Last Modified : Tue, 22 Mar 22

- Advanced Engineering Mathematics; Syllabus: 2013 Δ, 2011 Δ, 2009 Δ, 2008 Δ, UIUC; Listing: Campus/ECE-493/MATH-487
- Calendars: Class, Campus; Time/place, Website, Contact, Text,: etc
- Office Hours: Tuesday 2-3PM, 330M Everitt Lab (ECE Building), ; and by apt, on Monday 4-5 2137BI; TA: Sarah (srrobin2@illinois.edu); Instructor: Prof. Jont Allen (netID jontalle; Office 2061BI).
- This week's schedule; Final

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

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

0/3 | M | 1/14 | Classes start |

1/3 | T | 1/15 | L1: T25. The fundamental Thm of Vector Fields \(\mathbf{F}(x,y,z) = \nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)\)The frequency domain: Complex \(Z(s) = R(s)+iX(s)\) as a function of complex frequency \(s=\sigma+i\omega\); e.g., \(Z,s \in \mathbb{C}\)), phasors and delay \(e^{-i\omega T}\), \(\log(z)\), \(\sum z^n\)Read: [Ch. 21.1-21.4]HW0: Evaluate your present state of knowledge (not graded) Assignment: CV1 Complex Functions and Laplace transforms |

2 | R | 1/17 | L2: T 27. Differential calculus on \(\mathbb{C}\)compact sets Fréchet and related conceptsT 28. Cauchy-Riemann Eqs., Complex-Analytic functions are harmonic (i.e., irrotational vector fields, where \(\mathbf{A}=0\)) functionsT 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. |

0/4 | M | 1/21 | MLK Day; no class |

3 | T | 1/22 | L3: 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 Transformation (youtube, HiRes), pdf description T *Inverses of Analytic functions (Riemann Sheets and Branch cuts); Analytic coloring, dial-a-function and doc, Edgar, using zviz.m from www.mathworks.com/company/newsletters/news_notes/clevescorner/summer98.cleve.htmlT 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 fieldsRead: [16.10] pp. 826-838 & 841-843;Assignment: CV2; Analytic functions; 30. Integration of analytic functions33. Cauchy integral formula; Riemann Sheets and Branch cuts; Region of Convergence; inverse Laplace transforms |

4 | R | 1/24 | 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: Fundamental Theorem of Complex integration 32. Cauchy's Theorem; 37. Inverse Laplace transforms; 38. Rational fraction expansions, conservative fields; Read: [22.3] |

5/5 | T | 1/29 | L5: T 32.Cauchy's theorem;T 33.Cauchy's integral formula [23.5];T 35. Cauchy's Residue Theorem [24.5]Read: [23.3, 23.5];CV3; |

6 | R | 1/31 | 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. 841-843CV4; |

7/6 | T | 2/5 | L7: Hilbert Transforms and 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] |

8 | R | 2/7 | 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 | 2/12 | 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 Read: [24.2, 24.2] (power series and the ROC);CV5; |

10 | R | 2/14 | L10: 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)Read:'' [24.5, Appendix A]*Riemann zeta function: \(\zeta(s) = \sum_{n=1}^\infty 1/n^s\) *There is also a product form for the Riemann zeta function *Potpourri of other topics |

11/8 | T | 2/19 | NO CLASS due to Exam I Optional review and special office hours, of all the material, will be held in the class room 12:30-2PM 441 AH |

11/8 | T | 2/19 | Exam I Feb 19 Tuesday @ 7-9 PM; Place: 241 Altgeld |

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

1 | R | 2/21 | LA1: 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;LA1; (Solution) |

2/9 | T | 2/26 | LA2: 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 | 2/28 | LA3:*T5. The symmetric matrix: Eigenvectors; The significance of Reciprocity *Mechanics of determinates: \(B = P_n P_{n-1} \cdots P_1 A\) with permutation matrix \(P\) such that P1: (i) <- (i)+a(j); P2: (i) <-> (j); P3: (i)<- a(i) Read: 10.6-10.8, 11.4; LA2: Vector space; Schwartz and Triangular inequalities, eigenspaces |

1/10 | T | 3/5 | Move this to L1 of Vector Calculus (First lecture of 6 following spring Break) L1-VC: Vector dot-product \(A \cdot \)B, cross-product \(A \times \)B, triple-products \(A \cdot A \times \)B, \(A \times (B \times C)\)Gram-Schmidt proceedure* Read:'' 11.4 |

4 | R | 3/7 | L4: 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 LA3: Rank-n-Span; Taylor series; Vector products and fields |

0 | FS | 3/8-3/9 | Engineering Open House |

5/11 | T | 3/12 | L5: T 5. Asymmetric matrix; T; 8. Optimal approximation and least squares; Singular Value DecompositionRead: 9.10, Eigen-analysis and its applications |

6 | R | 3/14 | L6: Fourier/Laplace/Hilbert-space lecture: a detailed study of all the Fourier-like transforms Hilbert space and <bra|c|ket> notation |

0/12 | S | 3/16 | Spring Break |

0/13 | M | 3/25 | Instruction Resumes |

Part III: Vector Calculus (6 lectures) | |||

2/13 | T | 3/26 | L1: T9. Partial differentiation [Review: 13.1-13.4;]; T 10. Vector fields, Path, volume and surface integralsRead: 15-15.3 VC1: Topics: Rank-n-Span; Taylor series; Vector fields, Gradient Vector field topics (Due 1 week) |

3 | R | 3/28 | L2: 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: 13.6,15.4-15.6 |

4/14 | T | 4/2 | L3: 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)Read: 16.1-16.6 VC2: Key vector calculus topics (Due 1 week) |

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

6/15 | T | 4/9 | L5: Applications of Stokes and Divergence Thms: Maxwell's Equations;Potentials and Conservative fields; Review: 16 |

0/15 | R | 4/11 | Exam II @ 7-9 PM Room: 163 EVRT Lab (ECE) |

- | R | 4/11 | 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. | |||

1/16 | T | 4/16 | L1: T 1. PDE: parabolic, hyperbolic, elliptical, discriminantRead: Chapter 18.3; Look at: Emmy Noether, Noether's Thm. I; Examples of Symmetry in physicsBV1: Due Apr 25, 2013: Topic: Partial Differential Equations: Separation of variables, BV problems, use of symmetry |

2 | R | 4/18 | 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] |

3/17 | T | 4/23 | 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 | R | 4/25 | L3: T 20. Sturm-Liouville BV Theory 23. Special functions by Power Series: Bessel, Legendre Polynomials, Riemann Zeta Read: 20BV2: Sturm-Liouville, Boundary Value problems, Fourier and Laplace Transforms; Hints for problems 3+5 and 4. |

5 | T | 4/30 | L5: 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 solution to HW7T 40. ODE's with initial condition (vs. Boundary value problems) Di and Gilbert (1993)L6: T 24. Fourier: Integrals, Transforms, Series, DFTRead: 17.3-17-6Redo HW0: |

- | R | 5/1 | Instruction Ends |

- | F | 5/2 | Reading Day |

-/19 | R | 5/9 | Exam III 7:00-10:00+ PM, Room: 441AH (UIUC dictate) |

-/19 | F | 5/10 | 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 (Prof's Discuression), count for 10%.

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