Physics 27 (Methods of Theoretical Physics) Home Page, Fall 2010

Physics 27: Methods of Theoretical Physics


Welcome to the new semester!


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.


Times and places:


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:



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 8
Infinite series

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
Series / Complex numbers

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
Complex numbers / Linear algebra

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
Linear algebra

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
Linearity and linear transformations

Oct 4: Rotations and reflections: 2D and 3D

Oct 5: Exam 1

7-10 pm
Merrill 220
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
Eigenvalues and eigenvectors

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
Applications of similarity transformations

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
Multivariable calculus: differential calculus

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
Multivariable calculus: differential calculus

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
Multivariable calculus: integral calculus

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
Vector calculus

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
Vector calculus

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
Fourier series / ODEs

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
2nd order differential equations with constant coefficients

Dec 13: second order ODEs with constant coefficients

Dec 15: second order ODEs with constant coefficients