Physics 27 (Methods of Theoretical Physics) Home Page, Fall 2009

Physics 27: Methods of Theoretical Physics

Announcements

Welcome to the new semester!

Instructor

Course Information

Course Catalog Description:

The course will present the mathematical methods frequently used in theoretical physics. The physical context and interpretation will be emphasized. Topics covered will include vector calculus, complex numbers, ordinary differential equations (including series solutions), partial differential equations, functions of a complex variable, and linear algebra. Four class hours per week.

Schedule

Times and places:

Requisites

Mathematics 12 and Physics 17/24 or consent of the instructor

Course requirements

Statement of Intellectual Responsibility: particulars for this course

How to get the most from this class:

Grading:

Textbooks:

Required (I've ordered these from Amherst books): Additional useful references (NOT required):

Physics: Math books:

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.


Lecture Schedule
Week Notes Hmwk Other
1. September 7
Infinite series


Sept 8: Intro to infinite series

Geometric series (finite and infinite). Some useful series.
[Handouts: ]

Sept 9: Convergence (positive series)

Definition of convergent and divergent series. Illustrate dangers of manipulating divergent (and even some convergent) series. Convergence of series as limit of sequence of partial sums. Tests for absolute convergence of series/convergence of positive terms: (1) comparison test, (2) integral test.

Sept 10: Convergence (alternating series)

Convergence of series of positive terms (cont'd): (3) ratio test, (4) "special" comparison test. Alternating series. Conditionally convergent series. Convergence or divergence of series is not affected by multiplying by a nonzero constant or changing a finite number of terms. Two convergent series may be added term-by-term. Power series.

Sept 11: Power series

Range of values of x for which power series converges is the "interval of convergence." In the interval of convergence, the power series defines a function. Rules for manipulating power series (in the interval of convergence). Expanding functions in about some point in a Taylor series. If a function is make of simpler parts, we can expand the parts in power series and combine the expansions.

Read: Boas, Chap. 1

PS 1 -- Problems: Boas, 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
2. September 14
Complex numbers


Sept 14: Defining and representing complex numbers



Sept 16: Complex infinite series



Sept 17: Elementary functions of complex numbers



Sept 18: More elementary 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
3. September 21
Complex numbers / Linear algebra


Sept 21: Applications of complex numbers: AC circuits



Sept 23: Applications of complex numbers: n-source interference / Intro to linear algebra



Sept 24: Solving systems of linear equations by Gaussian elimination



Sept 25: Determinants



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, 3.5.13, 3.5.37, 3.5.44 [It's optional to check your problems by computer for those problems in which you're told to do so.]
4. September 28
Linear algebra


Sept 28: Basics of vectors



Sept 30: Analytic geometry with vectors / Matrix operations



Oct 1: Matrices: multiplication, inverses, functions of matrices



Oct 2: Linearity and linear transformations



Read: Boas, Chap. 3

PS 4 -- Problems: Boas, 3.6.6, 3.6.17, 3.6.30, 3.7.25, 3.7.33, 3.8.16, 3.8.21, 3.9.15, 3.9.17, 3.11.16, 3.11.19, 3.11.30, [3.11.43, 3.11.51 --- these last two problems are carried over to next week's problem set.] (due Thurs. Oct. 16, 11:59 pm)
5. October 5
Linearity and linear transformations


Oct 5: Rotations and reflections: 2D and 3D

Oct 6: Exam 1 (7-10 pm)

Oct 7: Linear dependence and independence / Homogeneous equations

Oct 8: Homogeneous equations / Matrix trivia



Oct 9: Linear vector spaces



Read: Finish Boas, Chap. 3

PS 5 -- Problems: Boas, Chap. 3: [carried over from last problem set: 3.11.43, 3.11.51], 3.11.35, 3.11.46, 3.12.3, 3.12.7, 3.12.14, 3.12.18
6. October 12
Eigenvalues and eigenvectors


Oct 12: Break

Oct 14: Eigenvalues, eigenvectors, diagonalizing matrices

Oct 15: Geometric interpretation of similarity transformations and diagonalization

Oct 16: Diagonaizing hermitian matrices

Read:

Problems:
7. October 19
Applications of similarity transformations


Oct 19: Orthogonal rotations in 3D / Powers of matrices



Oct 21: Simplifying equations for conic sections / Normal modes of vibrating systems



Oct 22: General vector spaces



Oct 23: Introduction to multivariable calculus / power series in two variables



Read: Start Boas, Chap. 4

PS 6 -- Problems: Boas: 3.14.7, 3.14.15, 4.1.5, 4.1.14, 4.1.20, 4.1.22, 4.2.6, 4.4.1, 4.4.9, 4.4.15 (dues Tuesday, Oct. 27)
8. October 26
Multivariable calculus: differential calculus


Oct 26: Total differentials for functions of one and two independent variables



Oct 28: Chain rule / implicit differentiation



Oct 29: Implicit differentiation / reciprocals of derivatives



Oct 30: Extremum problems: one variable, two variables, and extrema with constraints



Read: Boas, Chap. 4

PS 7 -- Problems: Boas: 4.5.6, 4.6.9, 4.7.6, 4.7.16, 4.7.23, 4.7.25, 4.8.5, 4.9.9, 4.10.5, 4.11.2
9. November 2
Multivariable calculus: differential calculus


Nov 2: Lagrange multipliers



Nov 4: Lagrange multipliers



Nov 5: Change of variables / Differentiating integrals



Nov 6: Multiple integrals



Read: Boas, Chap. 4, start Chap. 5

PS 8 -- Problems: Boas: 4.11.5, 4.11.10, 4.12.5, 4.12.6, 4.12.16, 5.2.6, 5.2.10, 5.2.22, 5.2.40, 5.2.48
10. November 9
Multivariable calculus: integral calculus


Nov 9: Applications of multiple integrals

Nov 11: Applications of multiple integrals / Change of variable in integrals (2D)

Nov 12: Cylindrical and spherical coordinates / Jacobian determinants



Nov 13: Jacobian determinants / Surface integrals / Triple scalar product



Read: Boas, Chap. 6, and read "div, grad, curl, and all that"

PS 9 -- Problems: Boas 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, 6.8.7 [due Tues. Dec. 1, 11:59 pm]
11. November 16
Vector calculus


Nov 16: Vector triple products / differentiating vectors

Nov 17 (evening): Exam 2

Nov 18: Differentiating vectors / directional derivatives and gradients

Nov 19: Gradients / the "del" operator



Nov 20: Vector calculus



Read:

Problems:

12. November 30
Vector calculus


Nov 30: curls, gradients, and path independence of line integrals

Dec 2: Calculating potentials / Green's theorem

Dec 3: Green's theorem / flux



Dec 4: Divergence theorem



Read: Boas, Chap. 6; div, grad, curl, and all that

PS 10 -- Problems: Boas, 6.8.13, 6.8.16, 6.9.3, 6.9.12, 6.10.6, 6.10.9, 6.11.8, 6.11.14, 6.11.21, 6.12.30

13. December 7
Fourier series / ODEs


Dec 7: Stokes theorem

Dec 9: Stokes theorem / Fourier series

Dec 10: Fourier series / first order ODEs

Dec 11: first order ODEs / second order homogeneous ODEs with constant coefficients



Read: Boas, Chap 7.1-7.11 and Chap. 8.1-8.7

Problems: suggested problems (not to turn in): 7.2.6, 7.4.16, 7.5.9, 7.7.1, 7.8.15, 7.9.11, 8.2.16, 8.3.5, 8.4.10, 8.5.12, 8.6.25, 8.6.36, 8.7.5
14. December 14
2nd order differential equations with constant coefficients


Dec 14: second order ODEs with constant coefficients



Read:

Problems: