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  • Advanced Engineering Mathematics: UIUC; Syllabus: 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
    *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: T 27. Differential calculus on {$\mathbb{C}$}
T 28. Cauchy-Riemann Eqs., Complex-Analytic functions are harmonic
T 34. 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
T 28. Visualizing complex analytic and harmonic functions using zviz Examples from Cleve's Corner, Summer 98,
Analytic coloring, dial-a-function and doc, Edgar
T 30. Analytic function integration: Fundamental theorems of Calculus (FTC) and complex calculus (FTCC)
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 Transformations (youtube, HiRes), pdf description.
T 33. 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: 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
T 32. 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: T 32.Cauchy's theorem;
T 33.Cauchy's integral formula [23.5];
T 35. Cauchy's Residue Theorem [24.5]
Read: Handout
CV3 Riemann Sheets, Branch cuts, Filters, PosDef operators, 2-port circuits (ABCD)
CV2 due
6R3/1L6a: 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: T 37. 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) $} Cleve's Corner: Summer 1998;Reflectance vs Impedance
Read: Lec 27, 28: Horns
CV5 Transmission lines
CV4 due
10R2/15L10: T *38. Rational Impedance (Pade) approximations: {$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)$} Boas, R.P., Invitation to Complex Analysis (Boas Ch 4)
Read: Conservation of Energy
CV5 due 2/20 7PM
11/8T2/20NO CLASS due to Exam I Optional review and special office hours of all CV material. Exam I in room 343AH, 7-10PM
11T2/20 Exam I
    Part II: Linear (Matrix) Algebra (7 Lectures)
1R2/22LA1: T 1. Basic definitions: Work, energy/power/work, impedance = Elementary linear operations;
T 2. Gauss Elimination
Review Exam I;
8.1-2, 10.2;
LA1 matrix algebra, Gaussian Elimination; eigen analysis
2/9T2/27LA2: T 3. Solutions of {$Ax=b$} by Gaussian elimination, T 4. Matrix inverse {$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$} such that
P1: (i) <- (i)+a(j); P2: (i) <-> (j); P3: (i)<- a(i);
Eigenvectors; The significance of Reciprocity
Read: ...;
LA2: Vector space; Schwartz and Triangular inequalities, eigen-space; Vandermonde analysis
LA1 Due
4/10T3/6LA4: T 7. Vector spaces in {$\mathbb{R}^n$}; Inner-product+Norms; Ortho-normal; Span and Perp ({$\perp$}); Schwarz and Triangular inequalities Complete+closed vs open set (mean minimum error (RMS) solution); T 6. Transformations (change of basis) Gram-Schmidt proceedure
Read: : ...
5R3/8L5: T 5. Asymmetric matrix; T; 8. Fat/thin and least squares; Eigen-function decomposition; Singular Value Decomposition of Fat/thin systems (SVD, pdf)
Read: ... ; Leykekhman Lecture 9
Version 1.21 LA3: Fat/Thin systems; Rank-n-Span; Operator symmetry
LA2 Due
0FS 3/9-3/10 Engineering Open House
6/11 T3/13L6: Taxonomy of matricies; examples of the use of Matrix analysis in engineering: Control theory, MIMO systems
Read: ... Matrix Taxonomy, Eigen-analysis and its applications
7R3/15L7: Scalar-product {$A \cdot B$}, vector-product {$A \times B$}, triple-products {$A \cdot A \times B$}, {$A \times (B \times C)$}; discussion of Vandermode systems
Fourier/Laplace/Hilbert-space lecture: a detailed study of all the Fourier-like transforms and their generalizations
Hilbert space and <bra|c|ket> inner product notation
LA3 Due
0/12S3/19 Spring Break
0/13M3/26 Instruction Resumes
    Part III: Vector Calculus (5 Lectures)
1/13T3/27L1: T9. Partial differentiation [Review: 13.1-13.4;]; T 10. Vector fields, Path, volume and surface integrals
Read: 15-15.3
VC1: Topics: Rank-n-Span; Taylor series; Vector fields, Gradient Vector field topics (Due 1 week)
2R3/29L2: Vector fields: {${\bf R}(x,y,z)$}, Change of variables under integration: Jacobians {$\frac{\partial(x,y,z)}{\partial(u,v,w)}$}
Review 3.5; Read: TBD
3/14T4/3L3: Gradient {$\nabla$}, Divergence {$\nabla \cdot$}, Curl {$\nabla \times$}, Scaler (and vector) Laplacian {$\nabla^2$}
Vector identies in various coordinate systems; Allen's Vector Calculus Summary (partial-pdf, pdf, djvu)
T25. The fundamental Thm of Vector Fields {$\mathbf{F}(x,y,z) = \nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)$}
T 29. incompressible [p. 839-840]: i.e., {$\nabla \cdot \mathbf{u} =0$} and irrotational [p. 826] {$\nabla \times \mathbf{w} =0$} vector fields
Read: 16.1-16.6
VC2: Key vector calculus topics (Due 1 week)
VC1 Due
4R4/5L4: Integral and conservation laws: Gauss, Green, Stokes, Divergence
Read: 16.8-16.10
5/15T4/10L5: Applications of Stokes and Divergence Thms: Maxwell's Equations;
Potentials and Conservative fields;
Review: 16
VC2 Due
0/15R4/12 Exam II @ 7-9 PM Room: 343 Alt Hall
 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)
   Outline: Ch. 17 Fourier Trans.; Ch. 18: Diffusion Eq.; Ch. 19: Wave Eq.; Ch. 20. Laplace's Eq.
1/16T4/17L1: 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
2R4/19L2: 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]
3/17T4/24L3: 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]
BV2: Sturm-Liouville, Boundary Value problems, Fourier and Laplace Transforms; Hints for problems 3+5 and 4.
BV1 Due
4R4/26L4: T 20. Sturm-Liouville BV Theory
23. Special functions by Power Series: Bessel, Legendre Polynomials, Riemann Zeta
Read: 20
5/18T5/1L5: 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 T 40. ODE's with initial condition (vs. Boundary value problems)
L6: T 24. Fourier: Integrals, Transforms, Series, DFT
Read: 17.3-17-6
Redo HW0:
BV2 Due
-R5/2 Instruction Ends
-F5/3 Reading Day
-/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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