 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)
WeekMWF
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
(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; zoom+Gradescope (Code: D5G555)
 L/W D Date Part I: Complex algebra (15 Lectures) Instruction begins 1/34 M 8/23 L1: Introduction + History; Map of mathematics; Understanding size requires an imaginationAssignment: HW0: (pdf); Evaluate your knowledge (not graded);Assignment: NS1; Due Lec 4: NS2; Due Lec 7: NS3; Due Lec 13 2 W 8/25 L2: Newton's method (p. 74) for finding roots of a polynomial $$P_n(s_k)=0$$ Newton's method; All m files: Allm.zip 3 F 8/27 L3: 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 4/35 M 8/30 L4: Eigenanalysis I: Eigenvalues and vectors of a matrix Singular-value analysis;Assignment: AE1, (Due 1 wk);Solution: AE1-solSolution: NS1-sol, NS1 due 5 W 9/1 L5: Taylor series 6 F 9/3 L6: Analytic functions; Complex analytic functions; Brune ImpedanceResidue expansions of ratios of polynomials: $$Z(s)=N(s)/D(s)$$ -/36 M 9/6 Labor day: Holiday 7 W 9/8 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: AE2; NS2, AE1 due in 1 week; Sol: AE2-sol; Sol: NS2-sol 8 F 9/10 L8: Analytic geometry of two vectors (generalized scalar product)Inverse of 2x2 matrix 9/37 M 9/13 L9: Gaussian Elimination; Permutation matricies; Matrix Taxonomy 10 W 9/15 L10: Transmission and impedance matricies Assignment: AE3, Due in 1 week;AE2 dueSol: NS3-sol;Sol: AE3-sol 11 F 9/17 L11: Fourier transforms of signals; Predicting tides 12/38 M 9/20 L12: Laplace transforms of systems;System postulates 13 W 9/22 L13: Comparison of Laplace and Fourier transforms; Colorized plots; View: Mobius/bilinear transform videoNS3, AE3 due 14 F 9/24 L14: Review for Exam I -/39 M 9/27 Exam 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)
WeekMWF
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)
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/W D Date Part II: Scalar (ordinary) differential equations (10 Lectures) 15 W 9/29 L15: The fundamental theorems of scalar and complex calculus Assignment: DE1-F21.pdf, (Due 1 wk); DE1-sol.pdf 16 F 10/1 L16: Complex differentiation and the Cauchy-Riemann conditionsProperties of complex analytic functions (Harmonic functions);Taylor series of complex analytic functions 17/40 M 10/4 L17: 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; View: Mobius/bilinear transform video 18 W 10/6 L18: 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-F21.pdf, (Due 1 wk); DE2-sol.pdf; 19 F 10/8 L19: Multi-valued complex analytic functionsBranch 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 20/41 M 10/11 L20: Three Cauchy integral theorems: CT-1, CT-2, CT-3How to calculate the residue 21 W 10/13 L21: Inverse Laplace transform ($$t<0$$), Application of CT-3DE2 DueAssignment: DE3-F21.pdf, (Due 1 wk); DE3-sol.pdf 22 F 10/15 L22: 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; 23/42 M 10/18 L23: Properties of the Laplace transform: Linearity, convolution, time-shift, modulation, derivative etc; Introduction to the Train problem and why it is important. 24 W 10/20 L24: Solving differential equations: Train problem (DE3, problem 2, p. 206) Fig. 4.11)DE3 Due
Part III: Reading assignments: Vector calculus (9 lectures)
WeekMWF
42  L25: 5.1.1 (p. 227) Fields and potentials
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, 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)
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);
L34: 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) 25/42 F 10/22 L25: Properties of Fields and potentialsAssignment: VC1.pdf, Due Lec-37; VC1-sol.pdf 26/43 M 10/25 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 27 W 10/27 L27: Field evolution for partial differential equations $$\S$$ 5.2; bubbles of air in water; Vector fields; Poincare Conjucture: Proved 28 F 10/29 L28: Review Field evolution $$\S$$5.2.1,.2 & Scalar wave equation (e.g., Acoustics) $$\S$$5.2.3 29/44 M 11/1 L29: Webster Horn equation Three examples of finite length horns; Solution methods; Eigen-function solutions (Tesla acoustic valve) 30 W 11/3 L30: Solution methods; Integral forms of $$\nabla()$$, $$\nabla\cdot()$$ and $$\nabla \times()$$ 31 F 11/5 L31: Integral form of curl: $$\nabla \times()$$ and Wedge-product (p. 269) 32/45 M 11/8 L32: Helmholtz decomposition theorem for scalar and vector potentials; (p. 270); Prandt Boundary Layers, 33 W 11/10 L33: Second order operators DoG, GoD, gOd, DoC, CoG, CoC 34 F 11/12 L34: Unification of E & M; terminology (Tbl 5.4); Review for Exam II

Part IV: Reading assignments: Maxwell's equations + solutions (7 lectures)
WeekMWF
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/W D Date Part IV: Maxwell's equation with solutions -/46 M 11/15 Exam II; No Class Assignment: VC-1 Due Lec 37 35 W 11/17 L35: Derivation of the wave equation from Eqs: EF and MF; Webster Horn equation: vs separation of variables method + integration by parts; 36 F 11/19 L36: 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 11/22 Thanksgiving Break 37/48 M 11/29 L37: 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 38 W 12/1 L38: 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, 39 F 12/3 L39: Derivation of the vector wave equation 40/49 M 12/6 L40: Physics and Applications; ME vs quantum mechanics; Video demos re ME 41 W 12/8 L41: Review of entire course (very brief)VC-2 due; VC2-sol
 - R 12/9 Reading Day - R 12/9 Optional Q&A Review for Final (no lec): 9-11 Room: 3017 ECEB + Zoom + Gradescope - M 12/13 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 -/50 F 12/17 Finals End 12/24 Final grade analysis