 ## ECE493-S18

• Advanced Engineering Mathematics: UIUC; Syllabus: pdf, pdf; Listing: ECE-493, (Math-487) Campus: ECE/Math
• Calendars: Class, Campus;
• Time/place: Altgeld 441, T,R 12:30, ECE-493;
• Matlab Tutorial pdf
• Text: An Invitation to Mathematical Physics and its History pdf; Supplemental reading: Greenberg
*Office Hours: Monday 3-4:45PM 3034 ECEB; Friday 3-5PM, 4036 ECEB
• Instructor: Prof. Jont Allen (netID jontalle; Office 2061BI); TA: Felix Wang (netID fywang2)
• This week's schedule; Final

### ECE 493/MATH-487 Daily Schedule Spring 2018

L/WDDateIntegrated Lectures on Mathematical Physics
Part I: Complex Variables (10 Lectures)
0/3M1/15 MLK Day; no class
0/3M1/16 Classes start
1/3T1/16L1: The frequency domain: Complex $$Z(s) = R(s)+iX(s)$$ as a function of complex frequency $$s=\sigma+i\omega$$, e.g., $$Z,s \in \mathbb{C}$$
1-node KCL network example $$(\Sigma_k i_k = \dot{\Psi})$$; Phasors, delay $$e^{-i\omega T}$$, $$\log(z)$$, $$\sum z^n$$
Assignment: CV1 Due 1 week: Complex algebra, functions and Laplace transform basics
2R1/18L2: T27. Differential calculus on $$\mathbb{C}$$
T28. Cauchy-Riemann Eqs., Complex-Analytic functions are harmonic
T34. Series: Maclaurin, Taylor, Laurent [24.3]; Frobenius power series method of solving differential equations
Read: JBA: 58-62; JPD: p. 63-71
3/4T1/23L3: rescheduled for Wed, Jan 24, 7-8:30PM Room 443AH
T28. Visualizing complex analytic and harmonic functions using zviz Examples from Cleve's Corner, Summer 98,
Analytic coloring, dial-a-function and doc, Edgar
T30. Analytic function integration: Fundamental theorems of Calculus (FTC) and complex calculus (FTCC)
T26. Singularities (poles) and Partial fractions (p. 1263-5): $$Z(s) = A + Bs + \sum_{k=1}^K a_k/(s-s_k)$$ and
Mobius Transformations (youtube, HiRes), pdf description.
T33. Cauchy integral formula; Riemann Sheets and Branch cuts; Region of Convergence; inverse Laplace transforms
CV2 calculus on analytic functions + CR conditions; Riemann sheets;
CV1 due
4R1/25L4: T30. Integral calculus on $$\mathbb{C}$$
T31. $$\int z^{n-1} dz$$ on the unit circle
Continue discussion of examples of analytic functions: Fundamental Theorem of Complex integration
T32. Cauchy's Theorem; 37. Inverse Laplace transforms; 38. Rational fraction expansions, conservative fields;
Inverses of Analytic functions (Riemann Sheets and Branch cuts);
5/5T1/30L5: T32. Cauchy's theorem;
T33. Cauchy's integral formula [23.5];
T35. Cauchy's Residue Theorem [24.5]
CV3 Riemann Sheets, Branch cuts, Filters, PosDef operators, 2-port circuits (ABCD)
CV2 due
6R2/1L6a: 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);
7/6T2/6L7: 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;
CV4 RoC, Fourier/Laplace transforms; $$\zeta(s)$$
CV3 due
8R2/8L8: Cauer synthesis, Bode plots, Network theory (Brune Positive-real (PR) impedance functions)
Read: Impedance (Kennely 1893, Schelkunoff 1938)
9/7T2/13L9: T37. More on Inverse Transforms: Laplace $${\cal L}^{-1}$$ and Fourier $${\cal F}^{-1}$$;
Analytic continuation by inverse Laplace Transform; Properties of the Log-derivative
The multi-valued $$i^s$$, $$\tanh^{-1}(s) = \frac{1}{2}\ln \left( \frac{1+s}{1-s} \right)$$ Cleve's Corner: Summer 1998;Reflectance vs Impedance
CV5 Transmission lines
CV4 due
10R2/15L10: T38. Rational Impedance (Pade) approximations: $$Z(s)={a+bs+cs^2}/({A+Bs})$$
*Properties of Brune impedance
*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)
CV5 due 2/20 7PM
11/8T2/20NO CLASS due to Exam I Optional review and special office hours of all CV material. Exam I in room 343AH, 7-10PM
11T2/20 Exam I
 Part II: Linear (Matrix) Algebra (7 Lectures) 1 R 2/22 LA1: T1. Basic definitions: Work, energy/power/work, impedance = Elementary linear operations;T2. Gauss Elimination Review Exam I;Read: 8.1-2, 10.2; LA1 matrix algebra, Gaussian Elimination; eigen analysis 2/9 T 2/27 LA2: T3. Solutions of $$Ax=b$$ by Gaussian elimination, T4. Matrix inverse $$x=A^{-1}b$$; Augmented matrix; Gaussian elimination ($$n^3$$); determinants ($$n!$$); Cramer's Rule Read: Linear Alg Handout (pdf) 3 R 3/1 LA3: Allen out of town T5. Mechanics of Gaussian elimination: $$B = P_n P_{n-1} \cdots P_1 A$$ with permutation matrix $$P$$ such thatP1: (i) <- (i)+a(j); P2: (i) <-> (j); P3: (i)<- a(i);Eigenvectors; The significance of Reciprocity Read: ...; LA2: Vector space; Schwartz and Triangular inequalities, eigen-space; Vandermonde analysisLA1 Due 4/10 T 3/6 LA4: T7. Vector spaces in $$\mathbb{R}^n$$; Inner-product+Norms; Ortho-normal; Span and Perp ($$\perp$$); Schwarz and Triangular inequalities Complete+closed vs open set (mean minimum error (RMS) solution); T6. Transformations (change of basis) Gram-Schmidt proceedureRead: : ... 5 R 3/8 L5: T5. Asymmetric matrix; T8. Fat/thin and least squares; Eigen-function decomposition; Singular Value Decomposition of Fat/thin systems (SVD, pdf)Read: ... ; Leykekhman Lecture 9 Version 1.21 LA3: Fat/Thin systems; Rank-n-Span; Operator symmetryLA2 Due 0 FS 3/9-3/10 Engineering Open House 6/11 T 3/13 L6: Taxonomy of matricies; examples of the use of Matrix analysis in engineering: Control theory, MIMO systemsRead: ... Matrix Taxonomy, Eigen-analysis and its applications 7 R 3/15 L7: Scalar-product $$A \cdot B$$, vector-product $$A \times B$$, triple-products $$A \cdot A \times B$$; Discussion of Vandermode systems Fourier/Laplace/Hilbert-space; Hilbert space and inner product notationLA3 Due 0/12 S 3/19 Spring Break 0/13 M 3/26 Instruction Resumes
 Part III: Vector Calculus (5 Lectures) 1/13 T 3/27 L1: T10. Potential fields: $$\Phi(x,y,z,t)$$; Notation; scalar & vector field products; Read: Morse-Feshbach Vol 1: FieldsVC1: Topics: Implicit Function Thm; Vector Cross products Taylor series; Vector fields(Due 1 week) 2 R 3/29 L2: T9. Partial differentiation; smooth vs. analytic functions; Taylor Series; Implicit function Thm, Line surface & volume integrals; Jacobians $$\frac{\partial(x,y,z)}{\partial(u,v,w)}$$ as volume conserving transformationsRead: pdf 3/14 T 4/3 L3: Gradient $$\nabla$$, Divergence $$\nabla \cdot$$, Curl $$\nabla \times$$, Laplacian $$\nabla^2$$Integral vs differential definitions;Integral and conservation laws: Gauss, Green, Stokes, Divergence Vector identies in various coordinate systems; (partial-pdf, pdf)T25. The fundamental thm of vector calculus: $$\mathbf{F}(x,y,z) = \nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)$$ (DoC & CoG)Read: Lec 39 pdfVC1 Due; VC2: Vector calculus topics 4 R 4/5 L4: Differential & integral forms of Grad, Div, Curl; Conservation theorems (Gauss's and Stokes's Laws);T29. incompressible: i.e., $$\nabla \cdot \mathbf{u} =0$$ and irrotational $$\nabla \times \mathbf{w} =0$$ vector fields Read: Handout Lec 40 pdf 5/15 T 4/10 L5: Maxwell's Equations; Physics of ME; Applications to ME of DoC and CoG Thms. How big is $$\mathbf{B}$$ in Teslas?VC2 Due 0/15 R 4/12 Exam II @ 7-9 PM Room: 343 Alt Hall scores R 4/12 NO Lecture due to Exam II; Class time will be converted to optional Office hours, to review home work solutions and discuss exam
 Part IV: Boundary value problems (5 Lectures) 1/16 T 4/17 L1: T 15. PDEs T 21. Special Equations of Physics: Laplace, Diffusion, Wave; Parabolic, hyperbolic, elliptical: discriminant,Read: Notes Lecture 34, Sect. 1.5.3 ; Symmetry in physics BV1: Topic: Partial Differential Equations 2 R 4/19 L2: T 17. Derivation of the wave equation from 2 first-order equations (mass+stiffness)T 18. Webster Horn equation: vs separation of variables method; integration by partsRead: 3/17 T 4/24 L3: T16. Transmission line theory: Lumped parameter approximation: Diffusion line, Telegraph equationRead: T17. $$2^{nd}$$ order PDE: Lecture on: HornsReview: System Postulates BV2: Topics:' Sturm-Liouville; Boundary Value problems; Fourier and Laplace Transforms; BV1 Due 4 R 4/26 L4: T 19, 20. Sturm-Liouville BV Theory T22. Solutions for 1, 2, 3 dimensions Impedance Boundary conditions; The reflection coefficient and its properties;Read: 5/18 T 5/1 L5: WKB solution of Horn Equation, T34. ODE's with initial condition (vs. Boundary value problems)L6: T 24. Fourier: Integrals, Transforms, Series, DFT; History: Newton, d'Alembert, Bernoullis, Euler Redo HW0:BV2 Due
 - W 5/2 Instruction Ends - R 5/3 Reading Day - 5/5 Review for Final: 2-4 PM Room 106B3 in Engineering Hall. -/19 R 5/8 Final Exam 1:30-4:30 PM, Room: 441 (UIUC Final Exam Schedule) -/19 F 5/10 Finals End

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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%.