 ## ECE493-F20

• Advanced Engineering Mathematics: Syllabus: pdf-2018, pdf-2009; Listing: ECE-493, (Math-487) Campus: ECE/Math
• Calendars: Class, Campus;
• Time/place: Zoom and 1090 Lincoln Hall, 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 Zoom; Friday 3-4:30PM, Zoom
• Instructor: Prof. Jont Allen (netID jontalle; Office 3062ECEB); TA: TBD
• COVID-19 on-campus testing twice per week is required for ALL faculty, staff, and students who are on-campus for any length of time.
Mandatory testing begins on August 10 for faculty and staff and on August 16 for students. See a list of on-campus testing sites here.
• This week's Reading schedule; Final

## ECE-493/MATH-487 Daily Schedule Spring 2020

Part I: Reading assignments: Complex algebra (15 Lec)
WeekMWF
1L1: 1, 3.1 (p. 1-11) Intro + historyL2: 3.1,.1,.2 (p. 51-61) Roots of polynomials; Newton's method.L3: 3.1.3,.4 (p.61-64) Companion maxtrix
2L4: 3.2,.1,.2, B1, B3 EigenanalysisL5: 3.2.3 Taylor seriesL6: 3.2,.4,.5 Analytic Functions
3Labor dayL7: 3.4,.1,.2 Anal Geom, ResiduesL8: 3.5.1-.4 $$\cdot, \times, \wedge$$ scalar products
4L9: 3.5.5, 3.6,.1-.5 Gauss Elim; Matrix algebra of systemsL10: 3.8,.1-.4 Thevenin parameters; Transmission lines; impedance matrixL11: 3.9,.1 $$\cal FT$$ of signals
5L12: 3.10,.1-.3 $$\cal LT$$ of systems + postulatesL13: 3.11,.1,.2 Complex analytic color maps; Riemann sphere; Bilinear transformL14: Review for Exam I
6 Exam I
L/WDDateLectures on Mathematical Physics and its History
Part I: Complex algebra (15 Lectures)
-/35M8/24 Instruction begins
1M8/24L1: Introduction + History
Assignment: NS1, Problems 1,2,4, 7; p. 36, Due 1 week:
2W8/26L2: Newton's method for finding roots of a polynomial $$P_n(s_k)=0$$ (p.56).
3F8/28L3: The companion matrix and its characteristic polynomial
Working with Octave/Matlab: 3.1.4 (p. 63-64) zviz.m
3.11 Brief introduction to colorized plots of complex mappings (p. 123-125)
4/36M8/31L4: Eigenanalysis I: Eigenvalues and vectors of a matrix
Assignment: AE1 (p. 75-78) Probs: 1-11 (Due 1 wk)
NS1 due
5W9/2L5: Taylor series
6F9/4L6: Analytic functions; Complex analytic functions; Brune Impedance
Residue expansions of ratios of polynomials: $$Z(s)=N(s)/D(s)$$
-/37M9/7 Labor day: Holiday
7W9/9L7: Analytic geomerty: Vectors and their dot $$\cdot$$, cross $$\times$$ and wedge $$\wedge$$ products. Residues.
Colorized plots of complex mappings
Assignment: AE2, Due 1 week
AE1 due
View: Mobius/bilinear transform video
8F9/11L8: Analytic geometry of two vectors (generalized scalar product)
Inverse of 2x2 matrix
9/38M9/14L9: Gaussian Elimination; Permutation matricies
10W9/16L10: Transmission and impedance matricies
Assignment: AE3
AE2 due
11F9/18L11: Fourier transforms of signals
12/39M9/21L12: Laplace transforms of systems;System postulates
13W9/23L13: Comparison of Laplace and Fourier transforms; Colorized plots; Probability -- basic definitions
AE3 due
14F9/25L14: Review for Exam I
15/40M9/28NO Class; Exam I, 7-10PM Rm ??447AH?? Exam 1 Grade Distribution F19
Part II: Reading assignments: Scalar differential equations (10 Lec)
WeekMWF
6Exam I brief discussionL1: 4.1,4.2,.1 (p. 133) Fundmental Thms of $$\mathbb R, \mathbb C$$ scalar calculusL2: 4.2.2 Cauchy-Riemann Eqs. CR-1, CR-2, CR-3, CR-4
7L3: 4.4 Brune impedance/admittanceL4: 4.4,.1,.2 Complex analytic ImpedanceL5: 4.4.3 Multi-valued functions, Branch cuts
8L6: 4.5,.1,.2 Cauchy's complex integration thms CT1, CT2, CT3L7: 4.7,.1,.2 Inv $${\cal LT} (t<0, t=0)$$L8: 4.7.3 Inv $${\cal LT} (t > 0)$$
9L9: 4.7.4 Properties of the $$\cal LT$$L10: 4.7.5 Solving LTI (simple) Diff. Eqs. with the $$\cal LT$$

Edits 2 here

 Part II: Scalar (ordinary) differential equations (10 Lectures) 1 W 9/30 L1: The fundamental theorems of scalar and complex calculus Assignment: DE1 2 F 10/2 L2: Complex differentiation and the Cauchy-Riemann conditionsProperties of complex analytic functions (Harmonic functions)Taylor series of complex analytic functions 3/41 M 10/5 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 functions 4 W 10/7 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 Due 5 F 10/9 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 6/42 M 10/12 L6: Three Cauchy integral theorems: CT-1, CT-2, CT-3How to calculate the residue 7 W 10/14 L7: Inverse Laplace transform ($$t<0$$), Application of CT-3DE2 DueAssignment: DE3 8 F 10/16 L8: Inverse Laplace transform ($$t\ge0$$) CT-3 9/43 M 10/19 L9: Properties of the Laplace transform Linearity, convolution, time-shift, modulation, derivative etcDifferences between the FT and LT; System postulates 10 W 10/21 L10: Solving differential equations: Train problem (DE3, problem 2, p. 167-8, Fig. 4.11)DE3 Due
Part III: Reading assignments: Vector calculus (10 lectures)
WeekMWF
10 L1: 5.1.1 (p. 171) Fields and potentials
11L2: 5.1.2,.3 $$\nabla(), \nabla \cdot(), \nabla \times(), \nabla \wedge(), \nabla^2()$$L3: 5.2 Field evolutionL4: 5.2.1 Scalar wave Eq.
12L5: 5.2.2,.3,5.4.1-.3 HornsL6: 5.5.1 Solution methodsL7: 5.6.1-.4 Integral forms of $$\nabla(), \nabla \cdot(), \nabla \times()$$
13L8: 5.6.5 Helmholtz decomposition thm
$$\vec{E} = -\nabla\phi +\nabla \times \vec A\$$
L9: 5.6.6 2d-order scalar operators: $$\nabla^2 = \nabla \cdot \nabla()$$,
vector operators: $${\mathbf\nabla}^2 = \nabla \cdot \mathbf\nabla(), \nabla \nabla \cdot(), \nabla \times \nabla()$$;
null operators: $$\nabla \cdot \nabla \times()=0, \nabla \times \nabla ()=0$$
Exam II
 Part III: Vector Calculus (10 Lectures) 1 F 10/23 L1: Properties of Fields and potentialsAssignment: VC1 2/44 M 10/26 L2: Gradient $$\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$$ dimensions 3 W 10/28 L3: Field evolution for partial differential equations 4 F 10/30 L4: Scalar wave equation (Acoustics) 5/45 M 11/2 L5: Webster Horn equation Three examples of finite length hornsSolution methods; Eigen function solutionsAssignment: VC2; VC1 Due 6 W 11/4 L6: Integral forms of $$\nabla()$$, $$\nabla\cdot()$$ and $$\nabla \times()$$ 7 F 11/6 L7: Helmholtz decomposition 8/46 M 11/9 L8: second order operators DoG, GoD, gOd, DoC, CoG, CoC 9 W 11/11 L9: review home work solutions and discuss exam II F 11/13 No Class: Exam II @ 7-10 PM; Room: ??447AH?? (Alt Hall) Exam2 Grade Distribution
Part IV: Reading assignments: Maxwell's equations + solutions (7 lectures)
WeekMWF
13L1: 5.7 (p. 203) Unification of E&M: terminologyL2: 5.7.1 Maxwell's equationsL3: 5.7.2 Derivation of ME
14Thanksgiving Holiday
15L4: 5.8 Use of Helmholtz' Thm on MEL5: 5.8 Helmholtz solutions of MEL6: 5.8 Analysis of simple impedances (Inductors & capacitors)
16L7: TEM, TE, TM modes in waveguides (horns)L8: Wave-filters; Final ReviewThur: Optional Review for Final; Reading Day
 Part IV: Maxwell's equation with solutions 1/47 M 11/16 L1: Unification of E & M; terminology (Tbl 5.4) View: Symmetry in physics 2 W 11/18 L2: Derivation of the wave equation from Eqs: EF and MF Webster Horn equation: vs separation of variables method + integration by parts 3 F 11/20 L3: Transmission line theory: Lumped parameter approximation:Diffusion line, Telegraph equation, Wave equation (Parabolic, hyperbolic, elliptical) -/47 S 11/21 Thanksgiving Break -/49 M 11/30 Instruction Resumes 4 M 11/30 L4: Helmholtz' Thm: The fundamental thm of vector calculus: $$\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 fieldsVC2 Due 5 W 12/2 L5: Properties of 2d-order operators 6 F 12/4 L6: Derivation of the vector wave equation 7/50 M 12/7 L7: Physics and Applications; ME vs quantum mechanics 8 W 12/9 L8: Sturm-Liouville Horn Theory Solutions for 1, 2, 3 dimensions (seperation of variables)Impedance Boundary conditions; The reflection coefficient and its properties;
 - R 12/10 Reading Day - R 12/10 Review for Final: 2-4 PM Room: 106B3 in Engineering Hall 12/?? Exam 3 grade distribution - F Friday Dec 13 7-10 PM Final Exam: Room: 445AH ( -/51 F 12/18 Finals End F 12/18 Final grade distribution; Letter Grade: 100-93 A+; 92=84 A; 83-79 A-; 78-75 B+

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