Last Modified : Wed, 05 Aug 15

- 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

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

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

0/4 | M | 1/19 | MLK Day; no class |

0/4 | M | 1/20 | Classes start |

1/4 | T | 1/20 | L1: 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 $e^{-i\omega T}$, $\log(z)$, $\sum z^n$ Read: [Ch. 21.1-21.4]HW0: Evaluate your present state of knowledge (not graded) Assignment: CV1: Complex Functions and Laplace transforms CV1-sol.pdf |

2 | R | 1/22 | L2: T 27. Differential calculus on $\mathbb{C}$T 28. Cauchy-Riemann Eqs., Complex-Analytic functions are harmonic (i.e., irrotational vector fields (i.e., $\mathbf{A}=0$) 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. |

3/5 | T | 1/27 | L3: 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 fieldsRead: [16.10] pp. 826-838 & 841-843;Assignment: CV2; |

4 | R | 1/29 | L4: T 30.Integral calculus on $\mathbb{C}$; T 30. Integration of analytic functionsT 31. $\int z^{n-1} dz$ on the unit circle; T 32. Cauchy's TheoremT 33. Cauchy integral formula [23.5]; Riemann Sheets and Branch cuts; Region of ConvergenceT 37. Inverse Laplace transforms; Residue theoremRead: [22.3] Bilinear (M"obius) transformation |

5/6 | T | 2/3 | L5: 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 Read: [23.3, 23.5];CV3 |

6 | R | 2/5 | L6a: 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 |

7/7 | T | 2/10 | L7: 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 Read: [17.1-17.10]CV4 |

8 | R | 2/12 | L8: 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) |

9/8 | T | 2/17 | L9: 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 Read: [24.3]CV5 |

10 | R | 2/19 | 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]*Potpourri of other topics |

11/9 | T | 2/24 | NO CLASS due to Exam I Optional review and special office hours, of all the material 12:30-2PM 441 AH |

11/9 | T | 2/24 | Exam I Feb 24 Tuesday @ 7-9 PM; Place: 441 Altgeld |

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

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

1/9 | R | 2/26 | L1: T 1. Basic definitions, Elementary operations;T 2. Inverse Matrix, Aug Matrix and Gauss Elimination; VandermondeReview Exam I; Read: 8.1-2, 10.2;Assignment: LA1 |

2/10 | T | 3/3 | LA2: T 3. Solutions to $Ax=b$ by Gaussian elimination; DeterminantsT 4. Matrix inverse $x=A^{-1}b$; Cramer's Rule; Gram-Schmidt proceedureRead: 8.3, 10.4 ; |

3 | R | 3/5 | L3:*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) Read: 10.6-10.8, 11.4; Assignment: LA2: Vector space; Schwartz and Triangular inequalities, eigenspaces |

4/11 | T | 3/10 | 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 Assignment: LA3: Rank-n-Span; Taylor series; Vector products and fields |

5 | R | 3/12 | L5: 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 |

FS | 3/13-14 | Engineering Open House | |

6/12 | T | 3/17 | L6: 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/W | D | Date | Integrated Lectures on Mathematical Physics |
---|---|---|---|

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

1/12 | R | 3/19 | L1: 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: $F = -\nabla \Phi + \nabla \times A$ Read: 11.4 |

0/12 | S | 3/21 | Spring Break |

0/14 | M | 3/30 | Instruction Resumes |

2 | T | 3/31 | L2: T 9. 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 | 4/2 | L3: 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 |

4/15 | T | 4/7 | L4: 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 | 4/9 | L5: Integral and conservation laws: Gauss, Green, Stokes, DivergenceLook at: Emmy Noether, Noether's Thm. I; Examples of Symmetry in physicsRead: 16.8-16.10 |

6/16 | T | 4/14 | L6: Applications of Stokes and Divergence Thms: Maxwell's EquationsPotentials and Conservative fields; Review: 16 |

0 | R | 4/16 | Exam II April 16 @ 7-9 PM Room: TBD |

0 | R | 4/16 | NO Lecture due to Exam II;Class time will be converted to optional Office hours, to review home work solutions and discuss exam |

L/W | D | Date | Integrated 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. | |||

1/17 | T | 4/21 | L1: T 1. PDE: parabolic, hyperbolic, elliptical, discriminantRead: Chapter 18.3Assignment: BV1: Due Apr 30, 2015: Topic: Partial Differential Equations: Separation of variables, BV problems, use of symmetry |

2 | R | 4/23 | L2: 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 |

3/18 | T | 4/28 | L3: T 16. Transmission line theory: Lumped parameter approximations; 1D wave equations17. $2^{nd}$ order PDE: Lecture on: HornsIntegration 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. |

4 | R | 4/30 | L4: T 20. Sturm-Liouville BV Theory 23. Special functions by Power Series: Bessel, Legendre Polynomials, Riemann Zeta Read: 20Read: (optional) Levine and Schwinger (1948) pdf'Assignment:'' BV3: (not assigned) |

5/19 | T | 5/1 | L5: 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)Read: 17.3-17-6Redo HW0: |

-/19 | R | 5/6 | Instruction Ends |

- | F | 5/7 | Reading Day |

- | R | 5/14 | Exam III 7:00-10:00 PM, Room: 1AH-441 (UIUC Official) |

- | F | 5/15 | 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 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%.

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