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ECE-493/MATH-487 Daily Schedule Fall 2021: 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.F21@5:00 min; (L1-Z.F20@8:27 min) The size of things;
L2: \(\S\)3.1,.1,.2 (p. 69-84) Roots of polynomials; Newton's method.
L2.F21@11:30 min; (L2-360.F20@3:25 min)
L3: \(\S\)3.1.3,.4 (p.84-88) Companion matrix
L4-Z.F20@0:30 (Eigen-analysis); @1:04 (Companion Matrix)
L3.F21-360 (Audio died after 2 sec; (L3-Z.F20@10:35 min)
35L4: Eigenanalysis: \(\S\)3.2,.1,.2; Pell & Fibonocci sol.; \(\S\)B1,B3 L4.F21-360; (L4-Z.F20@0:38)L5: \(\S\)3.2.3 Taylor series
L5-360.F21 @9:00, (L5-Z.F20) (L5@14:50)
L6: Impedance; residue expansions \(\S\)3.4.2
L6-360.F21; (L6-360.F20)
36Labor dayL7: \(\S\)3.5, Anal Geom, Generalized scalar products (p. 112-121)
L7-360.F21@5:15, (L7-360.F20@2:20 min)
L8: \(\S\)3.5.1-.4 \(\cdot, \times, \wedge \) scalar products
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; zoom+Gradescope (Code: D5G555) 
L/WDDate Part I: Complex algebra (15 Lectures)
    Instruction begins
1/34M8/23L1: Introduction + History; Map of mathematics; Understanding size requires an imagination
Assignment: HW0: (pdf); Evaluate your knowledge (not graded);
Assignment: NS1; Due Lec 4: NS2; Due Lec 7: NS3; Due Lec 13
2W8/25L2: Newton's method (p. 74) for finding roots of a polynomial \(P_n(s_k)=0\) Newton's method; All m files: Allm.zip
3F8/27L3: The companion matrix and its characteristic polynomial:Working with Octave/Matlab: \(\S\)3.1.4 (p. 86) zviz.m or zvizMay31V4.m
3.11 (p. 167) Introduction to the colorized plots of complex mappings
4/35M8/30L4: Eigenanalysis I: Eigenvalues and vectors of a matrix Singular-value analysis;
Assignment: AE1, (Due 1 wk);Solution: AE1-sol
Solution: NS1-sol, NS1 due
5W9/1L5: Taylor series
6F9/3L6: Analytic functions; Complex analytic functions; Brune Impedance
Residue expansions of ratios of polynomials: \( Z(s)=N(s)/D(s) \)
-/36M9/6 Labor day: Holiday
7W9/8L7: 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: AE2; NS2, AE1 due in 1 week; Sol: AE2-sol; Sol: NS2-sol
8F9/10L8: Analytic geometry of two vectors (generalized scalar product)
Inverse of 2x2 matrix
9/37M9/13L9: Gaussian Elimination; Permutation matricies; Matrix Taxonomy
10W9/15L10: Transmission and impedance matricies
Assignment: AE3, Due in 1 week;AE2 dueSol: NS3-sol;Sol: AE3-sol
11F9/17L11: Fourier transforms of signals; Predicting tides
12/38M9/20L12: Laplace transforms of systems;System postulates
13W9/22L13: Comparison of Laplace and Fourier transforms; Colorized plots; View: Mobius/bilinear transform video
NS3, AE3 due
14F9/24L14: Review for Exam I
-/39M9/27Exam I; The exam will be in ECEB-3017 from 9-12, no online local attendance.A paper copy of the exam will be provided. One student in China will be online.
Part II: Reading assignments and videos: Complex algebra (10 Lecs)
39 Exam IL15: 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)
L16: 4.2.2 Cauchy-Riemann Eqs. CR-1, CR-2, CR-3, CR-4
Lec16-360.F21; (LEC-16-360.S20)
40L17: 4.4 Brune impedance/admittance
LEC-17-360.F21, (LEC-17-360.S20)
L18: 4.4,.1,.2 Complex analytic Impedance; Lec-18-360.F21@0:25, (LEC-18-360.S20@2:50,, zoom)
L19: 4.4.3 Multi-valued functions, Branch cuts; LEC-19-360.S21@0:10, (LEC-19-360.S20@0:40,, zoom)
41L20: 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)L21: 4.7,.1,.2 Inv \({\cal LT} (t<0, t=0)\) Lec21-360-S21 @00:30, (LEC-21-360.S20,, zoom)L22: 4.7.3 Inv \({\cal LT} (t > 0) \) LEC-22-360-S21@0:45, (LEC-22-360.S20)
42L23: 4.7.4 Properties of the \(\cal LT\); Lec-23-360.F21@00:40, (LEC-23-360.S20,, zoom)L24: 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)
15W9/29L15: The fundamental theorems of scalar and complex calculus
Assignment: DE1-F21.pdf, (Due 1 wk); DE1-sol.pdf
16F10/1L16: Complex differentiation and the Cauchy-Riemann conditions; Life of Cauchy
Properties of complex analytic functions (Harmonic functions);Taylor series of complex analytic functions
17/40M10/4L17: Brune impedance/admittance and complex analytic
Ratio of polynomials of similar degree: \( Z(s) = {P_n(s)}/{P_m(s)} \) with \(n,m \in {\mathbb N}\)
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
18W10/6L18: 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-F21.pdf, (Due 1 wk); DE2-sol.pdf;
19F10/8L19: 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 \(k\in{\mathbb N}\) is the sheet index; Balakrishnan Lecture \(F(s)=\sqrt{s}\)
20/41M10/11L20: Three Cauchy integral theorems: CT-1, CT-2, CT-3
How to calculate the residue
21W10/13L21: Inverse Laplace transform (\(t<0\)), Application of CT-3
DE2 Due
Assignment: DE3-F21.pdf, (Due 1 wk); DE3-sol.pdf
22F10/15L22: 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;
23/42M10/18L23: Properties of the Laplace transform: Linearity, convolution, time-shift, modulation, derivative etc; Introduction to the Train problem and why it is important.
24W10/20L24: Solving differential equations: Train problem (DE3, problem 2, p. 206) Fig. 4.11)
DE3 Due
Part III: Reading assignments: Vector calculus (9 lectures)
42  L25: 5.1.1 (p. 227) Fields and potentials (VC-1);
LEC-26-360.F21 @2:00 min, (LEC-25-zoom.S20)
43L26: 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)L27: 5.2 Field evolution \(\S\) 5.2 (p. 242) Lec-27-360.F21; Cont of Lec 26, (LEC-27-360.S20@3:22), (zoom)L28: 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)
44L29: 5.2.2,.3,5.4.1-.3 (p. 248) Horns
Lec29-360.F21 @0:15, (LEC-29-360.S20)
L30: 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)
L31: 5.6.3-.4 Integral forms of \(\nabla()\) \(\nabla \cdot()\), \(\nabla \times() \) (LEC-31-360.S20)
45L32: 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)
L33: 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); DE1,2,3 solutions.zip
L34: Unification of E & M; terminology (Tbl 5.4, p. 288); View: Symmetry in physics
L/WDDate Part III: Vector Calculus (10 Lectures)
25/42F10/22L25: Properties of Fields and potentials
Assignment: VC1.pdf, Due Lec-37; VC1-sol.pdf
26/43M10/25L26: 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
27W10/27L27: Field evolution for partial differential equations \(\S\) 5.2; bubbles of air in water; Vector fields; Poincare Conjucture: Proved
28F10/29L28: Review Field evolution \(\S\)5.2.1,.2 & Scalar wave equation (e.g., Acoustics) \(\S\)5.2.3
29/44M11/1L29: Webster Horn equation Three examples of finite length horns; Solution methods; Eigen-function solutions (Tesla acoustic valve)
30W11/3L30: Solution methods; Integral forms of \(\nabla()\), \(\nabla\cdot()\) and \(\nabla \times()\)
31F11/5L31: Integral form of curl: \(\nabla \times()\) and Wedge-product (p. 269)
32/45M11/8L32: Helmholtz decomposition theorem for scalar and vector potentials; (p. 270); Prandt Boundary Layers,
33W11/10L33: Second order operators DoG, GoD, gOd, DoC, CoG, CoC
34F11/12L34: Unification of E & M; terminology (Tbl 5.4); Review for Exam II

Part IV: Reading assignments: Maxwell's equations + solutions (7 lectures)
46 Exam II @ 8-11 AM; Gradescope+Zoom + Room 3017 ECEB
L35: 5.7.1-.3 Maxwell's equations LEC-35-360.F21 @1&36 min, (LEC-35-360.S20 @00:30)L36: Derivation of ME \(\S\)5.7.4,.8, p. 281-5; LEC-36-360.F21 @1, (LEC-36-360.S20 @2)
47Thanksgiving Holiday
48L37: 5.8 Use of Helmholtz' Thm on ME LEC-37-360.F21,, (LEC-37-360.S20)L38: 5.8 Helmholtz solutions of ME Lec-38-360.F21; (LEC-38-360.S20) \(\S5.6.5\) Tbl 5.3L39: 5.8 Analysis of simple impedances (Inductors & capacitors) Lec-29-360.F21, (LEC-39-360.S20)
49L40: Stokes's Curl theorem & Gauss's divergence theorem; LEC-40-360.F21 @3:30, (LEC-40-360.S20)L41: Review (LEC-41-360.S20)Thur: Optional Review for Final; Reading Day
L/WDDate Part IV: Maxwell's equation with solutions
-/46M11/15 Exam II; No Class
Assignment: VC-1 Due Lec 37
35W11/17L35: Derivation of the wave equation from Eqs: EF and MF; Webster Horn equation: vs separation of variables method + integration by parts;
36F11/19L36: 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
-/47S11/22 Thanksgiving Break
37/48M11/29L37: 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: \(\nabla \times \mathbf{w} =0\)
VC-1 Due; VC2.pdf VC-2: due Lec: 41; VC2-sol
38W12/1L38: Properties of 2d-order operators; \(\S5.6.5\) Table 5.3 p. 270; OliverHeaviside, Nyquist proof of 4ktB noise floor (Add a HW problem on Thermal Noise in resistors), L10: Summary,
39F12/3L39: Derivation of the vector wave equation
40/49M12/6L40: Physics and Applications; ME vs quantum mechanics; Video demos re ME
41W12/8L41: Review of entire course (very brief)
VC-2 due; VC2-sol
-R12/9 Reading Day
-R12/9Optional Q&A Review for Final (no lec): 9-11 Room: 3017 ECEB + Zoom + Gradescope
Final Exam: Zoom + Room: 3017 ECEB; UIUC Official Final Exam Schedule: If the class starts on Monday at 10:00 AM: The exam is scheduled for Dec 13, Monday, 1:00-5:00 PM
-/50F12/17 Finals End
  12/24Final grade analysis

Edits 2 here Block-comments

 ||- ||F||5/50||  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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