Why do the homework?
Since you're sophomores and juniors, you've all figured this out by now. But, I'll say it anyway...
I can't emphasize enough the importance of working the problems. In some of your classes homework is primarily evaluative; the point is for you to demonstrate what you've learned from the readings and lectures. In physics the homeworks are primarily instructional; you learn physics primarily by doing working problems. You must work the problems, think about the results, and understand any mistakes you've made if you wish to attain the type of understanding of the subject required of a working physicist. In at nutshell: If you can't work problems you don't know physics. I (or a grader) will grade the problems, and I'll hand out solutions. I encourage you to read the solutions and understand any mistakes immediately. If something doesn't make sense, ask me about it right away---don't wait until right before an exam.
If you've got a good reason why you need an extension, come talk to me in advance. I'll usually grant the extension for some additional reasonable amount of time that we agree upon. However, I will not grant a homework extension without penalty if you ask for it on the day the homework is due, so don't ask for one. In general, life will be easier for both of us if you do your best to finish the problem set on time and hand in as much as you've been able to complete by the deadline. [If you need such a last-minute or post-facto extension due to extenuating circumstances (e.g. death in the family, sudden illness, travel problem), consult the Dean of Students or your Class Dean formally make such a request to me and suggest a rescheduled due date. You should also take this route if you need an extension but you don't want to tell me why (say, it's for personal or legal reasons). If you explain your reason to a Dean and the Dean tells me it's OK, that's good enough for me.]
The College requires that all written work for a course except for a final be submitted by 5 pm on the last day of classes. The physics department takes this deadline seriously. After that day/time, no homework will be accepted.
The roles of lectures and textbooks
Lecture will not be a regurgitation of the text, a summary of all you need to know for the course, or a how-to guide for the homework. Rather, I'll try delve deeper into selected points. In lecture I'll cover material and do demonstrations related to the readings, but I won't feel obliged to be comprehensive in those places where I feel the text is adequate and I may focus only on a few points that I feel are particularly interesting or subtle. You shouldn't expect to understand what's going on without close study of the readings, and you should come to class with questions you have on the readings. Further, after we settle into the semester a bit, I expect the classes will become less lecture-oriented and more participatory; it will be difficult to reap the maximum benefit from that format if you're not sufficiently prepared to fully participate.
For the problems you can't solve, talk to classmates, attend the problem sessions, or ask me. When you ask me, either try to give you just enough of a hint to get you through, or I'll guide you through the problem with a series of leading questions. I'll never just tell you how to do it. If you run out of time and don't finish the set, start earlier next week. When the solutions come out, look over them right away, before you've forgotten all of the points you were confused about. You think you'll just get clear on it before the next exam, but there's never as much time as you think.
On the other hand, if you find the class too slow for your liking, if you have questions that you aren't getting answers to, if you'd like more detail, if you are frustrated that we aren't digging deeply enough, if you crave more applications, come talk to me. I'm very happy to provide you with additional materials or explanations that will will stimulate you and challenge you at whatever level you can handle.
One word of warning: Amherst College students tend to have lots of extracurriculars of all types. I support this, and I am occasionally willing to be flexible to facilitate your participation in range of activities, but don't let your extracurriculars overshadow your academics. If you become concerned that your courses are getting in the way of your extracurriculars, you've got the wrong mindset. Remember why you're here.
We may use Mathematica in the homework, to obtain numerical solutions to problems that are not analytically solvable and to simplify plotting of results. If you've never used Mathematica before, or haven't used it much, the tutorials will help you get started. They were written by Professor Emeritus Bob Hilborn and revised by Rebecca Erwin '02. If you download the file and save it to the desktop with a .nb suffix in the name, your computer will recognize it as a Mathematica notebook and will start up Mathematica automatically when you double-click on the icon, provided you have Mathematica installed. Mathematica is installed on lots of the college's public machines, including on the computers in the Physics Department computer lab. Alternately, you can pay the $140 or so to buy the student version.
|1. September 5||
Sept 7: Course Logistics / Intro to Infinite Series
Geometric series (finite and infinite). Some useful series.
Sept 8: Convergence (positive series)
Convergent and divergent series defined. Convergence defined via a limit of partial sums. Test for convergence: Preliminary test. Absolutely convergent series defined. Tests for convergence of series of positive terms: (1) Comparison test (2) Integral test
Sept 9: Convergence (positive and alternating series)
Tests for convergence of series of positive terms: (2) Integral test, (3) Ratio test, (4) "Special" comparison test. Alternating series test. Introduce conditionally convergent series. Conditionally convergent series can be rearranged to sum to any value (Riemann series theorem).
Read: Boas, Chap. 1
Problems: none this week (no grader yet!)
|2. September 12||
Sept 12: Power Series
Power series defined. Convergence of power series. Interval of convergence. Allowed manipulations of power series. Taylor series expansions around the origin.
Sept 14: Power series / Defining and representing complex numbers
Taylor series expansions around a general point. Complex numbers from solutions to the quadratic equation. The imaginary number i. General complex number as real part + imaginary part. Complex numbers as points in the Argand diagram. Polar representation. Complex conjugate of a complex number.
Sept 15: Complex numbers: algebra, infinite series
Addition, subtraction, multiplication, and division of complex numbers. Modulus of a complex number. Complex equations. Partial sums of complex series. Convergence, absolute convergence of a complex series defined. An absolutely convergent series is convergent. Tests for convergence. Complex power series.
Sept 16: Complex power series
Disc of convergence generalizes the interval of convergence. Rules for manipulating complex power series are similar to those for real power series. Euler's formula.
Read: Boas, Chap. 2
PS1 -- Problems: 1.2.6, 1.4.6, 1.5.4, 1.6.30, 1.9.22, 1.15.30, 1.15.31, 1.15.32, 1.16.2, 1.16.10, 1.16.14, 1.16.18 [due 11:59 pm Tuesday Sept. 20, 2011]
|3. September 19|
Sept 19: Elementary functions of complex numbers
Powers and roots of complex numbers. Exponential functions, trig functions, and hyperbolic trig functions of complex numbers.
Sept 21: Elementary functions of complex numbers
Logs of complex numbers. Complex roots and powers of complex numbers. Inverse trig and inverse hyperbolic trig functions of complex numbers. Application: Simple harmonic oscillator using complex numbers.
Sept 22: Complex numbers in physics applications
Simple harmonic oscillator using complex numbers. Why is the harmonic oscillator so important and ubiquitous in physics? Because for a potential with a stable equilibrium point, for sufficiently small excursions around the equilibrium point the potential the potential looks like a harmonic oscillator. Show this explicitly with Taylor series expansion of potential about stable equilibrium point. Set up the damped, sinusodally driven (AC) LRC series circuit problem.
Sept 23: AC circuits using complex numbers
Damped, sinusodally driven (AC) LRC series circuit using complex numbers. Talk about the importance of resonance phenomena generally in physics.
Read: Boas, finish Chap. 2, start Chap. 3
PS2 -- Problems: Boas, 2.5.21, 2.5.48, 2.5.60, 2.6.13, 2.7.15, 2.10.25, 2.11.18, 2.16.9, 2.16.10, 2.16.12 [due 11:59 pm Tuesday Sept. 27, 2011]
|4. September 26||
Sept 26: Matrices and Gaussian elimination
Matrices, matrix notation, transpose of a matrix. Start to talk about solving systems of linear equations using row reduction Gaussian elimination. Express systems of linear equations in matrix form. Solving systems of linear equations using Gaussian elimination. Possible outcomes: no solutions, unique solution, infinitely many solutions.
Sept 28: Linear equations and determinants
Relate categories of possible outcomes to relationships among (rank of M, rank of A, number of unknows). Calculate determinant of nxn square matrix, where n=1, n=2, and n general. Some relations to help calculate determinants more quickly.
Sept 29: Vectors
Cramer's rule for solving systems of linear equations using determinants. Algebra of vectors, geometric and in terms of components. Cartesian unit vectors. Dot product: its calculation and its properties. Cross product: its calculation. Einstein summation convention, Kronecker delta function.
Sept 30: Analytic geometry with vectors / Matrix operations
Read: Boas, Chap. 3
PS 3 -- Problems: Boas, 3.2.13, 3.2.14, 3.2.18, 3.3.4, 3.3.17, 3.4.20, 3.4.23
|5. October 3||
Oct 3: Matrix multiplication / Inverse of a matrix
Oct 5: Functions of matrices / linearity
Oct 5: Exam 1
Oct 6: Transformations in the plane: general linear and orthogonal
Oct 7: Rotations and reflections in 2D and 3D / Linear independence
Read: Boas, Chap. 3
PS 4 -- Problems: Boas, 3.5.13, 3.5.37, 3.5.44, 3.6.6, 3.6.17, 3.6.30, 3.7.25, 3.7.33, 3.8.16, 3.8.21
|6. October 10||
Oct 10: Break
Oct 12: Title
Oct 13: Title
Oct 14: Title
Read: Boas, Chap. 3
PS 5 -- Problems: Boas, 3.9.15, 3.9.17, 3.10.2, 3.10.4, 3.10.10, 3.11.16, 3.11.19, 3.11.30
|7. October 17||
Oct 17: Title
Oct 19: Title
Oct 20: Title
Oct 21: Title
Read: Start Boas, Chap. 4
PS 6 -- Problems: Boas 3.11.35, 3.11.43, 3.11.46, 3.11.51, 3.11.60, 3.11.62
|8. October 24||
Oct 24: Title
Oct 26: Title
Oct 27: Title
Oct 28: Title
Read: Boas, Chap. 4
PS 7 -- Problems: see Problem set 7
|9. October 31||
Oct 31: Title
Nov 2: Title
Nov 3: Title
Nov 4: Title
Read: Boas, Chap. 4, start Chap. 5
PS 8 -- Problems: 4.1.5, 4.1.14, 4.1.20, 4.1.22, 4.2.6, 4.4.1, 4.4.9, 4.4.15, 4.5.6, 4.6.9 [due Wednesday, November 9, 11:59 pm]
|10. November 7||
Nov 7: Title
Nov 9: Title
Nov 10: Title
Nov 11: Title
Read: Boas, Chap. 6, and read "div, grad, curl, and all that"
PS 9 -- Problems: 4.7.6, 4.7.16, 4.7.23, 4.7.25, 4.8.5, 4.9.9, 4.10.5, 4.11.2, 4.11.5, 4.11.10, 4.12.5, 4.12.6, 4.12.16, 5.2.6, 5.2.10 [due 11:59 pm, Tuesday November 29, 2011]
|11. November 14||
Nov 14: Title
Nov 16: Title
Nov 16 (evening): Exam 2
Covering through the end of Chap. 4.
Nov 17: Title
Nov 18: Title
Problems: (carried over from last week)
|12. November 28||
Nov 28: Title
Nov 30: Title
Dec 1: Title
Dec 2: Title
Read: Boas, Chap. 6; div, grad, curl, and all that
PS 10 -- Problems: 5.2.22, 5.2.40, 5.2.48, 5.3.30, 5.4.13, 5.5.10, 5.6.11, 6.3.18, 6.4.6, 6.6.3, 6.6.13, 6.7.8 [due 11:59 pm, Tuesday December 6, 2011] Problem set 10
|13. December 5||
Dec 5: Title
Dec 7: Title
Dec 8: Title
Dec 9: Title
Read: Boas, Chap 6, and Div, Grad, Curl, and All That
PS 11 -- Problems: 6.8.16, 6.8.18, 6.8.19, 6.8.20, 6.9.3, 6.9.12, 6.10.6, 6.10.9, 6.11.8, 6.11.14, 6.11.21, 6.12.26, 6.12.30 [due 11:59 pm, Tuesday December 13, 2011]
|14. December 12||
Dec 12: Title
Dec 14: Title