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

Part I: Reading assignments and videos: Complex algebra (15 Lecs)
WeekMWF
1L1: 1, 3.1 (Read p. 1-17) Intro + history
(Lec 1)
L2: 3.1,.1,.2 (p. 69-84) Roots of polynomials; Newton's method.L3: 3.1.3,.4 (p.84-88) Companion maxtrix
(Lec 3)
2L4: 3.2,.1,.2, B1, B3 Eigenanalysis
(Lec 4)
L5: 3.2.3 Taylor series
(Lec 5)
L6: 3.2,.4,.5 Analytic Functions, Residues, Convolution
(Lec 6)
3Labor dayL7: 3.5, Anal Geom, Generalized scalar products
(Lec 7)
L8: 3.5.1-.4 $$\cdot, \times, \wedge$$ scalar products (Lec 8)
4L9: 3.5.5, 3.6,.1-.5 Gauss Elim; Matrix algebra (systems)
(Lec 9)
L10: 3.8,.1-.4 Thevenin parameters; Transmission lines; impedance matrix
(Lec 10 No audio!)
L11: 3.9,.1 $${\cal FT}$$ of signals
(Lec 11)
5L12: 3.10,.1-.3 $$\cal LT$$ of systems + postulates
(Lec 12-zoom, Lec 12-360)
L13: 3.11,.1,.2 Complex analytic color maps; Riemann sphere; Bilinear transform
(Lec 13-zoom, Lec 13-360)
L14: Review for Exam I
Lec 14-zoom Lec 14-360
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, p. 26, Problems 1, 2, 4, 7; Due 1 week: NS1-sol
2W8/26L2: 3.1.2 (p. 74) Newton's method for finding roots of a polynomial $$P_n(s_k)=0$$ Newton's method; All m files: Allm.zip
3F8/28L3: The companion matrix and its characteristic polynomial
Working with Octave/Matlab: 3.1.4 (p. 86) zviz.m
3.11 (p. 167) Brief introduction to colorized plots of complex mappings
4/36M8/31L4: Eigenanalysis I: Eigenvalues and vectors of a matrix
Assignment: AE1.pdf, Probs: 1-11 (Due 1 wk); Soluton: AE1-sol
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.pdf, Due 1 week; Soluton: AE2-sol
AE1 due
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.pdf Due 1 week;Soluton: AE3-sol
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;View: Mobius/bilinear transform video
AE3 due
14F9/25L14: Review for Exam I
15/40M9/28Exam I; Start time: Any 2 hour period between 8AM-10AMStop time: 11AM; 3017ECEB for locals; Zoom for remotes NO 360
Part II: Reading assignments: Scalar differential equations (10 Lec)
WeekMWF
6Exam I brief discussionL1: 4.1,4.2,.1 (p. 178) Fundmental Thms of $$\mathbb R, \mathbb C$$ scalar calculus
(Lec 1-II: zoom)
L2: 4.2.2 Cauchy-Riemann Eqs. CR-1, CR-2, CR-3, CR-4
(Lec 2-II: 360)
(Lec 3-II: 360)
L4: 4.4,.1,.2 Complex analytic Impedance
(Lec 4-II: 360, zoom)
L5: 4.4.3 Multi-valued functions, Branch cuts
(Lec 5-II: 360, 360dir, zoom)
8L6: 4.5,.1,.2 Cauchy's complex integration thms CT1, CT2, CT3
(Lec 6-II: 360, zoom)
L7: 4.7,.1,.2 Inv $${\cal LT} (t<0, t=0)$$
(Lec 7-II: 360, zoom)
L8: 4.7.3 Inv $${\cal LT} (t > 0)$$
(Lec8-II:360dir, (Lec 8-II: 360)
9L9: 4.7.4 Properties of the $$\cal LT$$ (Lec 9-II: 360, zoom)L10: 4.7.5 Solving LTI (simple) Diff. Eqs. with the $$\cal LT$$ (Lec10-II:360dir (@5:30), zoom)
 Part II: Scalar (ordinary) differential equations (10 Lectures) 1 W 9/30 L1: The fundamental theorems of scalar and complex calculus Assignment: DE1.pdf, (Due 1 wk); DE1-sol.pdf 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: DE2.pdf, (Due 1 wk); DE2-sol.pdf; 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.pdf, (Due 1 wk); DE3-sol.pdf 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. 206) Fig. 4.11)DE3 Due
Part III: Reading assignments: Vector calculus (10 lectures)
WeekMWF
10 L1: 5.1.1 (p. 227) Fields and potentials (VC-1)
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$$ (VC-1 due)
Exam II
 Part III: Vector Calculus (10 Lectures) 1 F 10/23 L1: Properties of Fields and potentialsAssignment: VC1.pdf, Due 3 weeks; 2/44 M 10/26 L2: Gradient $$\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$$ dimensions 3 W 10/28 L3: Field evolution for partial differential equations 4 F 10/30 L4: Scalar wave equation (Acoustics) 5/45 M 11/2 L5: Webster Horn equation Three examples of finite length hornsSolution methods; Eigen function solutionsAssignment: VC1.pdf, Due 1 week; 6 W 11/4 L6: Integral forms of $$\nabla()$$, $$\nabla\cdot()$$ and $$\nabla \times()$$ 7 F 11/6 L7: Helmholtz decomposition 8/46 M 11/9 L8: second order operators DoG, GoD, gOd, DoC, CoG, CoC 9 W 11/11 L9: review home work solutions and discuss exam II VC-1 due F 11/13 No Class: Exam II @ 7-10 PM; Room: 3017 ECEB
Part IV: Reading assignments: Maxwell's equations + solutions (7 lectures)
WeekMWF
13L1: 5.7 (p. 276) Unification of E&M: terminology (VC-2)L2: 5.7.1 Maxwell's equationsL3: 5.7.2 Derivation of ME
14Thanksgiving Holiday
15L4: 5.8 Use of Helmholtz' Thm on MEL5: 5.8 Helmholtz solutions of MEL6: 5.8 Analysis of simple impedances (Inductors & capacitors)
16L7: TEM, TE, TM modes in waveguides (horns)L8: Wave-filters; Final Review (VC-2 due)Thur: Optional Review for Final; Reading Day
 Part IV: Maxwell's equation with solutions 1/47 M 11/16 L1: Unification of E & M; terminology (Tbl 5.4) View: Symmetry in physicsAssignment: VC-2 (Due 4 weeks) 2 W 11/18 L2: Derivation of the wave equation from Eqs: EF and MF Webster Horn equation: vs separation of variables method + integration by parts 3 F 11/20 L3: Transmission line theory: Lumped parameter approximation:Diffusion line, Telegraph equation, Wave equation (Parabolic, hyperbolic, elliptical) -/47 S 11/21 Thanksgiving Break -/49 M 11/30 Instruction Resumes 4 M 11/30 L4: Helmholtz' Thm: The fundamental thm of vector calculus: $$\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; VC-2 due
 - R 12/10 Reading Day - R 12/10 Review for Final: 2-4 PM Room: 3017 ECEB + Zoom - F Friday Dec 13 7-10 PM Final Exam: Zoom + Room: 3017 ECEB UIUC Final Exam Schedule) -/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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