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- Text:
**An Invitation to Mathematical Physics**pdf; ebook Sep 12.20; Greenberg: Advanced Eng Mathematics Greenberg: supplementary text - Instructor: Prof. Jont Allen (netID jontalle; Office 3062ECEB); TA: TBD
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Mandatory testing begins on August 10 for faculty and staff and on August 16 for students. See a list of on-campus testing sites here.

- COVID-19 on-campus testing twice per week is required for ALL students, 1 per week kfor staff for those who are on-campus for any length of time.
- This week's Reading schedule; Final

Week | M | W | F |
---|---|---|---|

1 | L1: 1, 3.1 (Read p. 1-11) Intro + history(Lec 1) | L2: 3.1,.1,.2 (p. 51-61) Roots of polynomials; Newton's method. | L3: 3.1.3,.4 (p.61-64) Companion maxtrix(Lec 3) |

2 | L4: 3.2,.1,.2, B1, B3 Eigenanalysis(Lec 4) | L5: 3.2.3 Taylor series(Lec 5) | L6: 3.2,.4,.5 Analytic Functions(Lec 6) |

3 | Labor day | L7: 3.4,.1,.2 Anal Geom, Residues(Lec 7) | L8: 3.5.1-.4 \(\cdot, \times, \wedge \) scalar products (Lec 8) |

4 | L9: 3.5.5, 3.6,.1-.5 Gauss Elim; Matrix algebra (systems)(Lec 9) | L10: 3.8,.1-.4 Thevenin parameters; Transmission lines; impedance matrix(Lec 10 No audio!) | L11: 3.9,.1 \({\cal FT}\) of signals(Lec 11) |

5 | L12: 3.10,.1-.3 \(\cal LT\) of systems + postulates(Lec 12-zoom, Lec 12-360) | L13: 3.11,.1,.2 Complex analytic color maps; Riemann sphere; Bilinear transform(Lec 13-zoom, Lec 13-360) | L14: Review for Exam I Lec 14-zoom Lec 14-360 |

6 | Exam I |

L/W | D | Date | Lectures on Mathematical Physics and its History |
---|---|---|---|

Part I: Complex algebra (15 Lectures) | |||

-/35 | M | 8/24 | Instruction begins |

1 | M | 8/24 | L1: Introduction + HistoryAssignment: HW0: pdf Evaluate your knowledge (not graded) Assignment: NS1, Problems 1,2,4, 7; p. 21-22, Due 1 week: NS1-sol |

2 | W | 8/26 | L2: Newton's method for finding roots of a polynomial \(P_n(s_k)=0\) (p.56). Newton's method; All m files: Allm.zip |

3 | F | 8/28 | L3: The companion matrix and its characteristic polynomial Working with Octave/Matlab: 3.1.4 (p. 63-64) zviz.m 3.11 Brief introduction to colorized plots of complex mappings (p. 123-125) |

4/36 | M | 8/31 | L4: Eigenanalysis I: Eigenvalues and vectors of a matrix Assignment: AE1.pdf, Probs: 1-11 (Due 1 wk); Soluton: AE1-sol NS1 due |

5 | W | 9/2 | L5: Taylor series |

6 | F | 9/4 | L6: Analytic functions; Complex analytic functions; Brune ImpedanceResidue expansions of ratios of polynomials: \( Z(s)=N(s)/D(s) \) |

-/37 | M | 9/7 | Labor day: Holiday |

7 | W | 9/9 | L7: 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-solAE1 due |

8 | F | 9/11 | L8: Analytic geometry of two vectors (generalized scalar product)Inverse of 2x2 matrix |

9/38 | M | 9/14 | L9: Gaussian Elimination; Permutation matricies |

10 | W | 9/16 | L10: Transmission and impedance matricies Assignment: AE3.pdf Due 1 week;Soluton: AE3-sol |

11 | F | 9/18 | L11: Fourier transforms of signals |

12/39 | M | 9/21 | L12: Laplace transforms of systems;System postulates |

13 | W | 9/23 | L13: Comparison of Laplace and Fourier transforms; Colorized plots;View: Mobius/bilinear transform video |

AE3 due ||

14 | F | 9/25 | L14: Review for Exam I |

15/40 | M | 9/28 | Exam I; Start time: Any 2 hour period between 8AM-10AMStop time: 11AM; 3017ECEB for locals; Zoom for remotes NO 360 |

Week | M | W | F |
---|---|---|---|

6 | Exam I brief discussion | L1: 4.1,4.2,.1 (p. 133) Fundmental Thms of \(\mathbb R, \mathbb C\) scalar calculus | L2: 4.2.2 Cauchy-Riemann Eqs. CR-1, CR-2, CR-3, CR-4 |

7 | L3: 4.4 Brune impedance/admittance | L4: 4.4,.1,.2 Complex analytic Impedance | L5: 4.4.3 Multi-valued functions, Branch cuts |

8 | L6: 4.5,.1,.2 Cauchy's complex integration thms CT1, CT2, CT3 | L7: 4.7,.1,.2 Inv \({\cal LT} (t<0, t=0)\) | L8: 4.7.3 Inv \({\cal LT} (t > 0) \) |

9 | L9: 4.7.4 Properties of the \(\cal LT\) | L10: 4.7.5 Solving LTI (simple) Diff. Eqs. with the \(\cal LT\) |

Edits 2 here

Part II: Scalar (ordinary) differential equations (10 Lectures) | |||

1 | W | 9/30 | L1: The fundamental theorems of scalar and complex calculus Assignment: DE1 |

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 functions Calculus on complex analytic functions Assignment: DE2 DE1 Due |

5 | F | 10/9 | L5: 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 |

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 Due Assignment: DE3 |

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 etc Differences between the FT and LT; System postulates |

10 | W | 10/21 | L10: Solving differential equations: Train problem (DE3, problem 2, p. 167-8, Fig. 4.11)DE3 Due |

Week | M | W | F | ||
---|---|---|---|---|---|

10 | L1: 5.1.1 (p. 171) Fields and potentials | ||||

11 | L2: 5.1.2,.3 \(\nabla(), \nabla \cdot(), \nabla \times(), \nabla \wedge(), \nabla^2() \) | L3: 5.2 Field evolution | L4: 5.2.1 Scalar wave Eq. | ||

12 | L5: 5.2.2,.3,5.4.1-.3 Horns | L6: 5.5.1 Solution methods | L7: 5.6.1-.4 Integral forms of \(\nabla(), \nabla \cdot(), \nabla \times() \) | ||

13 | L8: 5.6.5 Helmholtz decomposition thm\( \vec{E} = -\nabla\phi +\nabla \times \vec A\ \) | 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: \(\nabla \cdot \nabla \times()=0, \nabla \times \nabla ()=0 \) | Exam II |

Part III: Vector Calculus (10 Lectures) | |||

1 | F | 10/23 | L1: Properties of Fields and potentialsAssignment: VC1 |

2/44 | 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 |

4 | F | 10/30 | L4: Scalar wave equation (Acoustics) |

5/45 | M | 11/2 | L5: Webster Horn equation Three examples of finite length horns Solution methods; Eigen function solutions Assignment: VC2; VC1 Due |

6 | W | 11/4 | L6: Integral forms of \(\nabla()\), \(\nabla\cdot()\) and \(\nabla \times()\) |

7 | F | 11/6 | L7: Helmholtz decomposition |

8/46 | M | 11/9 | L8: second order operators DoG, GoD, gOd, DoC, CoG, CoC |

9 | W | 11/11 | L9: review home work solutions and discuss exam II |

F | 11/13 | No Class: Exam II @ 7-10 PM; Room: ??447AH?? (Alt Hall) Exam2 Grade Distribution |

Week | M | W | F |
---|---|---|---|

13 | L1: 5.7 (p. 203) Unification of E&M: terminology | L2: 5.7.1 Maxwell's equations | L3: 5.7.2 Derivation of ME |

14 | Thanksgiving Holiday | ||

15 | L4: 5.8 Use of Helmholtz' Thm on ME | L5: 5.8 Helmholtz solutions of ME | L6: 5.8 Analysis of simple impedances (Inductors & capacitors) |

16 | L7: TEM, TE, TM modes in waveguides (horns) | L8: Wave-filters; Final Review | Thur: Optional Review for Final; Reading Day |

Part IV: Maxwell's equation with solutions | |||

1/47 | M | 11/16 | L1: Unification of E & M; terminology (Tbl 5.4)View: Symmetry in physics |

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: Transmission line theory: Lumped parameter approximation:Diffusion line, Telegraph equation, Wave equation (Parabolic, hyperbolic, elliptical) |

-/47 | S | 11/21 | Thanksgiving Break |

-/49 | M | 11/30 | Instruction Resumes |

4 | M | 11/30 | L4: Helmholtz' Thm: The fundamental thm of vector calculus: \(\mathbf{F}(x,y,z) = \nabla{\phi(x,y,z)} + \nabla \times \mathbf{A}(x,y,z)\) incompressible: i.e., \(\nabla \cdot \mathbf{u} =0\) and irrotational \(\nabla \times \mathbf{w} =0\) vector fieldsVC2 Due |

5 | W | 12/2 | L5: Properties of 2d-order operators |

6 | F | 12/4 | L6: Derivation of the vector wave equation |

7/50 | M | 12/7 | L7: Physics and Applications; ME vs quantum mechanics |

8 | W | 12/9 | L8: Sturm-Liouville Horn Theory Solutions for 1, 2, 3 dimensions (seperation of variables)Impedance Boundary conditions; The reflection coefficient and its properties; |

- | R | 12/10 | Reading Day |

- | R | 12/10 | Review for Final: 2-4 PM Room: 106B3 in Engineering Hall |

12/?? | Exam 3 grade distribution | ||

- | F | Friday Dec 13 7-10 PM | Final Exam: Room: 445AH ( UIUC Final Exam Schedule) |

-/51 | F | 12/18 | Finals End |

F | 12/18 | Final grade distribution; Letter Grade: 100-93 A+; 92=84 A; 83-79 A-; 78-75 B+ |

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