MATH 3423 - Physical Mathematics II, Section 001 - Fall 2007
TR 12:00-1:15 p.m., 359 PHSC
Instructor:
Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu
Office Hours:
Mon 2:30-3:30 p.m., Tue 4:30-5:30 p.m., or by appointment.
Prerequisite:
2443 (Calculus and Analytic Geometry IV),
3413 (Physical Mathematics I).
Course catalog description:
The Fourier transform and applications,
a survey of complex variable theory,
linear and nonlinear coordinate transformations,
tensors, elements of the calculus of variations.
Duplicates one hour of 3333 and one hour of 4103. (Sp)
Text:
D. A. McQuarrie,
Mathematical Methods for Scientists and Engineers,
University Science Books, Sausalito, CA, 2003.
The course will cover (parts of) chapters 4-10, 17-20.
Homework
(solutions are deposited after the due date in the Chemistry-Mathematics
Library, 207 PHSC):
-
Homework 1, due Thu, Aug 30.
-
Homework 2, due Thu, Sep 6.
-
Homework 3, due Thu, Sep 13.
-
Homework 4, due Thu, Sep 27.
-
Homework 5, due Thu, Oct 4.
-
Homework 6, due Thu, Oct 11.
-
Homework 7, due Thu, Oct 18.
-
Homework 8, due Thu, Nov 1.
SOLUTIONS
-
Homework 9, due Tue, Nov 13.
SOLUTIONS
-
Homework 10, due Tue, Nov 27.
SOLUTIONS
-
Homework 11, due Thu, Dec 6.
Content of the lectures:
-
Lecture 1 (Tue, Aug 21):
Reminder: Fourier series:
Fourier series of a periodic function
(using complex exponents or sines and cosines),
even and odd funcitons, sine and cosine series,
convergence of Fourier series, Parseval's theorem,
physical interpretation
(Sec. 15.1-15.3).
Fourier transform:
definition of Fourier transform, examples
(pages 845-849 of Sec. 17.5).
-
Lecture 2 (Thu, Aug 23):
Fourier transform (cont.):
delta function and its derivatives,
Fourier transform of a delta function,
Fourier transform of eiωa,
shifting properties of the FT,
derivative properties of the FT,
convolution, convolution properties of the FT,
Parseval's theorem
(pages 850-854 of Sec. 17.5).
-
Lecture 3 (Tue, Aug 28):
Fourier transforms and partial differential equations:
solving the heat equation on the whole real line
by using Fourier transform
(for a general initial condition and for an
initial condition δ(x));
solving the wave equation on the whole real line
by using Fourier transform
(for a general initial condition and zero initial velocity),
physical interpretation of the solution.
(pages 856, 857, 859, 860 of Sec. 17.6).
-
Lecture 4 (Thu, Aug 30):
Fourier transforms and partial differential equations (cont.):
solving Laplace's equation in the upper half-plane
by using Fourier transform, Poisson's integral formula;
Fourier transform for odd ane even functions,
sine and cosine Fourier transform;
solving Laplace's equation in the first quadrant
by using sine Fourier transform;
multidimensional Fourier transform,
Fourier transform of a spherically symmetric function
in three spatial dimensions
(pages 858, 860, 861 of Sec. 17.6).
-
Lecture 5 (Tue, Sep 4):
Complex numbers and the complex plane:
complex numbers, real and imaginary parts of a complex number,
complex numbers as solutions of quadratic equations,
complex conjugate, addition, subtraction,
multiplication and division of complex numbers,
modulus and argument, the complex plane
(Sec. 4.1).
Functions of a complex variable:
definition and examples
(Sec. 4.2).
Euler's formula and the polar form of complex numbers:
derivation of Euler's formula
(pages 169, 170 of Sec. 4.3).
-
Lecture 6 (Thu, Sep 6):
Euler's formula and the polar form
of complex numbers (cont.):
Cartesian and polar coordinates in the complex plane,
arg and Arg, branch cuts in the complex plane,
multiple-valued functions,
de Moivre's formula,
integer powers and roots of complex numbers,
examples
(Sec. 4.3).
Trigonometric and hyperbolic functions:
definitions and examples
(Sec. 4.4).
The logarithms of complex numbers:
definition and elementary properties
(page 181 of Sec. 4.5).
-
Lecture 7 (Tue, Sep 11):
The logarithms of complex numbers (cont.):
examples
(pages 182-184 of Sec. 4.5).
Powers of complex numbers:
definition, examples: integer real powers,
rational real powers, complex powers
(Sec. 4.6).
Functions, limits, and continuity of complex-valued
functions of a complex variable:
a brief discussion of branch cuts;
limits and continuity, analogies
with functions of two real variables
(Sec. 18.1).
Differentiation: The Cauchy-Riemann equations:
computing derivatives of a complex-valued function
of a complex variable along different lines
in the complex plane
(pages 875-876 of Sec. 18.2).
-
Lecture 8 (Thu, Sep 13):
Differentiation: The Cauchy-Riemann equations (cont.):
derivation of Cauchy-Riemann equations,
analytic functions, entire functions,
regular points, isolated and non-isolated singularities,
poles of order n
(pages 876-878 of Sec. 18.2).
Complex integration: Cauchy's theorem:
definition and examples of integration
along a path in the complex plane
(pages 882-884 of Sec. 18.3).
-
Lecture 9 (Tue, Sep 18):
Complex integration: Cauchy's theorem (cont.):
more examples, Cauchy-Goursat theorem,
applications of the theorem
(pages 885-893 of Sec. 18.3).
-
Lecture 10 (Thu, Sep 20):
Hour exam 1.
-
Lecture 11 (Tue, Sep 25):
Cauchy's integral formula:
derivation of the formula, applications,
generalization to the case of non-simple poles,
examples
(Sec. 18.4).
Taylor series and Laurent series:
power series,
Taylor series of an analytic function,
circle of convergence and radius of convergence
(pages 901-904 of Sec. 18.5).
-
Lecture 12 (Thu, Sep 27):
Taylor series and Laurent series (cont.):
Laurent series, examples
(pages 905-909 of Sec. 18.5).
Residues and the residue theorem:
definition of residue,
integrals of Laurent series and residues
(pages 911-912 of Sec. 18.6).
-
Lecture 13 (Tue, Oct 2):
Residues and the residue theorem (cont.):
computing the residue of a function
at a pole of order N
(pages 913-917 of Sec. 18.6).
Evaluation of real definite integrals:
integrals of F(sinθ,cosθ)
from 0 to 2π, example;
integrals of a rational function F(x)
over the real line for F decaying fast enough
as x→∞, example
(pages 929-932 of Sec. 19.2).
-
Lecture 14 (Thu, Oct 4):
Evaluation of real definite integrals (cont.):
integrals of F(x)sin(mx)
or F(x)cos(mx)
for a rational function F(x)
that decays quickly enough as x→∞, example;
integrals of functions that have a branch cut
(when considered as a functions of a complex variable),
example
(pages 933-936 of Sec. 19.2).
Conformal mapping:
the real and the imaginary parts of an analytic funcion
are harmonic functions (i.e., satisfy Laplaces equation): if
f(z)=f(x+iy)=u(x,y)+iv(x,y),
then Δu(x,y)=0,
Δv(x,y)=0;
conformal maps, Mercator's projection
(pages 954-956 of Sec. 19.5).
-
Lecture 15 (Tue, Oct 9):
Conformal mapping:
examples of conformal mappings,
invariance of the Laplace's equation
under a conformal change of variables
(pages 957-968 of Sec. 19.5).
Conformal mapping and boundary value problems:
idea of the method, example:
solving Laplace's equation ΔΦ(u,v)=0
in the upper half plane v>0
with boundary conditions
Φ(u,0)=5 for u<0 and
Φ(u,0)=0 for u>0
by using the conformal (i.e., analytic) function
u+iv=w=f(z)=z2=(x+iy)2
by using the fact that the solution of
Laplace's equation ΔΨ(x,y)=0
in the infinite strip 0<y<a
with boundary conditions
Ψ(x,0)=5, Ψ(u,a)=0
is Ψ(x,y)=5(1-y/a);
other examples
(pages 970-972 of Sec.19.6).
Conformal mapping and fluid flow:
problem to be solved:
steady irrotational flow of inviscid incompressible fluid
in a simply connected two-dimensional domain;
velocity field
V=(V1,V2)=∇φ
where φ(x,y) is the velocity potential;
φ(x,y) is harmonic due to the
incompressibility;
analytic function
Ω(z)=&phi(x,y)+i&psi(x,y);
boundary condition: V is parallel to the boundary
(pages 977-980 of Sec. 19.7).
-
Lecture 16 (Thu, Oct 11):
Conformal mapping and fluid flow (cont.):
finding the stream function ψ(x,y)
and the complex potential Ωψ(z)
from the velocity potential φ(x,y),
example with
φ(x,y)=exsin(y),
computing the velocity field
V(x,y)=∇φ(x,y)
and the speed |V(x,y)|,
finding the domain in which this function
describes a fluid flow
(pages 981-983 of Sec. 19.7).
-
Lecture 17 (Tue, Oct 16):
Vector spaces and linear transformations:
vector spaces (linear spaces), basis,
change of basis, examples,
matrices, transposed matrix, addition of matrices,
multiplication of a matrix by a number,
matrices of the same size form a vector space,
matrix multiplication, determinants,
Levi-Civita tensor, singular and non-singular matrices,
diagonal matrix, unit matrix, zero matrix, inverse matrix,
trace of a matrix, properties of matrices and determinants,
linear transformations (operators),
matrix elements of a linear operator,
composition of linear operators corresponds
to multiplication of their matrices,
inner product vector spaces
(pages 436-440 of Sec. 9.5;
418-425 of Sec. 9.3,
400-403 of Sec. 9.1,
457-458 of Sec. 10.1).
-
Lecture 18 (Thu, Oct 18):
Vector spaces and linear transformations (cont.):
more on inner product vector spaces,
examples of inner products,
vector spaces of polynomials,
inner products in vector spaces of polynomials,
weight functions,
examples (Legendre, Jacobi, Chebyshev polynomials);
norm, normed vector spaces,
examples of norms: ||u||2,
||u||1 and ||u||∞,
Cauchy-Schwarz inequality (with proof), orthogonality;
quantum mechanics notations:
ket vectors |u> (ordinary vectors),
bra vectors <u|
(linear functionals on the space of ket vectors),
orthogonal projection operator
||u||-2|u> <u|,
(pages 444-447 of Sec. 9.6, handouts).
-
Lecture 19 (Tue, Oct 23):
Hour exam 2.
-
Lecture 20 (Thu, Oct 25):
Vector spaces and linear transformations (cont.):
linearity of the bra "vectors",
matrix elements of an operator in bra-ket notations,
Kronecker symbol, orthonormal basis,
completeness relation,
Gram-Schmidt orthogonalization,
examples of linear operators in the plane
(rotation by θ, multiplication by a constant,
diagonal operators),
writing a linear operator in as a double sum
of |i>aij<j|
(pages 447-448 of Sec. 9.6,
455-458 of Sec. 10.1).
-
Lecture 21 (Tue, Oct 30):
Miscellaneous:
equivalence of norms,
derivation of the equivalence
of the norms ||u||2
and ||u||∞
in R2.
Orthogonal transformations:
definition and examples of orthogonal transformations
in an inner product linear space,
orthogonal matrices.
Eigenvalues and eigenvectors:
definition, characteristic equation,
properties of eigenvalues and eigenvectors
of a symmetric (or Hermitean) matrix.
Diagonalization of matrices:
changing the basis to an eigenbasis
- idea and practical implementation,
function of a square matrix (defined via power series),
exponential of a matrix,
computing the exponential of a matrix
by first diagonalizing the matrix,
solution of an autonomous linear system
with constant coefficients
as an exponential of a matrix
(pages 462-468 of Sec. 10.2,
491-495 of Sec. 10.5).
-
Lecture 22 (Thu, Nov 1):
Diagonalization of matrices (cont.):
simultaneous diagonalization,
commutativity of matrices
as a necessary and sufficient condition
for simultaneous diagonalization;
application - solving a differential equation
by using linear algebra methods,
diagonalization of a symmetric matrix A
through an orthogonal change of basis
- the transformation matrix S is made of
the normalized eigenvectors written as columns
(so that S-1 is equal to ST)
and then
D=S-1AS=STAS
is diagonal, example of simultaneous diagonalization
of a system of two point masses attached to springs
(pages 471-477 of Sec. 10.3,
496-498 of Sec. 10.5).
-
Lecture 23 (Tue, Nov 6):
Vectors in plane polar coordinates:
a detailed discussion of vectors in the plane
in polar coordinates
(pages 355-357 of Sec. 8.2).
-
Lecture 24 (Thu, Nov 8):
Vectors in orthogonal curvilinear coordinates:
orthogonal curvilinear coordinates,
metric coefficients, metric tensor,
expressions for ds and |ds|
using the metric coefficients of the metric tensor,
length of a parametrized curve in orthogonal coordinates,
coordinate-free definition of the gradient of a scalar function,
derivation of the components of the gradient
in polar coordinates in R2,
divergence in polar coordinates in R2
(pages 357-359 of Sec. 8.2, 378-382 of Sec. 8.5).
-
Lecture 25 (Tue, Nov 13):
Vectors in R2
in general curvilinear coordinates:
definition, metric tensor
gij=(∂s/∂ui).(∂s/∂uj);
line element in terms of the components
of the metric tensor:
ds2=gijduiduj;
area element:
dA2=(det(gij))1/2du1du2;
example of explicit computation
of an area in Cartesian and in (nonorthogonal) curvilinear
coordinates.
-
Lecture 26 (Thu, Nov 15):
The Euler-Lagrange equation:
funcitonals, first variation of a functional,
extremum of a functional, derivation of
the Euler-Lagrange equation, examples
(pages 986-988, 993 of Sec. 20.1).
-
Lecture 27 (Tue, Nov 20):
More on orthogonal curvilinear coordinates:
a detailed example on constructing a system
of orthogonal curvilinear coordinates in
R2.
-
Lecture 28 (Tue, Nov 27):
The Euler equation:
particular cases of the Euler-Lagrange equation:
when F does not contain y,
when F does not contain x
(pages 988-990 of Sec. 20.1).
Two laws of physics in variational form:
Lagrangian function, Hamilton's principle,
equation of motion of a particle in R3
attracted to a fixed center by Coulomb attraction,
Fermat's principle of geometric optics
(pages 996, 998-999 of Sec. 20.2).
-
Lecture 29 (Thu, Nov 29):
Hour exam 3.
-
Lecture 30 (Tue, Dec 4):
Two laws of physics in variational form (cont.):
Euler-Lagrange equations for action
that is a function of more than one functions:
I{u1,...,un},
where
uj=un(x),
examples
(pages 996-998 of Sec. 20.2).
Multidimensional variational problems:
Euler-Lagrange equations for action that is a functional
of a function of several variables,
examples: derivation of the wave equation
and Laplace's equation
(pages 1015-1017 of Sec. 20.5).
-
Lecture 31 (Thu, Dec 6):
Multidimensional variational problems (cont.):
derivation and physical interpretation
of the wave equation for a heavy string in a homogeneous
gravity field, asymptotic shape of the string.
Variational problems with constraints:
definition, Lagrange multipliers,
example: heavy cable of constant length
hanging in a homogeneous gravity field, catenoid
(pages 1001-1004 of Sec. 20.3).
Good-bye, words of wisdom, etc.
-
Final exam:
Tuesday, Dec 11, 1:30-3:30 p.m.
Attendance:
You are required to attend class on those days when an
examination is being given;
attendance during other class periods is also
strongly encouraged.
You are fully responsible for the
material covered in each class, whether or not you attend.
Make-ups for missed exams will be given only if
there is a compelling reason for the absence,
which I know about beforehand
and can document independently of your testimony
(for example, via a note or a phone call from a
doctor or a parent).
You should come to class on time;
if you miss a quiz because you came late,
you won't be able to make up for it.
Homework:
It is absolutely essential
to solve a large number of problems on a regular basis!
Homework assignments will be given
regularly throughout the semester
and will be posted on this web-site.
Usually the homeworks will be due at the start
of class on Thursday.
Each homework will consist of several problems,
of which some pseudo-randomly chosen problems will be graded.
Your lowest homework grade will be dropped.
All homework should be written on a 8.5"×11" paper
with your name clearly written, and should be stapled.
No late homework will be accepted!
You are encouraged to discuss the homework problems
with other students.
However, you have to write your solutions clearly
and in your own words - this is the only way to
achieve real understanding!
It is advisable that you first write a draft
of the solutions and then copy them neatly.
Please write the problems in the same order
in which they are given in the assignment.
Shortly after a homework assignment's due date,
solutions to the problems from that assignment
will be placed on restricted reserve in
the Chemistry-Mathematics Library in 207 PHSC.
Quizzes:
Short pop-quizzes will be given in class at random times;
your lowest quiz grade will be dropped.
Often the quizzes will use material
that has been covered very recently
(even in the previous lecture),
so you have to make every effort to keep up
with the material and to study the corresponding
sections from the book right after
they have been covered in class.
Exams:
There will be three in-class midterms and a (comprehensive) final.
The approximate dates for the midterms are
September 18, October 23 and November 27.
The final is scheduled for Tuesday, December 11, 1:30-3:30 p.m.
All tests must be taken at the scheduled times,
except in extraordinary circumstances.
Please do not arrange travel plans that prevent you
from taking any of the exams at the scheduled time.
Grading:
Your grade will be determined by your performance
on the following coursework:
| Pop-quizzes (lowest grade dropped) |
15% |
| Homework (lowest grade dropped) |
15% |
| Three in-class midterms |
15% each |
| Final Examination |
25% |
Academic calendar for
Fall 2007.
Policy on W/I Grades :
Through September 23, you can withdraw
from the course with an automatic W. In addition,
it is my policy to give
any student a W grade,
regardless of his/her performance in the course,
through the extended drop period that ends on December 7.
However, after October 29, you can only drop
via petition to the Dean of your college.
Such petitions are not often granted.
Furthermore, even if the petition
is granted, I will give you a grade
of "Withdrawn Failing" if you are
indeed failing at the time of your petition.
The grade of I (Incomplete)
is not intended to serve as
a benign substitute for the grade of F.
I only give the I grade
if a student has completed the majority
of the work in the course
(for example everything except the final exam),
the coursework cannot be completed
because of compelling and verifiable problems
beyond the student's control, and the student expresses a clear
intention of making up the missed work as soon as possible.
Academic Misconduct: All cases of suspected academic misconduct will
be referred to the Dean of the College of Arts and Sciences for prosecution
under the University's Academic Misconduct Code. The penalties can be quite
severe. Don't do it!
For more details on the University's
policies concerning academic misconduct see
http://www.ou.edu/provost/integrity/.
See also the Academic Misconduct Code,
which is a part of the Student Code
and can be found at
http://www.ou.edu/studentcode/.
Students With Disabilities:
The University of Oklahoma is committed to providing reasonable accommodation
for all students with disabilities. Students with disabilities who require
accommodations in this course are requested to speak with the instructor
as early in the semester as possible. Students with disabilities must be
registered with the Office of Disability Services prior to receiving
accommodations in this course. The Office of Disability Services is located
in Goddard Health Center, Suite 166: phone 405-325-3852 or TDD only
405-325-4173.
Good to know: