Instructor

Professor William Loinaz
Office: 223 Merrill Science Center
Phone: (413) 542-7968
email: waloinaz [AT] amherst [DOT] edu
Office Hours:
MWTh 2:00-3:00 [subject to change as the semester goes on].
For my scheduled office hours, you can either drop in or make an appointment. To meet with me at some other time, it will be easiest if you email me for an appointment.
You're also welcome to drop by my office. If I'm in and free I'm happy to meet with you, but it's not so often during the day that I'm in and I'm free.

Grader: Titus Chirchir, Robert Flynn

Course Information

Course Catalog Description:

Schedule

• Class: MWF 10:00-10:50 am (Merrill 211), Th 1-1:50 pm, (Merrill 315)
• Problem Session (informal): Wednesdays 9 pm, Physics student lounge
  

Requisites

Course requirements

• Attendance: Come to class. Lectures and in-class discussion will generally NOT duplicate the text, so don't assume you can just "get it from the book". It's your affair how you allocate your time and I won't explicitly penalize you for being absent, but if you're going to be absent please do me the courtesy of letting me know in advance, if possible. It's a small class--I'll notice if you're missing. And if you're consistently absent I'll want to know why. If you arrive late, please enter quietly at the back of the classroom. Obviously it will be your responsibility to find out what happened in the class you missed.

• Homework: I'll assign weekly problem sets, due (tenatively) Thursdays @11:59 pm. You may leave your homework in my mailbox near the physics department office, or of course you may give it to me personally (please do NOT leave it in the box beside my office or stick it under my door). Homework submitted late without prior arrangement will receive a 50% penalty if submitted within three days of the due date (i.e. by Sunday @11:59 pm), and will receive no credit after that.

  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.

  Homework extensions
  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.]

• Exams: There will be two mid-term exams and a final, and these will serve as the primary evaluative mechanisms for the course. The exams will be closed-book, and all course material (lectures, handouts, text readings and homeworks) are fair game for the exam. I'll give you relevant formulae, so the exams will be not be memorization exercises. I tend to like problems that ask that you derive results utilizing key derivations from class, then use the result to do something new.

• Extensions: In the interest of fairness I will hold fast to the deadlines and policies above. In particular, I won't grant extensions to homeworks on the day they're due, and I won't reschedule the exam for an individual to a different day from the rest of the class. Should circumstances render such rescheduling unavoidable, you'll need to have the Dean of Students or your Class Dean contact me with a formal request for an extension and a suggested rescheduled time. Such matters should be taken care of before the exam if at all possible. If the matter is personal or confidential, the Deans need not provide me with any particulars, only their opinion that the rescheduling is reasonable.

  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.

Statement of Intellectual Responsibility: particulars for this course

• I strongly encourage you to discuss the homework problems with your classmates---discussing and debating the concepts and procedures is a very effective way to learn. However, the final write-ups of your assignments should be your own. For example, the copying homework solutions in part or in whole from other students (current or prior) or from other sources (commercial or otherwise) is unacceptable. I don't mind if you use other texts to learn from, and if the answer to a homework problem is worked as a example in some other text you may utilize it (although you should first try to solve it on your own as best you can). But in that case you should make an explicit reference to the source in your homework, and of course you still need to explain the solution to the problem in your own words. It goes without saying, but I'll say it anyway: tests are written without help.
• If you have any questions about what is or is not acceptable under this policy, please ask us. In all cases, the spirit of the Statement of Intellectual responsibility will take precedence over legalistic convolutions of the text.

How to get the most from this class:

• Come to class awake and on time.

• Do the relevant reading and read through the problems assigned for the week before the lecture (I'll announce the readings and problems in class and post them on the website). Even if you don't understand the readings fully at the time, try to get the broad outline of the material so you'll have some hooks on which to hang the ideas that we talk about in class.

  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.

• Ask questions: in class, after class, during office hours, in problem sessions, in the hallways. Ask the questions you think you're supposed to know the answer to, the questions you're sure everyone else knows the answer to (they don't), questions about the overarching logic or the minute details, questions about the points or conclusions that weren't made that seem like they should follow. If individual arguments, their interconnection, or the entire point of the lecture don't seem clear to you, ASK. The main benefit to taking a class, especially at a small school like Amherst, is that you can ask questions of the experts.

• Work the problems: Start early. Try all of them. Struggle with them awhile yourself---some will be clear, simple exercises meant to familiarize you with a procedure, others will be meant to stretch your understanding (some will be intentionally vague, so that you have to clarify the picture before solving the problem). Be sure you can state what ideas (mathematical and physical) are being utilized in each problem; even if you still can't solve the problem you're most of the way there. The ability to work problems is not sufficient to claim an understanding of the physics, but it is a necessary step.

  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.

• If things aren't going as you'd like, take an active role in improving your situation. Use all of the resources at your disposal. Step 1 is almost always to come talk to me. If you don't understand what's going on in lecture, the book, or the problems, if you feel like you can't keep up, come talk to me. I can help you, and I can point you to additional resources. Don't hide.

  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 doing it wrong.

Grading:

• Homework: 20%
• First Midterm (Wed. Oct. 10): 25%
• Second Midterm (Mon. Nov. 12): 25%
• Final: 30%

Textbooks:

• Mathematical Methods in the Physical Sciences (3rd edition)
  Author: Mary L. Boas
  ISBN-13: 978-0471198260
  Publisher: Wiley
  [a comprehensive introductory mathematical physics book. very popular around here. This will be the main course text. A copy of this book is on reserve at the Library.]
  
• Div, Grad, Curl, and All That: An Informal Text on Vector Calculus (4th edition)
  Author: H. M. Schey
  ISBN-13: 978-0393925166
  Publisher: W.W. Norton & Co.
  [A usable introduction to vector calculus in a style that physicists find comfortable. A copy of this book is on reserve at the Library.]

• Physics (vol. 1 & 2), Hans Ohanian [a good general text for review of introductory physics material. Used as Physics 23/24 text in the past.]
  
  
• Lectures in Physics, R. P. Feynman [Feynman's insights are always unique and worthwhile. You appreciate them more as you get older.]
• Electricity and Magnetism, E. M. Purcell [an elegant and sophisticated intro electromagnetism textbook. Includes a treatment of AC circuits using complex numbers.]

• Mathematical Handbook (Schaum's Outline Series), Murray R. Spiegel [a convenient compendium of useful mathematical formulae, including integrals and special functions at a low, low price.]
• Basic Training in Mathematics, R. Shankar [sometimes used as a textbook for Physics 27]

Mathematica Tutorials

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.

• Intro
  
• Functions
  
• SymbolicManipulation
  
• Plots
  
• Lists and Arrays
  
• Solving Differential Equations Numerically
  
• Complex Numbers
  
• Fourier Analysis
  
• Programming
  
• Statistics
  
• Input-Output Issues
  
• Solving Equations Numerically
  

Lecture Schedule

Week	Notes	Hmwk	Other
1. September 3	Infinite series Sept 5: Course Logistics / Intro to Infinite Series Geometric series (finite and infinite). Some useful series. Sept 6: 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 7: 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.	Read: Boas, Chap. 1 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 Thursday Sept. 13, 2012]
2. September 10	Series / Complex numbers Sept 10: Power Series Introduce conditionally convergent 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 12: Power series / Defining and representing complex numbers Taylor series expansions, about the origin and about a general point. Tips on expanding functions in power series. 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. Sept 13: Complex numbers: algebra, infinite series, power series Complex conjugate of a complex number. 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. Disc of convergence generalizes the interval of convergence. Rules for manipulating complex power series are similar to those for real power series. Sept 14: Elementary functions of complex numbers General discussion of extending functions of a real variable to a complex variable: what properties should (could) the extension preserve. Use power series to extend to functions of complex variable. Exponential function. Euler's formula. Powers of complex numbers. DeMoive's theorem. Roots of complex numbers. Square root of 1.	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 11:59 pm, Thursday Sept. 20, 2012]
3. September 17	Complex numbers Sept 17: Elementary functions of complex numbers Powers and roots of complex numbers. Exponential functions, trig functions, and hyperbolic trig functions of complex numbers. Logs of complex numbers. Sept 19: 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 20: 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 21: 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 PS3 -- Problems: Boas, 3.2.13, 3.2.14, 3.2.18, 3.3.4, 3.3.17, 3.4.20, 3.4.23 [due 11:59 pm, Thursday Sept. 27, 2012]
4. September 24	Linear algebra Sept 24: Matrices and Gaussian elimination Matrices, matrix notation, transpose of a matrix. Einstein summation convention. 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 26: 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 27: 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. Kronecker delta function. Cross product: its calculation and properties. convention. Sept 28: Analytic geometry with vectors Calculating the cross-product using the Levi-Civita symbol. Equation for a line parallel to a vector. Equation for a line perpendicular to a vector, in 2D. Equation for a plane with a specified normal vector.	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 [due 11:59 pm, Thursday Oct. 4, 2012]
5. October 1	Linearity and linear transformations Oct 1: Matrix operations Matrix multiplication / Inverse of a matrix Matrix equations, multiplying a matrix by a number, adding matrices. Multiplying matrices. Conformable matrices. Matrix multiplication is not commutative, but is associative and distributive. Commutator defined. Zero matrix, identity matrix defined. Determinant of product is product of determinants, for square matrices. Inverse of matrix defined. Oct 3: Inverse of Matrix / Functions of matrices / Linearity Calculating the inverse of a matrix. Expression for unique solution of n linear equations in n unknowns, in terms of inverse matrix. Powers and polynomials of matrices. Functions of matrices defined in terms of power series. Exponential of a matrix. Note that some of the properties of exponentials of numbers don't hold for exponentials of matrices, unless the matrices commute. BCH formula. Linearity: of vectors, of scalar functions of vectors, of vector functions of vectors (linear functions vs. affine functions). Linear operators defined. Derivative as linear operator on functions. Oct 5: Transformations in the plane: general linear and orthogonal / Rotations in 2D Matrices as linear operators transforming vectors to vectors. Active and passive transformation pictures. Orthogonal transformation defined as those that preserve the length of vector. Form of orthogonal matrix in 2D derived. Orthogonal matrix has determinant +1 or -1. For orthogonal M, det M=1 is rotation, det M=-1 is reflection. Derive form of rotation in 2D in active transformation picture, using complex numbers. Oct 6: Rotations and reflections in 2D and 3D / Linear independence 2D rotation matrix in passive transformation picture. Check that det M=1 for rotation matrix. For reflection, can find line of reflection using Cr=r. Rotations and reflections in 3D: rotations about z axis. Reflections through xy plane. Combination of this is product of the corresponding matrices. Claim any 3D orthogonal matrix with det=1 can be written same way by choosing rotation axis as z-axis. Rotation about y-axis. Rotation axis can be determined by solving Mr=r, reflection plane found by solving Lr=-r. Linear dependent and independent vectors. Row reduction of components of vectors to determine a basis. Rank of component matrix gives number of independent vectors.	Read: Boas, Chap. 3 PS 5 -- Problems: 3.7.33, 3.8.16, 3.8.21, 3.9.15, 3.9.17, 3.10.2, 3.10.4, 3.10.10 [due 11:59 pm, Friday Oct. 12, 2012]
6. October 8	Linear vector spaces Oct 8: Break Oct 10: Linear independence / Homogenous equations Linear (in)dependence of functions defined. Wronskian can be used to determine if a collection of functions is linearly independent. Define homogeneous equations. Sets of homogeneous equations can have a unique solutions (trivial solution), or an infinite set of solutions. A system of n homogeneous equations in n unknowns has nontrivial solutions IFF the determinant of the coefficient matrix is zero. Prove earlier statement about Wronskian. Geometry of non-trivial solutions to sets of (in)homogeneous equations: in an example, we show the solution to a set of inhomogeneous equations is a line passing through a point. The parallel line through the origin is the corresponding homogeneous solution, the point is a solution to the inhomogeneous equation (homogeneous + particular solution). Oct 10: Exam 1 7-10 pm location: Merrill 3 Covers through end of Oct. 6. Oct 11: Homogeneous equations / Matrix Trivia Eigenvalues/eigenvector problems. Find eigenvectors and eigenvalues of a 2x2 real symmetric matrix. Note that the geometry of solutions corresponds to orthogonal line through the origin. Matrix trivia: make observations about transpose of products and inverse of products of matrices, trace of matrix and trace of product, define hermitian conjugation, hermitian and unitary matrices, state that U=Exp[iH] is unitary if H is hermitian. Oct 12: Linear vector spaces Extend ideas from 2D and 3D vector space to n-dim Euclidean vector spaces. Vector space as all linear combinations of some set of vectors. Span, basis, and dimension of a vector space. Inner product, norm, orthogonality. Schwarz inequality. Gram-Schmidt procedure for obtaining an orthonormal basis. Define analogs on complex Euclidean space.	Read: Boas, Chap. 3 PS 6 -- Problems: Boas 3.11.16, 3.11.19, 3.11.30, 3.11.35, 3.11.43, 3.11.46, 3.11.51, 3.11.60, 3.11.62
7. October 15	Applications of similarity transformations Oct 15: Eigenvectors, Eigenvalues, Diagonalization Eigenvectors of a transformation are rescaled but not rotated by the transformation. In the 2x2 case, the two eigenvalues can be real and distinct, real and degenerate, or complex conjugates. Work out the eigenvectors and eigenvalues of a 2x2 real symmetric matrix M. This particular example has distinct eigenvalues and orthogonal eigenvectors. Can diagonalize the matrix M with a similarity transformation the columns of which are the normalized eigenvectors of M. The resulting diagonal matrix has eigenvalues as the diagonal entries. Oct 17: Geometrical significance of similarity transformations Note that similar matrices have same trace and determinant. Work out a 2x2 case in which similarity transformation has interpretation as a change of coordinates. The similar matrices represent that same transformation expressed in different coordinate systems. The columns of similarity transformation C are the new coordinate axes (expressed in the original coordinate system). When a similarity transformation C diagonalizes a matrix M, it amounts to transforming to a coordinate system in which the action of the transformation represented by M (in the original coordinate system) has a particularly simple form. If the eigenvectors of C are coordinates of orthonormal vectors, C is an orthogonal matrix (its transpose is its inverse) and can thus be interpreted at a rotation matrix (or rotation + reflection). The columns of C are the rotated coordinate axes (expressed in the original coordinate system). Claim this is possible IFF real matrix M is symmetric. Oct 18: More on similarity transformations Similarity transformations by orthogonal matrices are rotations. If columns of transformation matrix are not components of orthogonal vectors, transformation is to a new coord system whose axes are not orthogonal. If symmetric matrix has repeated eigenvalues, can choose eigenvectors of eigenspace to be orthonormal using Gram-Schmidt procedure. A matrix has real eigenvalues and can be diagonalized by a unitary similarity transformation IFF it is hermitian. Oct 19: Orthogonal transformations in 3D	Read: Boas, Chap. 4 PS 7 -- Problems: see Problem set 7
8. October 22	Multivariable calculus: differential calculus Oct 22: No Class Class cancelled: Hurricane Sandy Oct 24: Title Oct 25: Title Oct 26: No Class Class Cancelled: Day of Dialogue	Read: Boas, Chap. 4 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 11:59 pm, Thursday November 1, 2012]
9. October 29	Multivariable calculus: differential calculus Oct 29: Title Oct 31: Title Nov 1: Title Nov 2: Title	Read: Boas, Chap. 4, start Chap. 5 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 [due 11:59 pm, Thursday November 8, 2012]
10. November 5	Multivariable calculus: differential and integral calculus Nov 5: Title Nov 7: Title Nov 8: Title Nov 9: Title	Read: Boas, Chap. 6, and read "div, grad, curl, and all that" PS 10 -- Problems: 5.2.6, 5.2.10, 5.2.22, 5.2.40, 5.2.48, 5.3.30, 5.4.13, 5.5.10, 5.6.11 [due 11:59 pm, Thursday November 15, 2011]
11. November 12	Multivariable Calculus: integral calculus Nov 12: Title Nov 12 (7-10 pm): Exam 2 Covering through the end of Boas, Chapter 4. Nov 14: Title Nov 15: Title Nov 16: Title	Read: Div, Grad, Curl, and All That. Boas, Chap. 6 Problems: PS 11 -- Problems: Boas, 6.3.18, 6.4.6, 6.6.3, 6.6.13, 6.7.8, 6.8.16, 6.8.18, 6.8.19, 6.8.20 [due Friday, November 30, 2012, 11:59 pm]
12. November 26	Vector calculus Nov 26: Title Nov 28: Title Nov 29: Title Nov 30: Title	Read: Boas, Chap. 6; div, grad, curl, and all that Problems: PS 12 -- Problems: Boas, 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 Thursday, December 6, 2012, 11:59 pm]
13. December 3	Vector Calculus / ODEs Dec 3: Title Dec 5: Title Dec 7: Title Dec 8: Title	Read: Boas, Chap 6, and Div, Grad, Curl, and All That PS 11 -- Problems:
14. December 10	2nd order differential equations with constant coefficients Dec 10: Title Dec 12: Title Dec 13: Title	Read: Problems: