 ## ECE-493/MATH-487 Daily Schedule Fall 2022: An Introduction to Mathematical Physics and its History

Part I: Reading assignments and videos: Complex algebra (14 Lecs)
WeekMWF
34L1: $$\S$$1, 3.1 (Read p. 1-17) Intro + history;
(L1-Z.F20@8:27 min)
360 L1.F21@5:00 min; The size of things;
L2: $$\S$$3.1,.1,.2 (p. 69-84) Roots of polynomials; Newton's method.
L3 F20@11:45 min;
360 L2.F20@3:25 min 360 L2.F21@11:30 min;
L3: $$\S$$3.1.3,.4 (p.84-88) Companion matrix
360 L4.F20@1:30; 360 L3.F20@10:35 min
35L4: Eigenanalysis: $$\S$$3.2,.1,.2; Pell & Fibonocci sol.; $$\S$$B1,B3 (L4-Z.F20@0:38) 360 L4.F21;L5: $$\S$$3.2.3 Taylor series
L5-Z.F20 @14:50;
L5-360.F21 @9:00,
L6: Impedance; residue expansions $$\S$$3.4.2
L6-F20, Part I?, L6-F20, Part II;
L6-360.F21; (L6-360.F20)
36Labor dayL7: $$\S$$3.5, Anal Geom, Generalized scalar products $$\S3.5$$ (p. 114-121), L7-Z.F20: @2:20 min, L7-F21, @5:15 @12:10 (primes), @14:10 (Rat Expansion), @19:20 (Analytic Geom); , @45:00 Wedge prodsL8: $$\S$$3.5.1-.4 $$\cdot, \times, \wedge$$ scalar products, L8-360.F20
37L9: $$\S$$3.5.5, $$\S$$3.6,.1-.5 Gauss Elim; Matrix algebra (systems)
L9-360.F21@3:30-Not Zoomed, (L9-360.F20@4:30min)
L10: $$\S$$3.8,.1-.4 Thevenin parameters; Transmission lines; impedance matrix, Screen_Shot-1, Screen_Shot-2, L10-360.F21@6:10, (L10-360.F20 No audio),L11: $$\S$$3.9,.1 $${\cal FT}$$ of signals
L11-360.F21@13:05 min, (L11-360.F20@8:20 min)
38L12: $$\S$$3.10,.1-.3 $$\cal LT$$ of systems + postulates
L12-360.F21@1:30m, (L12-360.F20), L12-Z.F20
L13: $$\S$$3.11,.1,.2 Complex analytic color maps; Riemann sphere (pdf); Bilinear transform, L13-360.21, No Audio (L13-360.F20%red\$@13min), L12-Z.F20L14: Review for Exam I; L14-360.F21; L14-360.F21, L14-360.F20@25min, L14-Z, L14-Z.F20 Exam_Review
39 Exam I; Any 3 hours between 4-8Location 2017 ECEBOn paper; graded on Gradescope
 L/W D Date Part I: Complex algebra (15 Lectures) Instruction begins 1/34 M 8/22 L1: Introduction + History; Feynman video-lecture on Physical Laws; (Lec1 pdf) Map of mathematics; Understanding size requires an imaginationAssignment: HW0: (pdf); Evaluate your math knowledge (not graded); Assignment: NS1; Due Lec 4:,Sol: NS1-sol 2 W 8/24 L2: Newton's method (p. 74) for finding roots of a polynomial $$P_n(s_k)=0$$; Newton's method; All m files: Allm.zipl 3 F 8/26 L3: The companion matrix and its characteristic polynomial: The role of complex numbersl; Working with Octave/Matlab: $$\S$$3.1.4 (p. 86) zviz.m 3.11 (p. 167) Introduction to the colorized plots of complex mappings 4/35 M 8/29 L4: Eigenanalysis I: Eigenvalues and vectors of a matrix Singular-value analysis;Assignment: NS2; Due Lec 7: Sol: NS2-sol 5 W 8/31 L5: Taylor series 6 F 9/2 L6: Analytic functions; Complex analytic functions; Brune ImpedanceResidue expansions of ratios of polynomials: $$Z(s)=N(s)/D(s)$$ -/36 M 9/5 Labor day: Holiday 7 W 9/7 L7: Analytic geomerty: Vectors and their dot $$\cdot$$, cross $$\times$$ and wedge $$\wedge$$ products. Residues.Colorized plots of complex mappings; View: Mobius/bilinear transform video, As geometryAssignment: NS3; Due Lec 10 Sol: NS3-sol; 8 F 9/9 L8: Analytic geometry of two vectors (generalized scalar product)Inverse of 2x2 matrix 9/37 M 9/12 L9: Gaussian Elimination; Permutation matricies; Matrix TaxonomyContinued Fraction approximation (CFA) (Example: $$pi \approx$$ 22/7); Hamming Correction codeAssignment: AE1, (Due 1 wk); Sol: AE1-sol 10 W 9/14 L10: Transmission and impedance matricies 11 F 9/16 L11: Fourier transforms of signals; Predicting tides, Part II Choas 12/38 M 9/19 L12: Laplace transforms of systems;System postulates Assignment:AE3, Due in 1 week; Sol: AE3-sol 13 W 9/21 L13: Comparison of Laplace and Fourier transforms; Colorized plots; View: Mobius/bilinear transform video 14 F 9/23 L14: Review for Exam I; AE3 due 15/39 M 9/26 L15 Exam I; A paper copy of the exam will be provided. No calculators or crib-sheets.
Part II: Reading assignments and videos: Complex algebra (10 Lecs)
WeekMWF
39L15: Exam IL16: 4.1,4.2,.1 (p. 178) Fundmental Thms of calculus & complex $$\mathbb R, \mathbb C$$ scalar calculus (FTCC)
(LEC-15-360.S20@8:00min), (LEC-15-zoom.S20)
L17: 4.2.2 Cauchy-Riemann Eqs. CR-1, CR-2, CR-3, CR-4
Lec16-360.F21; (LEC-16-360.S20)
LEC-17-360.F21, (LEC-17-360.S20)
L19: 4.4,.1,.2 Complex analytic Impedance; Lec-18-360.F21@0:25, (LEC-18-360.S20@2:50,, zoom)
L20: 4.4.3 Multi-valued functions, Branch cuts; LEC-19-360.S21@0:10, (LEC-19-360.S20@0:40,, zoom)
41L21: 4.5,.1,.2 Cauchy's complex integration thms CT1, CT2, CT3; Lec-20-360.F21, (LEC-20-360.S20 @2:30, @15:45, @22:00)L22: 4.7,.1,.2 Inv $${\cal LT} (t<0, t=0)$$ Lec21-360-S21 @00:30, (LEC-21-360.S20,, zoom)L23: 4.7.3 Inv $${\cal LT} (t > 0)$$ LEC-22-360-S21@0:45, (LEC-22-360.S20)
42L24: 4.7.4 Properties of the $$\cal LT$$; Lec-23-360.F21@00:40, (LEC-23-360.S20,, zoom)L25: 4.7.5 Solving LTI (simple) Diff. Eqs. with the $$\cal LT$$
Lec-24-360.F21, Lec-24-360.S20 (LEC-24-360.S20, start @5:00 PM, zoom)

 L/W D Date Part II: Scalar (ordinary) differential equations (10 Lectures) 16/39 W 9/28 L16: The fundamental theorems of scalar and complex calculus Assignment: DE1-F22.pdf, (Due 1 wk); DE1-sol.pdf 17 F 9/30 L17: Complex differentiation and the Cauchy-Riemann conditions; Life of CauchyProperties of complex analytic functions (Harmonic functions);Taylor series of complex analytic functions 18/40 M 10/3 L18: Brune impedance/admittance and complex analyticRatio of polynomials of similar degree: $$Z(s) = {P_n(s)}/{P_m(s)}$$ with $$n,m \in {\mathbb N}$$ $$Z(s) = {P_n(s)}/{P_m(s)}$$ 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; View: Mobius/bilinear transform video 19 W 10/5 L19: Generalized impedanceBrune vs. generalized impedance/admittance functions (ratio of polynomials; branch cuts)Examples of Colorized plots of Generalized Impedance/Admittance functions; Calculus on complex analytic functionsAssignment: DE2-F22.pdf, (Due 1 wk); DE2-sol.pdf; 20 F 10/7 L20: Multi-valued complex analytic functions: Branch cuts and their properties (e.g., moving the branch cut); Examples of multivalued function;Colorized plots of multivalued functions: e.g.: $$F(s) = \sqrt{s e^{jk2\pi}}$$ where $$k=\{0,1\}\in{\mathbb N}$$ is the sheet index; Balakrishnan Lecture $$W(z)=\sqrt{z}$$; More detailed lecture of multivalued $$f(z)$$ 21/41 M 10/10 L21: Three Cauchy integral theorems: CT-1, CT-2, CT-3;How to calculate the residue $$R_k = \lim_{s \rightarrow s_k} (s-s_k)F(s)$$, assuming a pole in $$F(s)$$ at $$s_k$$ 22 W 10/12 L22: Inverse Laplace transform ($$t<0$$), Application of CT-3DE2 DueAssignment: DE3-F22.pdf, (Due 1 wk); DE3-sol.pdf 23 F 10/14 L23: Inverse Laplace transform ($$t\ge0$$) CT-3 Differences between the FT and LT; System postulates: P1, P2, P3, etc. $$\S 3.10.2$$, p. 162-164; 24/42 M 10/17 L24: Properties of the Laplace transform: Linearity, convolution, time-shift, modulation, derivative etc; Introduction to the Train problem, and why it is important. 25 W 10/19 L25: Solving differential equations: Train problem (DE3, problem 2, p. 206) Fig. 4.11)DE3 Due date delayed to monday 10/23/22

Part III: Reading assignments: Vector calculus (9 lectures)
WeekMWF
42  L26: 5.1.1 (p. 227) Fields and potentials (VC1 due L35);
LEC-26-360.F21 @2:00 min, (LEC-25-zoom.S20)
43L27: 5.1,.2,.3 (p. 229): $$\nabla()$$, $$\nabla \cdot()$$, $$\nabla \times()$$, $$\nabla \wedge()$$, $$\nabla^2()$$: Differential and integral forms (LEC-26-360.S20@3:00), (zoom)L28: 5.2 Field evolution $$\S$$ 5.2 (pp. 242-245) Lec-27-360.F21; Cont of Lec 26, (LEC-27-360.S20@3:22), (zoom)L29: 5.2: Field evolution $$\S$$5.2.1, 5.2.1.1, .2 (pp 242-246); & Scalar Wave Equation $$\S$$5.2.2 p. 246; Lec-28-360.F21@0:45, (LEC-28-360.S20@3:00min, Acoustics@24m)
44L30: 5.2.2,.3,5.4.1-.3 (p. 248) Horns
Lec29-360.F21 @0:15, (LEC-29-360.S20)
L31: 5.5.1 Solution methods; 5.6.1-.2 Integral forms of $$\nabla()$$, $$\nabla \cdot()$$, $$\nabla \times()$$ Lec 30-360-Review of HWs, Lec 30-360.F21 @0:15, (LEC-30-360.S20)
L32: 5.6.3-.4 Integral forms of $$\nabla()$$ $$\nabla \cdot()$$, $$\nabla \times()$$ (LEC-31-360.S20)
45L33: 5.6.5 Helmholtz decomposition thm.
$$\vec{E} = -\nabla\phi +\nabla \times \vec A$$, ( $$\S$$ 5.6.5, p. 270 ); LEC-32-360.F21 @1.45, (LEC-32-360.S20 @1:30)
L34: 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: $$\nabla \cdot \nabla \times()=0$$, $$\nabla \times \nabla ()=0$$ Lec-33-360.F21 @0:45, (LEC-33-360.S20);
L35: Unification of E & M; terminology (Tbl 5.4, p. 288); View: Symmetry in physics
(LEC-34-360.S20)
 L/W D Date Part III: Vector Calculus (10 Lectures) 26/42 F 10/21 L25: Properties of Fields and potentialsAssignment: VC1.pdf, Due Lec-35; VC1-sol.pdf 27/43 M 10/24 L26: 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 28 W 10/26 L27: Field evolution for partial differential equations $$\S$$ 5.2; bubbles of air in water; Vector fields; Poincare Conjecture: Proved 29 F 10/28 L28: Review Field evolution $$\S$$5.2.1,.2 & Scalar wave equation & WHEN p. 248 (e.g., Acoustics) $$\S$$5.2.3 30/44 M 10/31 L29: Webster Horn equation and Tesla inventions: Tesla valve; Tesla Turbine; Tesla electricity from earthquakes; Tesla High Voltage coil; Laser Diodes & how they work; Electric Flying spiders 31 W 11/2 L30: Solution methods; Integral forms of $$\nabla()$$, $$\nabla\cdot()$$; (pp. 116/139, 120, 265-268) 32 F 11/4 L31: Integral form of curl: $$\nabla \times()$$ and Wedge-product (p. 268-270) 33/45 M 11/7 L32: Helmholtz decomposition theorem for scalar and vector potentials (p. 270-274); Electrical (Stark) and Magnetic (Zemann) eigenvalue splitting; Prandt Boundary Layers - T 11/8 Election Day Holiday 34 W 11/9 L33: Second order operators DoG, GoD, gOd, DoC, CoG, CoC; p. 274 35 F 11/11 L34: Unification of E & M; terminology (Tbl 5.4); VC1 Due; Review for Exam II
Part IV: Reading assignments: Maxwell's equations + solutions (7 lectures)
WeekMWF
46L36: Exam II @ 4-8 pM; NO ZOOM ROOM CHANGE: 1015 ECEB
L37: 5.7.1-.3 Maxwell's equations (p. 277-280) LEC-35-360.F21 @1&36 min, (LEC-35-360.S20 @00:30)L38: Derivation of ME $$\S$$5.7.4,.8, p. 281-5; LEC-36-360.F21 @1, (LEC-36-360.S20 @2)
47Thanksgiving Holiday
48L39: 5.8 Use of Helmholtz' Thm on ME LEC-37-360.F21,, (LEC-37-360.S20)L40: 5.8 Helmholtz solutions of ME Lec-38-360.F21; (LEC-38-360.S20) $$\S5.6.5$$ Tbl 5.3L41: 5.8 Analysis of simple impedances (Inductors & capacitors) Lec-29-360.F21, (LEC-39-360.S20)
49L42: Stokes's Curl theorem & Gauss's divergence theorem; LEC-40-360.F21 @3:30, (LEC-40-360.S20)L43: Review (LEC-41-360.S20)Thur: Optional Review for Final; Reading Day
 L/W D Date Part IV: Maxwell's equation with solutions 36/46 M 11/14 L36 Exam II; 1015 ECEB, 4-8 PM; No Class Assignment: 493:/Assignments/VC2-F22.pdf 37 W 11/16 L37: Derivation of the wave equation from Eqs: EF and MF; Webster Horn equation: vs separation of variables method + integration by parts; 38 F 11/18 L38: Derivation of Maxwell's Equations $$\S$$ 5.7.4 (p. 280),Transmission line theory: Lumped parameter approximation: 1D & 2D vs. 3D: $$\S 5.7.4 (p. 280)$$ d'Alembert solution of wave Equation; Poynting vector; Problem of light bulb in series with a very long pair of wires (e.g., to the moon, or sun & further); Telegraph equation, Wave equation (Parabolic, hyperbolic, elliptical); Diffusion,, Role of the Mobius Transformation -/47 S 11/19 Thanksgiving Break 39/48 M 11/28 L39: Helmholtz' Thm: The fundamental thm of vector calculus $$\mathbf{F}(x,y,z) = \nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)$$; As applied to Maxwell's Equations. Recall: incompressible: $$\nabla \cdot \mathbf{u} =0$$ and irrotational: $$\nabla \times \mathbf{w} =0$$ VC2.pdf VC2-sol 40 W 11/30 L40: Properties of 2d-order operators; $$\S5.6.5$$ Table 5.3 p. 270; OliverHeaviside, Nyquist proof of 4ktB noise floor L10: Summary 41 F 12/2 L41: Derivation of the vector wave equation; Discuss VC2 solutions 42/49 M 12/5 L42: Physics and Applications; MaxEq vs quantum mechanics; Pauli-Heiseberg debate Video demos re ME; 43 W 12/7 L43: Review of entire course (Summary) CancelledVC2 due; VC2-sol
 - R 12/8 Reading Day - R 12/8 Optional Q&A Review for Final (no Lec): 9-11 Room: ?3017? ECEB + Gradescope - R 12/15 Final Exam: Time: 8AM-Noon; Room: Henry Admin Bld (Our Classroom); Date: Thursday Dec. 15 -/50 F 12/16 Finals End 12/23 Final grade analysis

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