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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)
34L1: \(\S\)1, 3.1 (Read p. 1-17) Intro + history;
(L1 F20@8:27 min)
360 to mediaspace L1.F21@5:00 min; 360-orig L1.F21@5:00 min; The size of things;
L2: \(\S\)3.1,.1,.2 (p. 69-84) Roots of polynomials; Newton's method.
Newton's Method (pdf)
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: \(\S\)3.2.3 Taylor series
L5 F20 @14:50;
L5-360.F21 @9:00,
L5: Eigenanalysis: \(\S\)3.2,.1,.2; Pell & Fibonocci sol.; \(\S\)B1,B3 (L4 F20@0:38) 360 L4.F21;L6: Impedance; residue expansions \(\S\)3.4.2
L6-F20, Part I @15 min, L6-F20, Part II @?? min;
L6-360.F21; (L6-360.F20)
36Labor dayL7: \(\S\)3.5, Anal Geom, Generalized scalar products \(\S3.5\) (p. 114-121), L7 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 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 F20L14: Review for Exam I; L14-360.F21; L14-360.F21, L14-360.F20@25min, L14 ], [[https://mediaspace.illinois.edu/media/1_l9tv8i7e|L14 F20 Exam_Review
39 Exam I; Any 3 hours between 4-8Location 2017 ECEBOn paper; graded on Gradescope
L/WDDate Part I: Complex algebra (15 Lectures)
    Instruction begins
1/34M8/21L1: Introduction + History; Feynman video-lecture on Physical Laws; (Lec1 pdf)
Map of mathematics; Understanding size requires an imagination
Assignment: HW0: (pdf); Evaluate your math knowledge (not graded);
Assignment: NS1; Due Lec 4:,Sol: NS1-sol
2W8/23L2: Newton's method (p. 74) for finding roots of a polynomial \(P_n(s_k)=0\); Newton's method; Able's Thm: No closed form solution for \(P_n(s)\) for \(n \ge 5\), All m files: Allm.zip
3F8/25L3: 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
4/35M8/28L4: Taylor series
5W8/30L5: Eigenanalysis I: Eigenvalues and vectors of a matrix Singular-value analysis;
Assignment: NS2; Due Lec 7: Sol: NS2-sol
6F9/1L6: Analytic functions; Complex analytic functions; Brune Impedance
Residue expansions of ratios of polynomials: \( Z(s)=N(s)/D(s) \)
-/36M9/4 Labor day: Holiday
7W9/6L7: 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 geometry
Assignment: NS3; Due Lec 10 Sol: NS3-sol;
8F9/8L8: Analytic geometry of two vectors (generalized scalar product)
Inverse of 2x2 matrix
9/37M9/11L9:Gaussian Elimination; Permutation matricies; Matrix Taxonomy
Continued Fraction approximation (CFA) (Example: \(pi \approx\) 22/7); Hamming Correction code
Assignment: AE1, (Due 1 wk); Sol: AE1-sol
10W9/13L10: Transmission and impedance matricies
11F9/15L11: Fourier transforms of signals; Predicting tides, Part II Choas
12/38M9/18L12: Laplace transforms of systems;System postulates
Assignment: AE3, Due in 1 week; Sol: AE3-sol
13W9/20L13: Comparison of Laplace and Fourier transforms; Colorized plots; View: Mobius/bilinear transform video
14F9/22L14: Review for Exam I; AE3 due
15/39M9/25L15 Exam I; A paper copy of the exam will be provided. No calculators or crib-sheets. Exam 1 Grade Distribution F23
Part II: Reading assignments and videos: Complex algebra (10 Lecs)
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)
40L18: 4.4 Brune impedance/admittance
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/WDDate Part II: Scalar (ordinary) differential equations (10 Lectures)
16/39W9/28L16: The fundamental theorems of scalar and complex calculus
Assignment: DE1-F22.pdf, (Due 1 wk); DE1-sol.pdf
17F9/29L17: Complex differentiation and the Cauchy-Riemann conditions; Life of Cauchy
Properties of complex analytic functions (Harmonic functions);Taylor series of complex analytic functions
18/40M10/2L18: Brune impedance/admittance and complex analytic
Ratio 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
19W10/4L19: Generalized impedance
Brune vs. generalized impedance/admittance functions (ratio of polynomials; branch cuts)
Examples of Colorized plots of Generalized Impedance/Admittance functions; Calculus on complex analytic functions
Assignment: DE2-F22.pdf, (Due 1 wk); DE2-sol.pdf;
20F10/6L20: 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 \(f(z)\)
21/41M10/9L21: 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 \(s_k\)
22W10/11L22: Inverse Laplace transform (\(t<0\)), Application of CT-3
DE2 Due
Assignment: DE3-F22.pdf, (Due 1 wk); DE3-sol.pdf
23F10/13L23: 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;
24/42M10/16L24: Properties of the Laplace transform: Linearity, convolution, time-shift, modulation, derivative etc; Introduction to the Train problem, and why it is important.
25W10/18L25: 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)
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,, .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
L/WDDate Part III: Vector Calculus (10 Lectures)
26/42F10/21L26: Properties of Fields and potentials
Assignment: VC1.pdf, Due Lec-35; VC1-sol.pdf
27/43M10/23L27: 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
28W10/25L28: Field evolution for partial differential equations \(\S\) 5.2;
Text Appendix B, Eigen analysis methods; bubbles of air in water; Vector fields; Poincare Conjecture: Proved
29F10/28L29: Review Field evolution \(\S\)5.2.1,.2 & Scalar wave equation & WHEN p. 248 (e.g., Acoustics) \(\S\)5.2.3
30/44M10/30L30: 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
31W11/1L31: Solution methods; Integral forms of \(\nabla()\), \(\nabla\cdot()\); (pp. 116/139, 120, 265-268)
32F11/3L32: Integral form of curl: \(\nabla \times()\) and Wedge-product (p. 268-270)
33/45M11/6L33: Helmholtz decomposition theorem for scalar and vector potentials (p. 270-274); Electrical (Stark) and Magnetic (Zemann) eigenvalue splitting; Prandt Boundary Layers
-T11/7Election Day Holiday
34W11/8L34: Second order operators DoG, GoD, gOd, DoC, CoG, CoC; p. 274
35F11/10L35: Unification of E & M; terminology (Tbl 5.4); VC1 Due; Review for Exam II

Part IV: Reading assignments: Maxwell's equations + solutions (7 lectures)
46L36: Exam II @ 4-8 PM; `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/WDDate Part IV: Maxwell's equation with solutions
36/46M11/13L36 Exam II; 4-8 PM; 1015 ECEB Review for tonight's Midterm II
37W11/15L37: Derivation of the wave equation from Eqs: EF and MF; Webster Horn equation: vs separation of variables method + integration by parts;
Assignment: 493:/Assignments/VC2-F22.pdf
38F11/17L38: 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
-/47S11/18 Thanksgiving Break
39/48M11/27L39: 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
40W11/29L40: Properties of 2d-order operators \(\S5.6.5\) Table 5.3 p. 270;
Oliver Heaviside and Maxwell's Equations, AIP, wikipedia,
Nyquist proof of 4kTB noise floor L10: Summary
41F12/1L41: Derivation of the vector wave equation; Discuss VC2 solutions
42/49M12/4L42: Physics and Applications; MaxEq vs quantum mechanics; Pauli-Heiseberg debate
Video demos re ME;
43W12/6L43: Review of entire course (Summary) VC2 due; VC2-sol
-/49R12/7 Reading Day
 R12/7Optional Q&A Review for Final (no Lec): 9-11 Room: ?3017? ECEB + Gradescope
-/50R12/14Final: 8-11 AM
 F12/15 Finals End
  12/22Final grade analysis

Final Exam: Time: 8AM-Noon; Room: Henry Admin Bld (Our Classroom); Date: Thursday Dec. 14

Edits 2 here Block-comments

 ||- ||F||?/??||  Backup: Exam III 7:00-10:00+ PM on HW1-HW11atest>><<

L= Lecture #
T= Topic #
W=week of the year, starting from Jan 1
D=day: T is Tue, W Wed, R Thur, S Sat, etc.
The somewhat random-ordered numbers in front of many (not all) topics are the topic numbers defined in 2009 Syllabus Δ:
ECE-493 is divided into 4 basic sections (I-IV), divided into 40 topics, delivered as 24=4*6 lectures. There are two mid-term exams and one final. There are 12 homework assignments, with a HW0 that does not count toward your final grade. Each exam (I, II and Final) will count as 30% of your final grade, while the Assignments (HW1-12) plus class participation (Prof's Discuression), count for 10%.

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