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  • 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 \(e^{-i\omega T}\), \(\log(z)\), \(\sum z^n\)
Read: [ JPD: p. 47-62]
HW0: Evaluate your present state of knowledge (not graded)
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
Read: Handout
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);
Read: [22.3]
5/5T1/30L5: T32. Cauchy's theorem;
T33. Cauchy's integral formula [23.5];
T35. Cauchy's Residue Theorem [24.5]
Read: Handout
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);
Read: pp. 841-843
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;
Read: [24.3]
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
Read: Lec 27, 28: Horns
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)
Read: Conservation of Energy
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)
1R2/22LA1: T1. Basic definitions: Work, energy/power/work, impedance = Elementary linear operations;
T2. Gauss Elimination
Review Exam I;
8.1-2, 10.2;
LA1 matrix algebra, Gaussian Elimination; eigen analysis
2/9T2/27LA2: 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)
3R3/1LA3: Allen out of town T5. Mechanics of Gaussian elimination: \(B = P_n P_{n-1} \cdots P_1 A\) with permutation matrix \(P\) such that
P1: (i) <- (i)+a(j); P2: (i) <-> (j); P3: (i)<- a(i);
Eigenvectors; The significance of Reciprocity
Read: ...;
LA2: Vector space; Schwartz and Triangular inequalities, eigen-space; Vandermonde analysis
LA1 Due
4/10T3/6LA4: 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 proceedure
Read: : ...
5R3/8L5: 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 symmetry
LA2 Due
0FS 3/9-3/10 Engineering Open House
6/11T3/13L6: Taxonomy of matricies; examples of the use of Matrix analysis in engineering: Control theory, MIMO systems
Read: ... Matrix Taxonomy, Eigen-analysis and its applications
7R3/15L7: 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 <bra|c|ket> inner product notation
LA3 Due
0/12S3/19 Spring Break
0/13M3/26 Instruction Resumes
    Part III: Vector Calculus (5 Lectures)
1/13T3/27L1: T10. Potential fields: \(\Phi(x,y,z,t)\); Notation; scalar & vector field products;
Read: Morse-Feshbach Vol 1: Fields
VC1: Topics: Implicit Function Thm; Vector Cross products Taylor series; Vector fields
(Due 1 week)
2R3/29L2: 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 transformations
Read: pdf
3/14 T4/3L3: 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 pdf
VC1 Due; VC2: Vector calculus topics
4R4/5L4: 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
5/15T4/10L5: Maxwell's Equations; Physics of ME; Applications to ME of DoC and CoG Thms. How big is \(\mathbf{B}\) in Teslas?
VC2 Due
0/15R4/12 Exam II @ 7-9 PM Room: 343 Alt Hall scores
 R4/12NO 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)
1/16T4/17L1: 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
2R4/19L2: 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 parts
3/17T4/24L3: T16. Transmission line theory: Lumped parameter approximation: Diffusion line, Telegraph equation
Read: T17. \(2^{nd}\) order PDE: Lecture on: HornsReview: System Postulates
BV2: Topics:' Sturm-Liouville; Boundary Value problems; Fourier and Laplace Transforms;
BV1 Due
4R4/26L4: T 19, 20. Sturm-Liouville BV Theory T22. Solutions for 1, 2, 3 dimensions
Impedance Boundary conditions; The reflection coefficient and its properties;
5/18T5/1L5: 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
-W5/2 Instruction Ends
-R5/3 Reading Day
- 5/5Review for Final: 2-4 PM Room 106B3 in Engineering Hall.
-/19 R5/8 Final Exam 1:30-4:30 PM, Room: 441 (UIUC Final Exam Schedule)
-/19F5/10 Finals End

 ||-     || F || 5/?? ||  Backup: Exam III 7:00-10:00+ PM on HW1-HW11atest>><<

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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