$$
Assignment: CV1 Complex Functions and Laplace transforms
2R1/17L2: T 27. Differential calculus on $$\mathbb{C}$$
compact sets Fréchet 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;
5/5T1/29L5: T 32.Cauchy's theorem;
T 33.Cauchy's integral formula [23.5];
T 35. Cauchy's Residue Theorem [24.5]
CV3;
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);
CV4;
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;
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);
CV5;
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
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) 1 R 2/21 LA1: T 1. Basic definitions, Elementary operations;T 2. Cramer's Rule, Determininants, Inverse Matrix, Aug Matrix and Gauss Elimination; Vandermonde Review Exam I;Read: 8.1-2, 10.2;LA1; (Solution) 2/9 T 2/26 LA2: 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 ; 3 R 2/28 LA3:*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 1/10 T 3/5 Move this to L1 of Vector Calculus (First lecture of 6 following spring Break) L1-VC: 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 proceedureRead:'' 11.4 4 R 3/7 L4: 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 0 FS 3/8-3/9 Engineering Open House 5/11 T 3/12 L5: T 5. Asymmetric matrix; T; 8. Optimal approximation and least squares; Singular Value DecompositionRead: 9.10, Eigen-analysis and its applications 6 R 3/14 L6: Fourier/Laplace/Hilbert-space lecture: a detailed study of all the Fourier-like transforms Hilbert space and notation 0/12 S 3/16 Spring Break 0/13 M 3/25 Instruction Resumes
 Part III: Vector Calculus (6 lectures) 2/13 T 3/26 L1: T9. Partial differentiation [Review: 13.1-13.4;]; T 10. Vector fields, Path, volume and surface integralsRead: 15-15.3 VC1: Topics: Rank-n-Span; Taylor series; Vector fields, Gradient Vector field topics (Due 1 week) 3 R 3/28 L2: 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 4/14 T 4/2 L3: 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) 5 R 4/4 L4: Integral and conservation laws: Gauss, Green, Stokes, Divergence Read: 16.8-16.10 6/15 T 4/9 L5: Applications of Stokes and Divergence Thms: Maxwell's Equations;Potentials and Conservative fields; Review: 16 0/15 R 4/11 Exam II @ 7-9 PM Room: 163 EVRT Lab (ECE) - R 4/11 NO 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. 1/16 T 4/16 L1: T 1. PDE: parabolic, hyperbolic, elliptical, discriminantRead: Chapter 18.3; Look at: Emmy Noether, Noether's Thm. I; Examples of Symmetry in physicsBV1: Due Apr 25, 2013: Topic: Partial Differential Equations: Separation of variables, BV problems, use of symmetry 2 R 4/18 L2: T 21. Special Equations of Physics: Diffusion (Ch. 18); Wave (Ch. 19); Laplace (Ch. 20)18. Separation of variables; integration by partsRead: [20.2-3] 3/17 T 4/23 L4: T 16. Transmission line theory: Lumped parameter approximations17. $$2^{nd}$$ order PDE: Lecture on: HornsRead:[17.7, pp. 887, 965, 1029, 1070, 1080] 4 R 4/25 L3: T 20. Sturm-Liouville BV Theory 23. Special functions by Power Series: Bessel, Legendre Polynomials, Riemann ZetaRead: 20BV2: Sturm-Liouville, Boundary Value problems, Fourier and Laplace Transforms; Hints for problems 3+5 and 4. 5 T 4/30 L5: R Solutions to several geometries for the wave equation (Strum-Liouville cases)WKB solution of the Horn EquationRead: 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, DFTRead: 17.3-17-6 Redo HW0: - R 5/1 Instruction Ends - F 5/2 Reading Day -/19 R 5/9 Exam III 7:00-10:00+ PM, Room: 441AH (UIUC dictate) -/19 F 5/10 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 (Prof's Discuression), count for 10%.