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 8||
Sept 7: Intro to infinite series
Geometric series (finite and infinite). Some useful series. Convergent and divergent series defined.
Sept 9: Convergence (positive series)
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.
Sept 10: 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.
Read: Boas, Chap. 1
PS 1 -- Problems: 1.2.6, 1.4.6, 1.5.4, 1.6.30, 1.9.22, 1.15.32, 1.16.2, 1.16.10, 1.16.14, 1.16.18 [due Tues. Sept. 14, 11:59 pm]
|2. September 13||
Sept 13: Conditionally convergent series / Power series
Conditionally convergent series can be rearranged to sum to any value (Riemann series theorem). Power series defined. Convergence of power series. Interval of convergence. Allowed manipulations of power series. Taylor series expansions around the origin.
Sept 15: 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. Addition, subtraction, and multiplication of complex numbers.
Sept 16: Complex infinite series
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. Disc of convergence generalizes the interval of convergence. Rules for manipulating complex power series are similar to those for real power series.
Sept 17: Elementary functions of complex numbers
Euler's formula. Powers and roots of complex numbers. Exponential functions, trig functions, and hyperbolic trig functions of complex numbers.
Read: Boas, Chap. 2
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 Tues. Sept. 21, 11:59 pm]
|3. September 20|
Sept 20: Elementary functions of complex numbers / Applications of complex numbers: SHO
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: Applications of complex numbers: SHO, AC circuits
Simple harmonic oscillator using complex numbers. Damped, sinusodally driven (AC) LRC series circuit using complex numbers.
Sept 23: Applications of complex numbers: n-source interference / Intro to linear algebra
n-source interference using complex numbers. 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.
Sept 24: Solving systems of linear equations by Gaussian elimination / Determinants
Solving systems of linear equations using Gaussian elimination. Possible outcomes: no solutions, unique solution, infinitely many solutions. Define rank of matrix. 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.
Read: Boas, start Chap. 3|
PS 3 -- Problems: see Problem set 3
|4. September 27||
Sept 27: Determinants / Basics of vectors
Sept 29: Analytic geometry with vectors / Matrix operations
Sept 30: Matrices: multiplication, inverses, functions of matrices
Oct 1: Linearity and linear transformations
Read: Boas, Chap. 3
PS 4 -- Problems: see Problem set 4
|5. October 4||
Oct 4: Rotations and reflections: 2D and 3D
Oct 5: Exam 1
Covers all we've done in class up through Friday Oct. 1 and everything in the book up to middle of p. 117.
Oct 6: Linear dependence and independence / Homogeneous equations
Oct 7: Homogeneous equations / Matrix trivia
Oct 9: Linear vector spaces
Read: Boas, Chap. 3
No new problem set
|6. October 11||
Oct 11: Break
Oct 13: Eigenvalues, eigenvectors, diagonalizing matrices
Oct 14: Geometric interpretation of similarity transformations and diagonalization
Oct 15: Diagonalizing hermitian matrices
Read: Boas, Chap. 3
PS 5 -- Problems: see Problem set 5
|7. October 18||
Oct 18: Orthogonal rotations in 3D / Powers of matrices
Oct 20: Simplifying equations for conic sections / Normal modes of vibrating systems
Oct 21: General vector spaces
Oct 22: Introduction to multivariable calculus / power series in two variables
Read: Start Boas, Chap. 4
PS 6 -- Problems: see Problem set 6
|8. October 25||
Oct 25: Total differentials for functions of one and two independent variables
Oct 27: Chain rule / implicit differentiation
Oct 28: Implicit differentiation / reciprocals of derivatives
Oct 29: Extremum problems: one variable, two variables, and extrema with constraints
Read: Boas, Chap. 4
PS 7 -- Problems: see Problem set 7
|9. November 1||
Nov 1: Lagrange multipliers
Nov 3: Lagrange multipliers
Nov 4: Change of variables / Differentiating integrals
Nov 5: Multiple integrals
Read: Boas, Chap. 4, start Chap. 5
PS 8 -- Problems: see Problem set 8
|10. November 8||
Nov 8: Applications of multiple integrals
Nov 10: Applications of multiple integrals / Change of variable in integrals (2D)
Nov 11: Cylindrical and spherical coordinates / Jacobian determinants
Nov 12: Jacobian determinants / Surface integrals / Triple scalar product
Read: Boas, Chap. 6, and read "div, grad, curl, and all that"
PS 9 -- Problems: see Problem set 9
|11. November 15||
Nov 15: Vector triple products / differentiating vectors
Nov 16 (evening): Exam 2
Nov 17: Differentiating vectors / directional derivatives and gradients
Nov 18: Gradients / the "del" operator
Nov 19: Vector calculus
Problems: (see last week)
|12. November 29||
Nov 29: curls, gradients, and path independence of line integrals
Dec 1: Calculating potentials / Green's theorem
Dec 2: Green's theorem / flux
Dec 3: Divergence theorem
Read: Boas, Chap. 6; div, grad, curl, and all that
PS 10 -- Problems: see Problem set 10
|13. December 6||
Dec 6: Stokes theorem
Dec 8: Stokes theorem / Fourier series
Dec 9: Fourier series / first order ODEs
Dec 10: first order ODEs / second order homogeneous ODEs with constant coefficients
Read: Boas, Chap 6, and Div, Grad, Curl, and All That
PS 11 -- Problems: see Problem set 11
|14. December 13||
Dec 13: second order ODEs with constant coefficients
Dec 15: second order ODEs with constant coefficients