Physics 451 Fall 2003
Methods of Theoretical Physics I


Instructor: Prof. James Wells (jwells@umich.edu)
Office: 3440 Randall Lab, 763-4478
Lectures: 12:10-1:00 MWF in 2325 Mason Hall
Office Hours: 3:00-3:45 M, 4:30-5:15 TR, 2:00-2:30 F
 
Grader: Meng Cui (mcui@umich.edu)
Office: 1485A Randall, 764-9578
 


Lectures and Homework

1. Wed Sep 3 §1. Vector Spaces and Inner Products
2. Fri Sep 5 §2. Linear Operators and Matrices
3. Mon Sep 8 §3. Matrix Theorems
4. Wed Sep 10 §3 continued [HW due for §1 (SG§1) and §2 (SG§2) ]
5. Fri Sep 12 §4. Coordinate Transformations
6. Mon Sep 15 §5. Eigenvalues, Eigenvectors and Diagonalization
7. Wed Sep 17 §6. Vector Differential Operators [HW due for §3 (SG§3) and §4 (SG§4) ]
8. Fri Sep 19 §7. Geometrical Vector Integral Theorems
9. Mon Sep 22 §8. General Coordinate Systems
10. Wed Sep 24 §9. Cartesian Tensors and Pseudotensors [HW due for §5 (SG§5), §6 (SG§6) and §7 (SG§7)]
11. Fri Sep 26 §10. Tensor Contractions and Dual Tensors
12. Mon Sep 29 §11. Algebraic Properties of Contravariant and Covariant Tensors
13. Wed Oct 1 §12. The Metric Tensor [HW due for §8 (SG§8) and §9 (SG§9) ]
14. Fri Oct 3 §13. The Covariant Derivative
15. Mon Oct 6 §14. The Riemann Curvature Tensor
16. Wed Oct 8 §14 Continued [HW due for §10 (SG§10) , §11 (SG§11) and §12 (SG§12) ]
17. Fri Oct 10 First Midterm
X. Mon Oct 13 No Class -- Fall Break
18. Wed Oct 15 §1. Convergent, Divergent and Oscillatory Series
19. Fri Oct 17 §2. Convergence Tests [HW due for §13 (SG§13), §14 (SG§14) ]
20. Mon Oct 20 §3. Alternating Series
21. Wed Oct 22 §4. Improving Convergence of a Series [HW due for §1, §2 (SG-HW7) ]
22. Fri Oct 24 §5. Uniform Convergence of a Series of Functions
23. Mon Oct 27 §6. Taylor Series, §7. Power Series
24. Wed Oct 29 §8. Asymptotic Series [HW due for §3, §4 , §5 (SG-HW8)]
25. Fri Oct 31 §9. Complex Algebra
26. Mon Nov 3 §10. Differentiability and the Cauchy-Riemann Conditions
27. Wed Nov 5 §11. Analytic Functions [ HW9 (SG-HW9)]
28. Fri Nov 7 §12. Conformal Mapping
29. Mon Nov 10 §13. Contour Integrals, §14. Taylor and Laurent Series Expansions
30. Wed Nov 12 §15. Multi-valued Functions: Branch Points and Cut Lines [ HW10 (SG-HW10)]
31. Fri Nov 14 §16. Singularities and the Calculus of Residues
32. Mon Nov 17 §17. Evaluating Definite Integrals
33. Wed Nov 19 §17. continued [ HW11 (SG-HW11)]
34. Fri Nov 21 Second Midterm
35. Mon Nov 24 §1. Motion of a Particle Experiencing Constant Force
§2. Evaporation of a Spherical Drop of Liquid
§3. Pressure of an Ideal Gas Undergoing Adiabatic Volume Expansion
36. Wed Nov 26 §4. Current in a Resistance-Inductance Circuit
§5. Radioactive Decay Chains [ HW12 (SG-HW12)]
X. Fri Nov 28 No Class -- Thanksgiving
37. Mon Dec 1 §6. General Solution of First Order Linear Differential Equation
38. Wed Dec 3 §7. Solutions Theorems of Second Order Linear Differential Equations
[ HW13 (SG-HW13)]
39. Fri Dec 5 §8. Damped Harmonic Oscillator
§9. Power Series Solutions about Ordinary Points
40. Mon Dec 8 §10. Power Series Solution of the Linear Oscillator
§11. Regular Singular Points and the Method of Frobenius
41. Wed Dec 10 § 12. Series Solution to Bessel's Equation
Suggested Final Problem Set

Final Exam is Wednesday December 17, 2003 from 1:30-3:30pm


Physics Plan

Recommended Reference Textbook: Arfken/Weber,
Mathematical Methods for Physicists, 5th ed.

This semester course will be broken into three roughly
equal length mini-courses:

"Vector Spaces and Tensor Analysis"
"Real and Complex Analysis"
"Differential Equations, Part I"

Next semester's course (Physics 452) will likely be
broken into these three equal length mini-courses:

"Differential Equations, Part II"
"Group Theory"
"Probability and Statistical Methods"



Grade Evaluation

Total of 100 points possible in course.
40 points awarded for perfect Homework.
15 points awarded for perfect 1st Midterm.
15 points awarded for perfect 2nd Midterm.
30 points awarded for perfect Final.

90 points or higher guarantees an A grade.
80 points or higher guarantees a B grade.
70 points or higher guarantees a C grade.
I might be more generous than this if warranted,
but the above scale is guaranteed.

All HW problems will be graded on the "3-point scale"
(aka, "the check plus minus scale"):
3 - correct and excellent showing,
2 - not correct but very good showing,
1 - weak showing (correct answer or not),
0 - no showing

Homework details: Homework is due at the beginning of class on
listed due date. Late homework is allowed up to the beginning of the next
class meeting, but the score will be halved. No homework will be accepted
after that. All homework will be counted. Petitioning for excused homework
must be made in writing to me no later than one week after the due date. I
generally do not excuse missed homework unless there is a very significant
reason (death in the family, sickness requiring doctor's care, etc.).

First Midterm details: Will cover mini-course on
"Vector Spaces and Tensor Analysis". Students are allowed to bring in
notes on one 3x5 inch index card only. Otherwise it is closed book.
Midterm will be in class. It will cover sections §1 through §12.

Second Midterm details: Will cover mini-course on
"Real and Complex Analysis." Students are allowed to bring in
notes on one 3x5 inch index card only. Otherwise it is closed book.
Midterm will be in class. It will cover sections §1 through §17.

Finals details: Approximately 1/2 of exam will be
directly on the mini-course "Differential Equations, part I".
The remaining 1/2 of exam will be a cumulative review exam of
the first two mini-courses, "Vector and Tensor Analysis" and
"Real and Complex Analysis." The review part of the exam will
be heavily weighted on direct understanding of the class lectures.
For an exam cribsheet, you may bring one 8.5 inch by
11 inch piece of paper (standard letter size) written on one
side only to the exam, and no other written materials.


Additional Document Links


1. World of Mathematics
2. Cofactors and the Inverse of a Non-Singular Matrix