 ## 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; Map of mathematics; (Lec1) The size of things;L2: 3.1,.1,.2 (p. 69-84) Roots of polynomials; Newton's method. (Lec2-360)
L3: 3.1.3,.4 (p.84-88) Companion maxtrix (Lec3)
2L4: 3.2,.1,.2, B1, B3 Eigenanalysis (Lec4)L5: 3.2.3 Taylor series (Lec5)L6: 3.2,.4,.5 Analytic Functions, Residues, Convolution (Lec6)
3Labor dayL7: 3.5, Anal Geom, Generalized scalar products (Lec7-360)L8: 3.5.1-.4 $$\cdot, \times, \wedge$$ scalar products (Lec8-360, Lec8-zoom @8min)
4L9: 3.5.5, 3.6,.1-.5 Gauss Elim; Matrix algebra (systems) (Lec9-360: audio on @ 4:30 min)L10: 3.8,.1-.4 Thevenin parameters; Transmission lines; impedance matrix Lec 10-360 No audioL11: 3.9,.1 $${\cal FT}$$ of signals (Lec11-360 @9min)
5L12: 3.10,.1-.3 $$\cal LT$$ of systems + postulates (Lec 12-360, -zoom)L13: 3.11,.1,.2 Complex analytic color maps; Riemann sphere; Bilinear transform (Lec13-360 @13 min, -zoom)L14: Review for Exam I (Lec14-360 @25min, Lec14-zoom, -zoom
L/WDDateLectures on Mathematical Physics and its History
Part I: Complex algebra (15 Lectures)
-/35M8/24 Instruction begins
1M8/24L1: Introduction + History; Understanding size requires an imagination
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 polynomialWorking with Octave/Matlab: 3.1.4 (p. 86) zviz.m or zvizMay30.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-10AM, Stop time: 11AM; 3017ECEB for locals; submit to Gradescope; Zoom for remotes NO 360
Part II: Reading assignments: Scalar differential equations (10 Lec)
WMWF
6Exam I brief discussionL1: 4.1,4.2,.1 (p. 178) Fundmental Thms of calculus & complex $$\mathbb R, \mathbb C$$ scalar calculus (FTCC) (L1-II @5 min, Lec1-II-zoom)L2: 4.2.2 Cauchy-Riemann Eqs. CR-1, CR-2, CR-3, CR-4 (Lec2-II-360,
7L3: 4.4 Brune impedance/admittance (Lec3-II-360)L4: 4.4,.1,.2 Complex analytic Impedance (Lec4-II-360, -zoom)
L5: 4.4.3 Multi-valued functions, Branch cuts (Lec5-360, -zoom)
8L6: 4.5,.1,.2 Cauchy's complex integration thms CT1, CT2, CT3 (Lec6-II-360, -zoom)L7: 4.7,.1,.2 Inv $${\cal LT} (t<0, t=0)$$ (Lec7-II-360, -zoom)L8: 4.7.3 Inv $${\cal LT} (t > 0)$$ (Lec8-II-360)
9L9: 4.7.4 Properties of the $$\cal LT$$ (Lec9-II-360, -zoom)L10: 4.7.5 Solving LTI (simple) Diff. Eqs. with the $$\cal LT$$ (Lec10-II-360, start @5:00 PM, -zoom)
 L/W 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); Lec1-III-zoom
11L2: 5.1.2,.3 $$\nabla(), \nabla \cdot(),$$ $$\nabla \times(), \nabla \wedge(), \nabla^2()$$ (Lec2-III-360, -zoom)L3: 5.2 Field evolution $$\S$$ 5.2 (p. 242) (Lec3-III-360, Lec3-III-zoom)L4: 5.2.1 Scalar wave Eq. (Lec4-III-360)
12L5: 5.2.2,.3,5.4.1-.3 Horns (Lec5-III-360)
L6: 5.5.1 Solution methods; 5.6.1-.2 Integral forms of $$\nabla(), \nabla \cdot(), \nabla \times()$$ (Lec6-III-360)
L7: 5.6.3-.4 Integral forms of $$\nabla(), \nabla \cdot(), \nabla \times()$$ (Lec7-III-360)
13L8: 5.6.5 Helmholtz decomposition thm
$$\vec{E} = -\nabla\phi +\nabla \times \vec A\$$ ($$\S$$ 5.6.5, p. 270)
Lec8-III-360
L9: 5.6.6 2d-order scalar operators: $$\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$$
(VC-1 due, VC1-sol.pdf), (Lec 9-III-360)
Exam II; Gradescope + Zoom; 8-11 AM Central
 L D Date Part III: Vector Calculus (10 Lectures) 1 F 10/23 L1: Properties of Fields and potentialsAssignment: VC1.pdf, Due 3 weeks; 2 M 10/26 L2: Gradient $$\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$$ dimensions 3 W 10/28 L3: Field evolution for partial differential equations $$\S$$ 5.2 Vector fields 4 F 10/30 L4: Scalar wave equation (Acoustics) 5 M 11/2 L5: Webster Horn equation (Tesla acoustic valve)Three examples of finite length hornsSolution methods; Eigen function solutionsAssignment: VC1.pdf, Due 1 week; 6 W 11/4 L6: Solution methods; Integral forms of $$\nabla()$$, $$\nabla\cdot()$$ and $$\nabla \times()$$ 7 F 11/6 L7: Integral form of curl: $$\nabla \times()$$ and Wedge-product (p. 269) 8 M 11/9 L8: Helmholtz decomposition theorem for scalar and vector potentials; 9 W 11/11 L9: Second order operators DoG, GoD, gOd, DoC, CoG, CoC VC-1 due F 11/13 No Class: Exam II @ 8-11 AM; Gradescope+Zoom + Room 3017 ECEB

Part IV: Reading assignments: Maxwell's equations + solutions (7 lectures)
WeekMWF
13L1: 5.7 (p. 276) Unification of E&M: terminology (Lec 1-IV-360)L2: 5.7.1-.3 Maxwell's equations (Lec2-IV-360)L3: 5.7.4,5.8 Derivation of ME $$\S$$ 5.7.4,5.8; (Lec3-IV-360)
14Thanksgiving Holiday
15L4: 5.8 Use of Helmholtz' Thm on ME (Lec4-IV-360)L5: 5.8 Helmholtz solutions of ME (Lec5-IV-360)L6: 5.8 Analysis of simple impedances (Inductors & capacitors) (Lec6-IV-360)
16L7: Stokes's Curl theorem & Gauss's divergence theorem (Lec7-IV-360)L8: Review (VC-2 due) (Lec8-IV-360)Thur: Optional Review for Final; Reading Day
 L D Date Part IV: Maxwell's equation with solutions 1 M 11/16 L1: Unification of E & M; terminology (Tbl 5.4) View: Symmetry in physicsAssignment: VC-2 (Due 4 weeks) VC-2 sol pdf 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: Derivation of Maxwell's Equations $$\S$$ 5.7.4 (p. 280)Transmission line theory: Lumped parameter approximation:Diffusion line, Telegraph equation, Wave equation (Parabolic, hyperbolic, elliptical) - S 11/21 Thanksgiving Break 4 M 11/30 L4: Helmholtz' Thm: The fundamental thm of vector calculus $$\mathbf{F}(x,y,z) = \nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)$$, applied to Maxwell's EquationsRecall: incompressible: $$\nabla \cdot \mathbf{u} =0$$ and irrotational: $$\nabla \times \mathbf{w} =0$$VC-2 Due 5 W 12/2 L5: Properties of 2d-order operators 6 F 12/4 L6: Derivation of the vector wave equation 7 M 12/7 L7: Physics and Applications; ME vs quantum mechanics 8 W 12/9 L8: Review of entire course (very brief)VC-2 due
 - R 12/10 Reading Day - R 12/10 Optional Q&A Review for Final (no lec): 9-11 Room: 3017 ECEB + Zoom + Gradescope - M Monday, Dec 14 7:00-11:59 AM Final Exam: Zoom + Room: 3017 ECEB UIUC Official Final Exam Schedule: If the class is on Monday at 10:00 AM: The exam is scheduled for 8:00am-11:59 AM, Monday, Dec. 14 -/51 F 12/18 Finals End 12/24 Final grade analysis

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