 • Advanced Engineering Mathematics: UIUC; Syllabus: 2015, 2013, 2011, 2009, 2008; Listing: ECE-493 Campus: ECE/Math
*Calendars: Class, Campus; Time/place, Website: etc; Text: Greenberg
• Office Hours: Sunday 3-4PM, 3036 ECE Building, or by apt 4-5 2061BI; TA: Michael Wei (netID: mwei5); Instructor: Prof. Jont Allen (netID jontalle; Office 2061BI).
*
This week's schedule; Exam I Feb 24; Exam II Apr 16; Final

### ECE 493/MATH-487 Daily Schedule Spring 2015

L/WDDateIntegrated Lectures on Mathematical Physics
Part I: Complex Variables (10 lectures)
0/4M1/19 MLK Day; no class
0/4M1/20 Classes start
1/4T1/20L1: T25. The fundamental Thm of Vector Fields (p. 842) $\mathbf{F}(x,y,z) = \nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)$
The frequency domain: Complex $Z(s) = R(s)+iX(s)$ as a function of complex frequency $s=\sigma+i\omega$, e.g., $Z,s \in \mathbb{C}$
1-node KCL network example $( \Sigma_k i_k = \dot{\Psi} )$; Phasors, delay $e^{-i\omega T}$, $\log(z)$, $\sum z^n$
Assignment: CV1: Complex Functions and Laplace transforms CV1-sol.pdf
2R1/22L2: T 27. Differential calculus on $\mathbb{C}$
T 28. Cauchy-Riemann Eqs., Complex-Analytic functions are harmonic (i.e., irrotational vector fields (i.e., $\mathbf{A}=0$) 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.
3/5T1/27L3: T 26. Singularities (poles) and Partial fractions (p. 1263-5): $Z(s) = A + Bs + \sum_{k=1}^K a_k/(s-s_k)$
T 38. Rational fraction expansions, conservative fields;
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;
4R1/29L4: T 30.Integral calculus on $\mathbb{C}$; T 30. Integration of analytic functions
T 31. $\int z^{n-1} dz$ on the unit circle; T 32. Cauchy's Theorem
T 33. Cauchy integral formula [23.5]; Riemann Sheets and Branch cuts; Region of Convergence
T 37. Inverse Laplace transforms; Residue theorem
5/6T2/3L5: T 32.Cauchy's theorem;
T 33.Cauchy's integral formula [23.5];
T 35. Cauchy's Residue Theorem [24.5]; Fundamental Theorems of complex integration (p. 1197); Analytic functions
*Inverses of Analytic functions (Riemann Sheets and Branch cuts); Analytic coloring, dial-a-function and doc, Edgar
*Mobius Transformation (youtube, HiRes), pdf description
*Usingzviz.m via matlab
CV3
6R2/5L6a: 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);
7/7T2/10L7: Review of Residues use in finding solutions of integrals (with examples);
*Bode plots, Network theory (Brune Positive-real (PR) impedance functions)
*The Fourier vs. Laplace step functions: $2{\tilde u}(t) \equiv 1 + sgn(t) \leftrightarrow 2\pi\delta(\omega) + 2/j\omega$ and $2u(t) \leftrightarrow 2/s$
*The definition of the Hilbert Transform vs. the Cauchy Integral formula
CV4
8R2/12L8: Frequency domain lecture: a detailed study of all Fourier-like transforms: FS, FT, DTFT, DFT, zT, LT
*ABCD method for modeling transmission lines
*
Transmission line equations George Campbell on Wave-Filters (1922)
Read: [24.2] (power series and the ROC)
9/8T2/17L9: T 37. More on Inverse Laplace and z Transforms;
*The multi-valued $i^s$, $\tanh^{-1}(s) = \frac{1}{2}\ln \left( \frac{1+s}{1-s} \right)$ and: $\cosh^{-1}(s) = \ln(s \pm \sqrt{s^2 -1} )$
*Riemann zeta function: $\zeta(s) = \sum_{n=1}^\infty 1/n^s$
*There is also a product form for the Riemann zeta function
*Analytic continuation
CV5
10R2/19L10: 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)
*Potpourri of other topics
11/9T2/24NO CLASS due to Exam I Optional review and special office hours, of all the material 12:30-2PM 441 AH
11/9T2/24 Exam I Feb 24 Tuesday @ 7-9 PM; Place: 441 Altgeld
L/WDDateIntegrated Lectures on Mathematical Physics
Part II: Linear (Matrix) Algebra (6 lectures)
1/9R2/26L1: T 1. Basic definitions, Elementary operations;
T 2. Inverse Matrix, Aug Matrix and Gauss Elimination; Vandermonde
Review Exam I;
8.1-2, 10.2;
Assignment: LA1
2/10T3/3LA2: T 3. Solutions to $Ax=b$ by Gaussian elimination; Determinants
T 4. Matrix inverse $x=A^{-1}b$; Cramer's Rule; Gram-Schmidt proceedure
3R3/5L3:*T 5. The symmetric matrix: Eigenvectors; The significance of Reciprocity
*Mechanics of determinates: $B = P_n P_{n-1} \cdots P_1 A$ with permutation matrix $P$:
P1: (i) <- (i)+a(j); P2: (i) <-> (j); P3: (i)<- a(i)
Assignment: LA2: Vector space; Schwartz and Triangular inequalities, eigenspaces
4/11T3/10L4: 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
Assignment: LA3: Rank-n-Span; Taylor series; Vector products and fields
5R3/12L5: T 5. The symmetric matrix; T 8. Optimal approximation and least squares; SVD Least-squares solutions to $Ax=b$: $x^\dagger = (A^\dagger A)^{-1} A^\dagger b$
Read: 9.10, Eigen-analysis and its applications
FS3/13-14Engineering Open House
6/12T3/17L6: Lower-Upper decomposition: $A = LU$; Cholesky decomposition (positive-definite A): $A=L L^\dagger$
Hilbert space (<bra|O|ket>) notation are Hilbert space weighted norms on operator $O$
Read 11.4, 11.6: Diagonalization of Matrices, when and how; Quadratic Forms (p. 589);
Assignment: LA4: Symmetric matricies; LU and Cholesky Decompositions
L/WDDateIntegrated Lectures on Mathematical Physics
Part III: Vector Calculus (6 lectures)
1/12R3/19L1: Vector dot-product $A \cdot B$, cross-product $A \times B$, triple-products $A \cdot A \times B$, $A \times (B \times C)$
Gradient: $E = -\nabla \Phi$, Divergence: $\nabla \cdot D = \rho$, Curl: $\nabla \times H = C$, $\nabla \times E = -\dot{B}$
Fundamental Theorem of Vector Calculus: $F = -\nabla \Phi + \nabla \times A$
0/12S3/21 Spring Break
0/14M3/30 Instruction Resumes
2T3/31L2: T 9. Partial differentiation [Review: 13.1-13.4;]; T 10. Vector fields, Path, volume and surface integrals
VC1: Topics: Rank-n-Span; Taylor series; Vector fields, Gradient Vector field topics (Due 1 week)
3R4/2L3: Vector fields: ${\bf R}(x,y,z)$, Change of variables under integration: Jacobians $\frac{\partial(x,y,z)}{\partial(u,v,w)}$
4/15T4/7L4: 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)
VC2: Key vector calculus topics (Due 1 week)
5R4/9L5: Integral and conservation laws: Gauss, Green, Stokes, Divergence
Look at: Emmy Noether, Noether's Thm. I; Examples of Symmetry in physics
6/16T4/14L6: Applications of Stokes and Divergence Thms: Maxwell's Equations
Potentials and Conservative fields;
Review: 16
0R4/16 Exam II April 16 @ 7-9 PM Room: TBD
0R4/16NO Lecture due to Exam II;
Class time will be converted to optional Office hours, to review home work solutions and discuss exam

L/WDDateIntegrated Lectures on Mathematical Physics
Part IV: Boundary value problems (5 lectures)
Outline: Ch. 18: Diffusion Eq.; Ch. 19: Wave Eq.; Ch. 20. Laplace's Eq.
1/17T4/21L1: T 1. PDE: parabolic, hyperbolic, elliptical, discriminant
Assignment: BV1: Due Apr 30, 2015: Topic: Partial Differential Equations: Separation of variables, BV problems, use of symmetry
2R4/23L2: T 21. Special Equations of Physics: Diffusion (Ch. 18); Wave (Ch. 19); Laplace (Ch. 20)
Read: 18.3, 19.2-3; 20.1-2: Separation of variables
3/18T4/28L3: T 16. Transmission line theory: Lumped parameter approximations; 1D wave equations
17. $2^{nd}$ order PDE: Lecture on: Horns
Integration by parts
Read:[17.7, pp. 887, 965, 1029, 1070, 1080]
'Assignment:''BV2: Sturm-Liouville, Boundary Value problems, Fourier and Laplace Transforms; Hints for problems and Problem 4.
4R4/30L4: T 20. Sturm-Liouville BV Theory
23. Special functions by Power Series: Bessel, Legendre Polynomials, Riemann Zeta
Read: (optional) Levine and Schwinger (1948) pdf
'Assignment:''BV3: (not assigned)
5/19T5/1L5: 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)
Redo HW0:
-/19R5/6 Instruction Ends
- R5/14 Exam III 7:00-10:00 PM, Room: 1AH-441 (UIUC Official)
-F5/15 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 2008 Syllabus:
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%.