Last Modified : Fri, 19 Oct 12

- Advanced Engineering Mathematics; Syllabus: 2011 Δ, 2009 Δ, 2008 Δ, UIUC; Listing: Campus/ECE-493/MATH-487
- Calendars: Class, Campus; Time/place, Website, Contact, Text: etc B106-B6 Eng Hall
- Office Hours: Friday 1-2 Room 330M Everitt Lab (ECE Building); TA: Po-Sen Huang (huang146@illinois.edu)
- This week's schedule

L/W | D | Date | Integrated Lectures on Mathematical Physics |
---|---|---|---|

0/4 | M | 1/17 | MLK Day; no class |

Part I: Complex Variables (10 lectures) | |||

1/4 | T | 1/18 | L1: T25. 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, Phasers and delay {$e^{-i\omega T}$}, {$\log(z)$}, {$\sum z^n$}T26. Singularities (i.e., poles, branch cuts and transformations);Mobius Transformation (youtube, HiRes), pdf description Read: [Ch. 21.1-21.4]HW0: Evaluate your present state of knowledge (not graded) HW8 Complex Functions and Laplace transforms (Solution) |

2 | R | 1/20 | L2: T 27. Differential calculus on {$\mathbb{C}$}T 28. Cauchy-Riemann Eqs., Analytic functions, Harmonic functionsRead: [21.5] and verify that you can do all the exercises on page 1113. |

3 | T | 1/25 | L3: Inverses of Analytic functions;T 29. Irrotational fields (e.g., velocity potential {$\mathbf{u} = \nabla \phi(x,y,z)$}) [p.~829];T 28. Discussion on CR conditions: Computing {$\Re f(z)$} and {$\Im f(z)$};Analytic coloring, via matlab, using zviz.m; Read: [16.10] pp. 826-838;HW9 Analytic functions; 30. Integration of analytic functions; 33. Cauchy integral formula; Riemann Sheets and Branch cuts; Region of Convergence; inverse Laplace transforms; (Solution) |

4 | R | 1/28 | L4: T 30.Integral calculus on {$\mathbb{C}$} T 31. {$\int z^{n-1} dz$} on the unit circleContinue discussion of examples of analytic functions, 33. Cauchy integral formula, 37. inverse Laplace transforms, 38. Rational fraction expansions, conservative fields; Boas' method for computing {$Z(s)$} given {$R(s) \equiv \Re Z(s)$} pdf Δ from R.P. Boas, Invitation to Complex Analysis Random House 1987Read: [22.3] |

5 | T | 2/1 | L5:T 32.Cauchy's theorem; T 33.Cauchy's integral formula [23.5]; Read: [23.3, 23.5];HW10 (Solution) |

6 | R | 2/3 | L6a: Contour integration and Inverse Laplace TransformsExamples 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. 841-843HW11; (Solution) |

7 | T | 2/8 | L7: Hilbert Transforms and the Cauchy Integral formula: The difference between the Fourier transform {$2{\tilde u}(t) \leftrightarrow 1+1/j\omega$} and the Laplace {$2u(t) \leftrightarrow 1/s$}Review of Residues (Examples) and their use in finding solutions to integrals; T 34. Series: Maclaurin, Taylor, Laurent [24.3]T 36.Jordan's LemmaRead: [24.3] |

8 | R | 2/10 | L8: 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 {$Z(s,x=0)$} |

9/7 | T | 2/15 | L9: 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 T 35. Cauchy's Residue Theorem [24.5]Read: [24.2, 24.2] (power series and the ROC);HW12; (Solution) |

10 | R | 2/17 | L10: T 38. Rational Impedance (Pade) approximations: {$Z(s)={a+bs+cs^2}/({A+Bs})$}Partial fraction: {$Z(s) = \sum a_i/(s-s_i)$} and Continued fractions: {$Z(s)=s+a/(s+b/(s + c/(s+\cdots)))$} expansions Read: [24.5] |

0/8 | T | 2/22 | NO CLASS Optional office hours for review, during class time |

0/8 | T | 2/22 | Exam I Feb 22 Tuesday @ 7-9 PM; Place: 1MEB 135 |

Part II: Linear (Matrix) Algebra (6 lectures) | |||

1 | R | 2/24 | L1: T 1. Basic definitions, Elementary operations;T 2. Cramer's Rule, Determininants, Inverse Matrix, Aug Matrix and Gauss Elimination; VandermondeReview Exam I; Read: 8.1-2, 10.2;HW1, (Solution) |

2/9 | T | 3/1 | L2: T 3. Solutions to {$Ax=b$} by Gaussian elimination, T 4. Matrix inverse {$x=A^{-1}b$}; Cramer's RuleRead: 8.3, 10.4 ; |

3 | R | 3/3 | L3: T 5. The symmetric matrix: Eigenvectors; T 6. Transformations (change of basis);Read: 10.6-10.8HW2: Vector space; Schwartz and Triangular inequalities, eigenspaces (Solution) |

4/10 | T | 3/8 | L4: T 7. Vector spaces in {$\mathbb{R}^n$}; Innerproduct+Norms; Ortho-normal; Span and Perp ({$\perp$}); Schwartz and Triangular inequalities Read: 9.1-9.6, 10.5, 11.1-11.3 |

5 | R | 3/10 | L5: Gram-Schmidt proceedure; Vector dot-product {$A \cdot B$}, cross-product {$A \times B$}, triple-products {$A \cdot A \times B$}, {$A \times (B \times C)$};Read: 9.10, 11.4 ; Leykekhman Lecture 9HW3 Rank-n-Span; Taylor series; Vector products and fields (Solution) |

0 | R | 3/11-3/12 | Engineering Open House |

6/11 | T | 3/15 | L6: T 5. Asymmetric matrix; T; 8. Optimal approximation and least squares; Singular Value Decomposition |

Part III: Vector Calculus (5 lectures) | |||

1 | R | 3/17 | L1: T9. Partial differentiation [Review: 13.1-13.5;]; T 10. Vector fields, Path, volume and surface integralsHW3-b:;Symmetric and non-symmetric matrices, eigenvectors, Singular value decomposition (Solution) Read: 15 |

0/ | S | 3/18 | Spring Break |

0/13 | M | 3/28 | Instruction Resumes |

2 | T | 3/29 | L2: Vector fields: {${\bf R}(x,y,z)$}, Change of variables under integration: Jacobians!! Read: 13.6 HW4: Key vector calculus topics (Solution) |

3 | R | 3/31 | L3: Gradient {$\nabla$}, Divergence {$\nabla \cdot$}, Curl {$\nabla \times$}, Scaler (and vector) Laplacian {$\nabla^2$};Vector identies in various coordinate systemsNotes (pdf, djvu)Read: 16.1-16.6 |

4/14 | T | 4/5 | L4: Integral and conservation laws: Gauss, Green, Stokes, DivergenceRead: 16.8-16.10 |

5 | R | 4/7 | L5: Applications of Stokes and Divergence Thms: Maxwell's Equations;Potentials and Conservative fields; Review: all 16 |

0/15 | T | 4/12 | Exam II Apr 12 Tues @ 7-9 PM Room: 163 Everitt Lab |

- | T | 4/12 | 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 | R | 4/14 | L1: T 1. PDE: parabolic, hyperbolic, elliptical, discriminantRead: Chapter 18.3; Look at: Emmy Noether, Noether's Thm. I; Examples of Symmetry in physicsHW6: Separation of variables, BV problems, symmetry (Solution) |

2/16 | T | 4/19 | L2: 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 | R | 4/21 | L4: T 16. Transmission line theory: Lumped parameter approximations17. {$2^{nd}$} order PDE: Lecture on: HornsRead:[17.7, pp.~ 887, 965, 1029, 1070, 1080] |

4/17 | T | 4/26 | L3: T 20. Sturm-Liouville BV Theory: Allen out of town; Prof. Levinson to lecture23. Special functions by Power Series: Bessel, Legendre Polynomials, Riemann Zeta Read: 20HW7: Sturm-Liouville, Boundary Value problems, Fourier and Laplace Transforms; (Solution); Hints for problems 3+5 and 4. |

5 | R | 4/28 | L5: T 24. Fourier: Integrals, Transforms, Series, DFTRead: 17.3-17-6 |

6/18 | T | 5/3 | L6: T Solutions to several geometries for the wave equation (Strum-Liouville cases)Read: Ch. 20, 5.1-5.3 + Review p.290-1; Study: the solution to HW7T 40. ODE's with initial condition (vs. Boundary value problems) Di and Gilbert (1993) ΔRedo HW0: |

- | W | 5/4 | Instruction Ends |

- | R | 5/5 | Reading Day |

- | T | 5/10 | Exam III 7:00-10:00+ PM on HW1-HW11 (Room: EH 106B3) |

-/19 | F | 5/13 | Finals End |

**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, count for 10%.

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