MATH 3413 - Physical Mathematics I, Section 001
- Spring 2006
MWF 9:30 - 10:20 a.m., 363 PHSC
Instructor:
Nikola Petrov, 802 PHSC, (405)325-4316, npetrov AT math.ou.edu.
Office Hours:
MWF 2-3 p.m., or by appointment, in 802 PHSC.
Prerequisite:
MATH 2443 (Calculus and Analytic Geometry IV)
or concurrent enrollment.
Text:
Mary L. Boas.
Mathematical Methods in the Physical Sciences,
3rd edition, John Wiley & Sons, 2005, ISBN 0471198269.
We will cover Chapter 2, 7, 8, 12 and Chapter 13.
Homework
(Solutions are deposited after the due date in the Chemistry-Mathematics
Library, 207 PHSC)
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Homework 1, due Mon, Jan 30.
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Homework 2, due Mon, Feb 6.
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Homework 3, due Mon, Feb 13.
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Homework 4, due Mon, Feb 27.
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Homework 5, due Mon, Mar 6.
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Homework 6, due Mon, Mar 20.
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Homework 7, due Mon, Mar 27.
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Homework 8, due Mon, Apr 10.
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Homework 9, due Mon, Apr 17.
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Homework 10, due Mon, Apr 24.
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Homework 11, due Mon, May 1.
Content of the lectures:
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Lecture 1 (Wed, Jan 18):
Complex numbers:
Introduction, real and imaginary part of a complex number,
the complex plane, Cartesian and polar coordinates,
modulus and argument of a complex number, complex conjugate
(Sec. 2.1, 2.2, 2.3, 2.4).
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Lecture 2 (Fri, Jan 20):
Complex algebra:
Addition, subtraction, multiplication and division
of complex numbers,
simplifying an expression to an x+iy form,
complex conjugate of an expression,
modulus of an expression, complex equations
(Sec. 2.5A-D).
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Lecture 3 (Mon, Jan 23):
Complex algebra (cont.):
More on complex equations,
graphs defined by complex equations and inequalities.
Complex infinite series:
convergence and absolute convergence of a complex series,
absolute convergence implies convergence
(of both real and complex series)
(Sec. 2.5E,D, 2.6).
Please refresh your memory about series with real terms,
in particular, geometric series, ratio test, alternating series.
One place where you can find all these topics is
Chapter 1 of the textbook.
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Lecture 4 (Wed, Jan 25):
Complex infinite series (cont.):
Examples of convergent and divergent series,
examples of the dangers in careless dealing
with infinite series.
Comples power series:
Real power series and interval of convergence,
comples power series and disk of convergence,
examples
(Sec. 2.6, 2.7).
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Lecture 5 (Fri, Jan 27):
Elementary functions of complex numbers:
functions involved only addition, subtraction,
multiplication and division,
defining a function as a sum of a complex power series,
exponential function - definition, properties;
binomial coefficients, binomial formula.
Euler's formula:
derivation and applications.
Powers and roots of complex numbers:
DeMoivre's theorem
(Sec. 2.8, 2.9, 2.10).
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Lecture 6 (Mon, Jan 30):
Powers and roots of complex numbers (cont.):
application of DeMoivre's theorem to computing
powers and roots of complex numbers, examples.
The exponential and trigonometric functions:
definitions of the trigonometric functions of a complex argument
through the complex exponential function, applications.
Hyperbolic functions:
definition, properties.
(Sec. 2.10, 2.11, 2.12).
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Lecture 7 (Wed, Feb 1):
Logarithms:
definition, principal value, examples.
Complex roots and powers:
definition (and the reasons behind it), examples
(Sec. 2.13, 2.14).
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Lecture 8 (Fri, Feb 3):
Inverse trigonometric and hyperbolic functions:
computing the value of arcsin or arccos
of an argument greater than 1 in absolute value;
an example of using hyperbolic functions to compute
an integral.
Introduction to Fourier series:
Taylor series as expansion in a basis of polynomials,
idea of Fourier series - expansion of a (periodis) function
in terms of sines and cosines;
periodic functions, examples
(Sec. 2.15, 7.2, 7.3).
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Lecture 9 (Mon, Feb 6):
Applications of periodic functions:
more examples of periodic and non-periodic functions;
wave processes - definition of amplitude,
wave number, angular frequency, initial phase,
wavelength, linear frequency, period, and speed
of a sinusoidal wave
(Sec. 7.2, 7.3).
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Lecture 10 (Wed, Feb 8):
Applications of periodic functions (cont.):
more examples and pictures.
Average value of a function:
definition, properties, examples.
Fourier coefficients:
definition
(Sec. 7.3, 7.4, 7.5).
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Lecture 11 (Fri, Feb 10):
Fourier coefficients (cont.):
derivation of the formulae for the Fourier coefficients,
examples, meaning of the term a0/2
(Sec. 7.5).
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Lecture 12 (Mon, Feb 13):
Dirichlet conditions:
Dirichlet Theorem on convergence of Fourier series,
examples.
Complex form of Fourier series:
derivation of expressions for the coefficients,
analogy with computing the components
of vectors in finite-dimensional vector spaces
by scalar multiplication
(Sec. 7.6, 7.7).
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Lecture 13 (Wed, Feb 15):
Hour exam 1.
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Lecture 14 (Fri, Feb 17):
Complex form of Fourier series (cont.):
scattered remarks.
Other intervals:
Fourier series of periodic functions of an arbitrary period.
Even and odd functions:
definition of even and odd functons,
decomposition of an arbitrary function
as a sum of an even and an odd function,
integrals of an even/odd function over an interval
symmetric around 0
(Sec. 7.8, 7.9).
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Lecture 15 (Mon, Feb 20):
Even and odd functions (cont.):
Fourier series of a function f(x)
defined on a finite interval (0,l)
- extending f(x) as an even 2l-periodic function
(obtaining a cosine Fourier series),
extending f(x) as an odd 2l-periodic function
(obtaining a sine Fourier series),
extending f(x) as an l-periodic function
- please read the Example on pages 367-369 of the book
(Sec. 7.9).
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Lecture 16 (Wed, Feb 22):
Parseval's Theorem:
relationship between the |v|2 for
a 3-dimensional vector
v=(v1,v2,v3)
and the sum of the squares of its components;
Parseval's theorem and its relation with the finite-dimensional case,
using Parseval's theorem to find sums of certain infinite series
(Sec. 7.11).
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Lecture 17 (Fri, Feb 24):
Introduction to ODEs:
nth order ODE and its general solution
(generally a n-parameter family of functions),
initial conditions (ICs) and initial value problems (IVPs),
boundary conditions (BCs) and boundary value problems (BVPs),
examples.
Separable ODEs:
definition and method of solution, examples
(Sec. 8.1, 8.2).
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Lecture 18 (Mon, Feb 27):
Separable ODEs (cont.):
singular solutions, example.
Linear first-order equations:
definition, integrating factor,
examples
(Sec. 8.2, 8.3).
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Lecture 19 (Wed, Mar 1):
Bernoulli equation:
y'+P(x)y=Q(x)yn,
where n can be any real number;
basic substitution: set z=y1-n
to be the new unknown function, and solve the resulting
linear equation for z(x).
Exact equations:
M(x,y)dx+N(x,y)dy=0,
condition for exactness:
∂M/∂y=∂N/∂x,
method of solving the equation to find
the solution defined implicitly by
F(x,y)=0
(Sec. 8.4).
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Lecture 20 (Fri, Mar 3):
Homogeneous equations:
basic substitution: v=y/x.
Homogeneous second order linear equations with constant coefficients:
definition, example of application: damped harmonic oscillator
with driving force
mx''=-kx-γx'+Acos(t)
(Sec. 8.4, 8.5).
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Lecture 21 (Mon, Mar 6):
Homogeneous second order linear equations with constant coefficients (cont.):
characteristic equation, general solutions in the different cases:
Case 1 (two distinct real roots r1 and r2 of the char. eqn.):
y(x)=C1er1x+C2er2x;
Case 2 (one double root r1 of the char. eqn.):
y(x)=(C1+C2x)er1x;
Case 3 (two non-real, necessarily complex conjugate, roots
r1=α+iβ
and
r2=α-iβ
of the char. eqn.):
y(x)=C1eαxcos(βx)+C2eαxsin(βx);
comments about the general solution of
a homogeneous linear equation of order n with constant coefficients
(Sec. 8.5).
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Lecture 22 (Wed, Mar 8):
Nonhomogeneous second order linear equations with constant coefficients:
(general solution of nonhomogeneous equation)
=
(general solution of homogeneous equation)
+
(particular solution of nonhomogeneous equation);
finding a particular solution of
Ly=f(x)
in the case
f(x)=ecxPn(x):
if c is a root of the characteristic equation
of multiplicity s,
then look for a particular solution of the nonhomogeneous equation
of the form
yp(x)=xsecxQn(x)
(Sec. 8.6).
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Lecture 23 (Fri, Mar 10):
Nonhomogeneous second order linear equations with constant coefficients (cont.):
finding a particular solution of
Ly=f(x)
in the case
f(x)=ecxPn(x)cos(dx) or f(x)=ecxPn(x)sin(dx):
if c+id is a root of the characteristic equation
of multiplicity s,
then look for a particular solution of the nonhomogeneous equation
of the form
yp(x)=xsecx[Qn(x)cos(dx)+
Rn(x)sin(dx)].
Here are some notes (in pdf format) on finding general solutions
of
homogeneous
and
nonhomogeneous
linear equations with constant coefficients
(Sec. 8.6).
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Lecture 24 (Mon, Mar 20):
Other second-order equations:
(A) Equations with the dependent variable y missing
(set p=y' and reduce the ODE to a 1st-order ODE for
p(x));
(B) Equations with the independent variable x missing
(set p=dy/dx and use the chain rule to show that
d2y/dx2=pdp/dy which reduces the ODE to a 1st-order ODE for p(y));
(C) Equations of the type y''+f(y)=0
(multiply by y' and integrate to obtain a 1st-order equation)
(Sec. 8.7).
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Lecture 25 (Wed, Mar 22):
Other second-order equations (cont.):
(D) Euler's equation
(change the independent variable from x
to z=ln(x))
(Sec. 8.7).
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Lecture 26 (Fri, Mar 24):
Other second-order equations (cont.):
using energy conservation law
- the example of a simple harmonic oscillator
(Problem 4 of Homework 7).
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Lecture 27 (Mon, Mar 27):
Review for Hour exam 2.
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Lecture 28 (Wed, Mar 29):
Hour exam 2.
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Lecture 29 (Fri, Mar 31):
Introduction to PDEs:
the concept of a PDE, order of a PDE,
amount of arbitrariness in the general
solution of an nth order PDE
for a function of k variables;
PDEs coming from physics
- Laplace, Poisson, heat (diffusion),
wave, Helmholtz, Schroedinger
(Sec. 13.1).
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Lecture 30 (Mon, Apr 3):
Laplace's equation in a rectangle;
separation of variables:
solving the boundary value problem
Δu(x,y)=0,
u(0,y)=0,
u(a,y)=0,
u(x,0)=0,
u(x,b)=f(x)
in the rectangle
0≤x≤a,
0≤y≤b;
separation of variables:
looking for a solution in the form
u(x,y)=X(x)Y(y),
obtaining a BVP
X''(x)-αX(x)=0,
X(0)=0, X(a)=0
for the function X(x);
finding values of α for which
this BVP has a nontrivial solution
(i.e., a solution that is not identically zero)
(Sec. 13.2).
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Lecture 31 (Wed, Apr 5):
Laplace's equation in a rectangle;
separation of variables (cont.):
complete solution of the BVP from last lecture,
example of a particular boundary condition f(x);
another BVP
Δu(x,y)=0,
ux(0,y)=0,
ux(a,y)=0,
u(x,0)=0,
u(x,b)=f(x)
in the rectangle
0≤x≤a,
0≤y≤b,
separation of variables,
BVP for X(x):
X''(x)-αX(x)=0,
X'(0)=0, X'(a)=0,
physical meaning of the BCs
(Sec. 13.2).
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Lecture 32 (Fri, Apr 7):
Laplace equation in a rectangle, separation of variables (cont.):
complete solution of the BVP from the last lecture.
Heat equation (diffusion equation):
heat equation, boundary and initial conditions for it,
physical meaning of the boundary and initial conditions;
separation of variables in heat equation
on an interval 0≤x≤a:
ut=σ2uxx,
u(0,t)=0,
u(a,t)=0,
u(x,0)=f(x)
(Sec. 13.2, 13.3).
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Lecture 33 (Mon, Apr 10):
Heat equation (cont.):
finishing the problem from last lecture,
discussion on the physical meaning
of the Dirichlet and Neumann boundary conditions
(Sec. 13.3).
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Lecture 34 (Wed, Apr 12):
Wave equation:
wave equation,
physical meaning of the Dirichlet, Neumann,
and mixed boundary conditions,
initial-boundary value problems for the wave equation,
separation of variables
in the (1+1)-dimensional wave equation
on 0≤x≤a, 0≤t
(Sec. 13.4).
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Lecture 35 (Fri, Apr 14):
Wave equation (cont.):
general solution of the (1+1)-dimensional wave equation
for -∞<x<∞:
u(x,t)=f(x-ct)+g(x+ct) and interpretation of the terms
f(x-ct)
g(x+ct)
as waves propagating to the right (resp. left)
at a speed c;
interpretation of the terms
in the solution of
the (1+1)-dimensional wave equation
on 0≤x≤a, 0≤t
by separation of variables
-- angular frequency ωn,
linear frequency fn,
period Tn,
wavelength λn;
remarks about effects in playing guitar
(Sec. 13.4).
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Lecture 36 (Mon, Apr 17):
Laplace transform (LT):
definition and examples.
Using LT to solve IVPs for ODEs:
LT of first and second derivatives of a function,
an example of using LT to solve
and IVP for a second-order constant-coefficient ODE
(Sec. 8.8, 8.9).
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Lecture 37 (Wed, Apr 19):
Using LT to solve IVPs for ODEs (cont.):
solving systems of ODEs with LT.
Convolution:
definition of a convolution,
commutativity property: f*g=g*f;
LT of a convolution:
L{f*g}=F(p)G(p)
(Sec. 8.9, 8.10).
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Lecture 38 (Fri, Apr 21):
Convolution (cont.):
using convolution to solve IVPs for ODEs,
transfer function,
representation of the solution of the IVP
as a convolution (Duhamel's principle).
The Dirac delta-function:
definition of a delta function
δa(t)
as a "limit" of functions
fa,ε(t)
as ε→0+,
"true" definition of a delta-function
(integral of
δa(t)g(t)
over t from -∞ to ∞
is equal to g(a) for
any continuous function g(x));
Laplace transform of
δa(t),
Heaviside function θa(t)
and its Laplace transform;
δa(t)
is the "derivative" of θa(t)
with respect to t
(Sec. 8. 10, 8.11).
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Lecture 39 (Mon, Apr 24):
The Dirac delta function (cont.):
derivatives of
δa(x).
Series solutions of differential equations:
introduction
(Sec. 8.11, 12.1).
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Lecture 40 (Wed, Apr 26):
Legendre's equation:
an intro to Legendre's equation and its series solution
(Sec. 12.2).
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Lecture 41 (Fri, Apr 28):
Facts about Legendre polynomials:
Legendre polynomials as solutions of Legendre's equation,
Rodrigues' formula,
generating function for Legendre polynomials,
physical applications
(Sec. 12.2, 12.4, 12.5).
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.
Pop-quizzes will be given in class at random times,
and all pop-quiz grades will be taken into account
in forming your final grade.
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 phone call from a
doctor or a parent).
Homework: Homework assignments will be set
regularly throughout the semester
and will be posted on this web-site.
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 hand-in assignments and must be submitted
in class on the due date.
No late homework will be accepted!
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.
Grading:
Your grade will be determined by your performance
on the following coursework:
-
three midterm exams, 50 minutes each
(each midterm contributes 17% to your overall grade);
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homework (20% of your overall grade);
-
pop-quizzes (5% of your overall grade);
-
final exam (24% of your overall grade).
Good to know:
Some Important Dates :
- Final Day to Register or Add a Class: Friday, January 20.
- Last day to drop a class with a refund: Monday, January 30.
- Last day to drop a class without recorded grade: Monday, January 30.
- Last day to withdraw with an automatic W:
Friday, February 24.
- Last day to withdraw with a W/F without permission
of the Dean: Friday, May 5.
Policy on W/I Grades : Through Friday, February 24, 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 Friday, May 5. However,
after May 5, 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. Thus it is in your
own best interest to drop the course on or before May 5 if you think there
is a reasonable chance that you will not want to see the course through to
the end.
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
click here.
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.