Last Modified : Mon, 25 Feb 19

- Advanced Engineering Mathematics: UIUC; Syllabus: pdf, 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; Supplemental reading: Greenberg

*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

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

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

0/3 | M | 1/15 | MLK Day; no class |

0/3 | M | 1/16 | Classes start |

1/3 | T | 1/16 | L1: 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 |

2 | R | 1/18 | L2: T27. Differential calculus on \(\mathbb{C}\)T28. Cauchy-Riemann Eqs., Complex-Analytic functions are harmonicT34. Series: Maclaurin, Taylor, Laurent [24.3]; Frobenius power series method of solving differential equationsRead: JBA: 58-62; JPD: p. 63-71 |

3/4 | T | 1/23 | L3: rescheduled for Wed, Jan 24, 7-8:30PM Room 443AHT28. Visualizing complex analytic and harmonic functions using zviz Examples from Cleve's Corner, Summer 98,Analytic coloring, dial-a-function and doc, Edgar T30. Analytic function integration: Fundamental theorems of Calculus (FTC) and complex calculus (FTCC)T26. 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. T33. Cauchy integral formula; Riemann Sheets and Branch cuts; Region of Convergence; inverse Laplace transformsRead: HandoutCV2 calculus on analytic functions + CR conditions; Riemann sheets;CV1 due |

4 | R | 1/25 | L4: T30. Integral calculus on \(\mathbb{C}\) T31. \(\int z^{n-1} dz\) on the unit circleContinue discussion of examples of analytic functions: Fundamental Theorem of Complex integration T32. 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] |

5/5 | T | 1/30 | L5: T32. Cauchy's theorem;T33. Cauchy's integral formula [23.5];T35. Cauchy's Residue Theorem [24.5]Read: HandoutCV3 Riemann Sheets, Branch cuts, Filters, PosDef operators, 2-port circuits (ABCD) CV2 due |

6 | R | 3/1 | 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. 841-843 |

7/6 | T | 2/6 | L7: 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 |

8 | R | 2/8 | L8: Cauer synthesis, Bode plots, Network theory (Brune Positive-real (PR) impedance functions)Read: Impedance (Kennely 1893, Schelkunoff 1938) |

9/7 | T | 2/13 | L9: T37. 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: HornsCV5 Transmission linesCV4 due |

10 | R | 2/15 | L10: T38. 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 EnergyCV5 due 2/20 7PM |

11/8 | T | 2/20 | NO CLASS due to Exam I Optional review and special office hours of all CV material. Exam I in room 343AH, 7-10PM |

11 | T | 2/20 | Exam I |

Part II: Linear (Matrix) Algebra (7 Lectures) | |||

1 | R | 2/22 | LA1: T1. Basic definitions: Work, energy/power/work, impedance = Elementary linear operations;T2. Gauss EliminationReview Exam I; Read: 8.1-2, 10.2;LA1 matrix algebra, Gaussian Elimination; eigen analysis |

2/9 | T | 2/27 | LA2: T3. Solutions of \(Ax=b\) by Gaussian elimination, T4. Matrix inverse \(x=A^{-1}b\); Augmented matrix; Gaussian elimination (\(n^3\)); determinants (\(n!\)); Cramer's RuleRead: Linear Alg Handout (pdf) |

3 | R | 3/1 | LA3: Allen out of town T5. Mechanics of Gaussian elimination: \(B = P_n P_{n-1} \cdots P_1 A\) with permutation matrix \(P\) such thatP1: (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 analysisLA1 Due |

4/10 | T | 3/6 | LA4: T7. 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); T6. Transformations (change of basis) Gram-Schmidt proceedureRead: : ... |

5 | R | 3/8 | L5: T5. Asymmetric matrix; T8. Fat/thin and least squares; Eigen-function decomposition; Singular Value Decomposition of Fat/thin systems (SVD, pdf)Read: ... ; Leykekhman Lecture 9Version 1.21 LA3: Fat/Thin systems; Rank-n-Span; Operator symmetryLA2 Due |

0 | FS | 3/9-3/10 | Engineering Open House |

6/11 | T | 3/13 | L6: Taxonomy of matricies; examples of the use of Matrix analysis in engineering: Control theory, MIMO systemsRead: ... Matrix Taxonomy, Eigen-analysis and its applications |

7 | R | 3/15 | L7: Scalar-product \(A \cdot B\), vector-product \(A \times B\), triple-products \(A \cdot A \times B\); Discussion of Vandermode systems Fourier/Laplace/Hilbert-space; Hilbert space and <bra|c|ket> inner product notation LA3 Due |

0/12 | S | 3/19 | Spring Break |

0/13 | M | 3/26 | Instruction Resumes |

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

1/13 | T | 3/27 | L1: T10. Potential fields: \(\Phi(x,y,z,t)\); Notation; scalar & vector field products; Read: Morse-Feshbach Vol 1: FieldsVC1: Topics: Implicit Function Thm; Vector Cross products Taylor series; Vector fields(Due 1 week) |

2 | R | 3/29 | L2: T9. Partial differentiation; smooth vs. analytic functions; Taylor Series; Implicit function Thm, Line surface & volume integrals; Jacobians \(\frac{\partial(x,y,z)}{\partial(u,v,w)}\) as volume conserving transformationsRead: pdf |

3/14 | T | 4/3 | L3: Gradient \(\nabla\), Divergence \(\nabla \cdot\), Curl \(\nabla \times\), Laplacian \(\nabla^2\)Integral vs differential definitions; Integral and conservation laws: Gauss, Green, Stokes, Divergence Vector identies in various coordinate systems; (partial-pdf, pdf) T25. The fundamental thm of vector calculus: \(\mathbf{F}(x,y,z) = \nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)\) (DoC & CoG)Read: Lec 39 pdfVC1 Due; VC2: Vector calculus topics |

4 | R | 4/5 | L4: Differential & integral forms of Grad, Div, Curl; Conservation theorems (Gauss's and Stokes's Laws);T29. incompressible: i.e., \(\nabla \cdot \mathbf{u} =0\) and irrotational \(\nabla \times \mathbf{w} =0\) vector fieldsRead: Handout Lec 40 pdf |

5/15 | T | 4/10 | L5: Maxwell's Equations; Physics of ME; Applications to ME of DoC and CoG Thms. How big is \(\mathbf{B}\) in Teslas?VC2 Due |

0/15 | R | 4/12 | Exam II @ 7-9 PM Room: 343 Alt Hall scores |

R | 4/12 | NO 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) | |||

1/16 | T | 4/17 | L1: T 15. PDEs T 21. Special Equations of Physics: Laplace, Diffusion, Wave; Parabolic, hyperbolic, elliptical: discriminant,Read: Notes Lecture 34, Sect. 1.5.3 ; Symmetry in physicsBV1: Topic: Partial Differential Equations |

2 | R | 4/19 | L2: T 17. Derivation of the wave equation from 2 first-order equations (mass+stiffness)T 18. Webster Horn equation: vs separation of variables method; integration by partsRead: |

3/17 | T | 4/24 | L3: T16. Transmission line theory: Lumped parameter approximation: Diffusion line, Telegraph equationRead: T17. \(2^{nd}\) order PDE: Lecture on: HornsReview: System Postulates BV2: Topics:' Sturm-Liouville; Boundary Value problems; Fourier and Laplace Transforms; BV1 Due |

4 | R | 4/26 | L4: T 19, 20. Sturm-Liouville BV Theory T22. Solutions for 1, 2, 3 dimensions Impedance Boundary conditions; The reflection coefficient and its properties;Read: |

5/18 | T | 5/1 | L5: WKB solution of Horn Equation, T34. ODE's with initial condition (vs. Boundary value problems)L6: T 24. Fourier: Integrals, Transforms, Series, DFT; History: Newton, d'Alembert, Bernoullis, EulerRedo HW0:BV2 Due |

- | W | 5/2 | Instruction Ends |

- | R | 5/3 | Reading Day |

- | 5/5 | Review for Final: 2-4 PM Room 106B3 in Engineering Hall. | |

-/19 | R | 5/8 | Final Exam 1:30-4:30 PM, Room: 441 (UIUC Final Exam Schedule) |

-/19 | F | 5/10 | Finals End |

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