 ## ECE493-F19

• Advanced Engineering Mathematics: Syllabus: pdf-2018, pdf-2009; Listing: ECE-493, (Math-487) Campus: ECE/Math
• Calendars: Class, Campus;
• Time/place: Altgeld 445, MWF 10:00-10:50, ECE-493;
• Matlab Tutorial pdf
• Text: An Invitation to Mathematical Physics pdf; Greenberg: Advanced Eng Mathematics Greenberg
* Office Hours: Tues 2:30-4:00PM 4036 ECEB; Friday 3-4:30PM, 4036 ECEB
• Instructor: Prof. Jont Allen (netID jontalle; Office 3062ECEB); TA: TBD
• This week's schedule; Final

### ECE-493/MATH-487 Daily Schedule Fall 2019

L/WDDateLectures on Mathematical Physics and its History
Part I: Complex algebra (15 Lectures)
-/35M8/26 Instruction begins
1M8/26L1: Algebraic Equation
Assignment: NS1, Problems 1,2,4 7; p. 36, Due 1 week:
2W8/28L2: Finding roots of polynomials
Read: 3.0, 3.1, 3.1.2 (pp. 71-81)
3F8/30L3: Matrix formulation of polynomials
Working with Octave/Matlab: 3.1.4 zviz.m
Read: 3.1.3-3.1.4 (pp. 81-84), 3.10 (pp. 143-147):
Brief introduction to colorized plots of complex mappings
-/36M9/2 Labor day: Holiday
4W9/4L4: Eigenanalysis I: Eigenvalues and vectors of a matrix
Assignment: AE1 Probs: 1, 2, 3, 4, and 3 questions each from probs 5-11?; p. 95; Due 1 wk
NS1 due
Read: 3.2, 3.2.1; B.1 (p. 267-269), B.3
5F9/6L5: Taylor series
6/37M9/9L6: Analytic functions; Complex analytic functions; Brune Impedance
Residue expansions of ratios of polynomials: $$Z(s)=N(s)/D(s)$$
7W9/11L7: Analytic geomerty: Vectors and their dot $$\cdot$$, cross $$\times$$ and wedge $$\wedge$$ products.
More on colorized plots of complex mappings
Assignment: AE2, Due 1 week
AE1 due
Read: 3.5, 3.5.1; 3.10 (colorized plots, p. 143) Mobius/bilinear transform video
8F9/13L8: Analytic geometry of two lines
Inverse of 2x2 matrix
9/38M9/16L9: Gaussian Elimination; Permutation matricies
10W9/18L10: Transmission and impedance matricies
Formulation of a transmission line (Figs. 3.9, 3.11, 4.10)
Assignment: AE3
AE2 due
11F9/20L11: 3.8: Fourier transforms of signals
12/39M9/23L12: 3.9: Laplace transforms of systems
System postulates
13W9/25L13: Comparison of Laplace and Fourier transforms (p. 134; 277-279)
What you need to know about probability (p. 142)
AE3 due
14F9/27NO Class Allen out of town
15/40M9/30L15: Review for Exam I; Exam I, 7-10PM Rm 447AH Exam 1 Grade Distribution
 Part II: Scalar (ordinary) differential equations (10 Lectures) 1 W 10/2 L1: The fundamental theorems of scalar and complex calculus Assignment: DE1 Read: 4.2, 4.2.1 (p. 154) 2 F 10/4 L2: Complex differentiation and the Cauchy-Riemann conditionsProperties of complex analytic functions (Harmonic functions)Taylor series of complex analytic functionsRead: 4.2.2 3/41 M 10/7 L3: Brune impedance/admittance and complex analyticRatio of polynomials of similar degree: $$Z(s) = {P_n(s)}/{P_m(s)}$$ with $$n,m \in {\mathbb N}$$ Basic properties of impedance functions (postulates) (e.g., causal, positive real) Complex analytic impedance/admittance is conservative (P3)Colorized plots of Impedance/Admittance functionsRead: 4.4+ 4 W 10/9 L4: Generalized impedanceBrune vs. generalized impedance/admittance functions (ratio of polynomials; branch cuts)Examples of Colorized plots of Generalized Impedance/Admittance functionsCalculus on complex analytic functionsAssignment: DE2DE1 DueRead: 4.4.1 5 F 10/11 L5: Multi-valued complex analytic functionsBranch cuts and their properties (e.g., moving the branch cut)Examples of multivalued functionColorized plots of multivalued functions: e.g.: $$F(s) = \sqrt{s e^{jk2\pi}}$$ where $$k\in{\mathbb N}$$ is the sheet index Read: 4.4.2, 4.4.3 6/42 M 10/14 L6: Three Cauchy integral theorems: CT-1, CT-2, CT-3How to calculate the residueRead: 4.5+ 7 W 10/16 L7: Inverse Laplace transform ($$t<0$$), Application of CT-3DE2 DueAssignment: DE3Read: 4.7+ 8 F 10/18 L8: Inverse Laplace transform ($$t\ge0$$) CT-3 Read: 4.7.1 9/43 M 10/21 L9: Properties of the Laplace transform Linearity, convolution, time-shift, modulation, derivative etcSolving differential equations: Train problem (DE3, problem 2, p. 191, Fig. 4.10)Read: 4.7.2, 4.7.3 10 W 10/23 L10: Differences between the FT and LT; System postulates 3.9.1DE3 Due
 Part III: Vector Calculus (9 Lectures) 1 F 10/25 L1: Properties of Fields and potentialsRead: 5.1Assignment: VC1 2/44 M 10/28 L2: 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;Laplacian in $$N$$ dimensionsRead: 5.1.1, 5.1.2 3 W 10/30 L3: Field evolution for partial differential equations Read: 5.2+ 4 F 11/1 L4: Scalar wave equation (Acoustics)Read: 5.3+ 5/45 M 11/4 L5: Webster Horn equation Three examples of finite length hornsSolution methods; Eigen function solutionsRead: 5.4+, 5.6+ Assignment: VC2VC1 Due 6 W 11/6 L6: Integral forms of $$\nabla()$$, $$\nabla\cdot()$$ and $$\nabla \times()$$ Read: 5.7-5.7.4 7 F 11/8 L7: Helmholtz decomposition Read: 5.7.5 8/46 M 11/11 L8: second order operators DoG, GoD, gOd, DoC, CoG, CoC Read: 5.7.6 W 11/13 NO Lecture due to Exam II; Class time will be converted to optional Office hours, to review home work solutions and discuss exam 9/46 W 11/13 Exam II @ 7-10 PM; Room: 447AH (Alt Hall) Exam2 Grade Distribution
 Part IV: Maxwell's equation and their solution 1 F 11/15 L1: Unification of Electricity and MagnitismRead: 5.8+, Symmetry in physics 2/47 M 11/18 L2: Derivation of the wave equation from 2 first-order equationsWebster Horn equation: vs separation of variables method; integration by partsRead: 5.8.2; Greenberg pp. ?? 3 W 11/20 L3: Transmission line theory: Lumped parameter approximation:Diffusion line, Telegraph equation, Wave equation (Parabolic, hyperbolic, elliptical)Read: $$2^{nd}$$ order PDE: Horns Sturm-Liouville (SL) Boundary Value (BV) problems; Read: Greenberg pp. ???, 5.2 4 F 11/22 L4: Sturm-Liouville BV Theory Solutions for 1, 2, 3 dimensions (seperation of variables)Impedance Boundary conditions; The reflection coefficient and its properties;Read: 4.4.1VC2 Due -/47 S 11/23 Thanksgiving Break -/49 M 12/2 Instruction Resumes 5/49 M 12/2 L6: The fundamental thm of vector calculus: $$\mathbf{F}(x,y,z) = \nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)$$ incompressible: i.e., $$\nabla \cdot \mathbf{u} =0$$ and irrotational $$\nabla \times \mathbf{w} =0$$ vector fieldsRead: 5.7.5 pages 228-230 6 W 12/4 L:7 Second order operatorsRead: 5.7.6 pages 229-233 7 F 12/6 L:8 Derivation of the vector wave equationRead: p. 233-234 8/50 M 12/9 L8: Maxwell's Equations: Physics and Applications; Some thoughts on quantum mechanics Read: p. 5.9 page 233-237 9 W 12/11 L9: Maxwell's Equations: Physics and Applications; Some thoughts on quantum mechanicsInstruction ends 9 W 12/21 Exam 3 grade distribution 9 W 12/22 Final grade distribution; Letter Grade: 100-93 A+; 92=84 A; 83-79 A-; 78-75 B+
 - R 12/12 Reading Day - 12/12 Review for Final: 2-4 PM Room: 106B3 in Engineering Hall - F Friday Dec 13 7-10 PM Final Exam: Room: 445AH ( -/51 F 12/20 Finals End

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