Miami Dade College
Mathematics Department
Review For Final Exam
MAT 1033 / ,_
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MULTIPLE CHOICE Choose the one alternative that best completes the statement or answers the question.
Solve using any appropria
Chapter 2 Activity
1. Consider the numbers 23.68 and 4.12. The sum of these numbers has _
significant figures, and the product of these numbers has _ significant figures.
A. 3, 3
B. 4, 4
C. 3, 4
D. 4, 3
E. 3, 2
2.
A.
B.
C.
D.
E.
Express the volume 622.6 c
THE LAPLACE EQUATION
The Laplace (or potential ) equation is the equation
u = 0.
where is the Laplace operator
2
in R
x22
2
=
+ 2
in R2
x2
y
2
2
2
+
+
in R3
=
x2
y 2
z 2
The solutions u of the Laplace equation are called harmonic functions and play
an imp
THE METHOD OF SEPARATION OF VARIABLES
To solve the BVPs that we have encountered so far, we will use separation of
variables on the homogeneous part of the BVP. This separation of variables leads
to problems for ordinary dierential equations (some with en
FOURIER SERIES PART I:
DEFINITIONS AND EXAMPLES
To a 2-periodic function f (x) we will associate a trigonometric series
a0
+
an cos(nx) + bn sin(nx) ,
2
n=1
or in terms of the exponential eix , a series of the form
cn einx .
nZ
For most of the functions
BESSEL EQUATIONS AND BESSEL FUNCTIONS
Bessel functions form a class of the so called special functions. They are important in math as well as in physical sciences (physics and engineering). They
are especially important in solving boundary values problems
FOURIER SERIES PART II:
CONVERGENCE
We have seen in the previous note how to associate to a 2-periodic function f
a Fourier series
a0
+
an cos(nx) + bn sin(nx) .
2
n=1
Now we are going to investigate how the Fourier series represents f . Let us rst
intro
FOURIER SERIES PART III:
APPLICATIONS
We extend the construction of Fourier series to functions with arbitrary periods,
then we associate to functions dened on an interval [0, L] Fourier sine and Fourier
cosine series and then apply these results to solve
THE WAVE EQUATION
The aim is to derive a mathematical model that describes small vibrations of a
tightly stretched exible string for the one-dimensional case, or of a tightly stretched
membrane for the dimensional case. The derivation of these models is m
LEGENDRE POLYNOMIALS AND APPLICATIONS
We construct Legendre polynomials and apply them to solve Dirichlet problems
in spherical coordinates.
1. Legendre equation: series solutions
The Legendre equation is the second order dierential equation
(1 x2 )y 2xy
THE HEAT EQUATION
The main equations that we will be dealing with are the heat equation, the wave
equation, and the potential equation. We use simple physical principles to show how
these equations are derived. We start the discussion with the heat equati
STURM-LIOUVILLE PROBLEMS:
GENERALIZED FOURIER SERIES
1. Regular Sturm-Liouville Problem
The method of separation of variables to solve boundary value problems leads
to ordinary dierential equations on intervals with conditions at the endpoints of
the inte
FOURIER-BESSEL SERIES AND
BOUNDARY VALUE PROBLEMS IN
CYLINDRICAL COORDINATES
The parametric Bessels equation appears in connection with the Laplace operator in polar coordinates. The method of separation of variables for problem with
cylindrical geometry
CLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR
SECOND ORDER LINEAR PDE
1. Linear Partial Differential Equations
A partial dierential equation (PDE) is an equation, for an unknown function
u, that involves independent variables, x, y, , the function u,
NONHOMOGENEOUS BOUNDARY VALUE PROBLEMS
AND PROBLEMS IN HIGHER DIMENSIONS
We illustrate how eigenfunctions expansions can be used to solve more general boundary value problems. These include some nonhomogeneous problems and
problems in higher dimensions.
1