REGIONS
North America-2
Western Balkans-6
The European Union-28
COUNTRY
GDP PER CAPITA 2013
FINAL CONSUMPTION EXPENDITURE PER CAPITA
Canada
37336
37336
United States
45665
37693
Bosnia and Herzegovi
6745
7567
Albania
8763
8814
The fyR of Macedoni
10335
91
Ordinary and Partial Dierential Equations
Lecture 8. Linear systems of dierential equations
Bibliography: Nagle, Sna, Sieder - Fundamentals of DEs and BVPs (6th ed.) - chapter 9
LINEAR SYSTEMS IN NORMAL FORM
A linear system of dierential equations can be
Ordinary and Partial Dierential Equations
Lecture 7. Higher order linear dierential equations
Bibliography: Nagle, Sna, Sieder - Fundamentals of DEs and BVPs (6th ed.) - chapter 6
INTRODUCTION
A linear dierential equation of order n is of the form
an (x)y
Ordinary and Partial Dierential Equations
Lecture 5. Linear second-order dierential equations with variable coecients
Homogeneous and nonhomogeneous equations. Cauchy-Euler equations.
Bibliography: Nagle, Sna, Sieder - Fundamentals of DEs and BVPs (6th ed
Ordinary and Partial Dierential Equations
Lecture 6. Introduction to systems and phase plane analysis
Bibliography: Nagle, Sna, Sieder - Fundamentals of DEs and BVPs (6th ed.) - chapter 5
DIFFERENTIAL OPERATORS
We dene the dierential operator D which acts
Ordinary and Partial Dierential Equations
Lecture 3. First-order dierential equations - Part 2.
Exact equations. Special integrating factors. Substitutions and transformations.
Bibliography: Nagle, Sna, Sieder - Fundamentals of DEs and BVPs (6th ed.) - se
Ordinary and Partial Dierential Equations
Lab 4. Mathematical Modeling: Consecutive Reactions for Batch Reactors
Bibliography:
[1] Nagle, Sna, Sieder - Fundamentals of DEs and BVPs (6th ed.) - chapter 5;
[2] Holland, C.D., Anthony R.G. - Fundamentals of C
Ordinary and Partial Dierential Equations
Lab 3. Analytical and graphical analysis of systems
Bibliography:
[1] Nagle, Sna, Sieder - Fundamentals of DEs and BVPs (6th ed.) - chapter 5;
[2] Perelson, Kerschner, DeBoer - Dynamics of HIV Infections of CD4+ T
Ordinary and Partial Dierential Equations
Lecture 1. Introduction
Bibliography: Nagle, Sna, Sieder - Fundamentals of DEs and BVPs (6th ed.) - sections 1.1, 1.2, 1.3
BACKGROUND
In the sciences and engineering, mathematical models are developed to aid in th
Ordinary and Partial Dierential Equations
Lab 2. Numerical solutions: Eulers method
Bibliography: Nagle, Sna, Sieder - Fundamentals of DEs and BVPs (6th ed.) - sections 1.4; 3.6.
Objectives:
1. To learn how to numerically approximate solutions using Euler
Ordinary and Partial Dierential Equations
Lab 1. Direction elds and numerical solutions of rst-order DEs
Bibliography: Nagle, Sna, Sieder - Fundamentals of DEs and BVPs (6th ed.) - section 1.3.
Objectives:
1. To learn how to use Mathematica to create the
CALCULUS HANDOUT 11 - LINE AND SURFACE INTEGRALS
ELEMENTARY CURVES
An elementary curve is a set of points C R3 for which there exists a closed interval [a, b] R and a function
: [a, b] C which is bijective on [a, b) and smooth (of class C 1 ).
The points
CALCULUS HANDOUT 9 - HIGHER ORDER DIFFERENTIABILITY. LOCAL EXTREMA.
Let f : A Rn Rm be a partially dierentiable function with respect to every variable xj , j = 1, n on A.
SECOND ORDER PARTIAL DERIVATIVES:
fi
f is two times partially dierentiable at a wi
CALCULUS HANDOUT 10 - DOUBLE AND TRIPLE INTEGRALS
JORDAN MEASURABLE SETS IN R2
Consider the set of one dimensional bounded intervals of the form (a, b), [a, b), (a, b], [a, b], where a, b R.
The cartesian product = I1 I2 of two intervals of this type will
CALCULUS HANDOUT 8 - PARTIAL AND DIRECTIONAL DERIVATIVES. DIFFERENTIA
BILITY. FRECHET DERIVATIVE. LOCAL INVERSION AND IMPLICIT FUNCTION THMS
PARTIAL DERIVATIVES
Let f : A Rn R1 be a real valued function of n variables and a = (a1 , a2 , ., an ) Int(A). Th
CALCULUS HANDOUT 7
FUNCTIONS OF SEVERAL VARIABLES. LIMITS AND CONTINUITY.
THE VECTOR SPACE Rn
Rn = cfw_(x1 , x2 , ., xn )|xi R1 , i = 1, 2, ., n. The elements of Rn are called vectors.
Rn is a n-dimensional vector space with respect to the sum and the sca
CALCULUS HANDOUT 6 - RIEMANN-DARBOUX INTEGRAL. FOURIER SERIES
THE RIEMANN DARBOUX INTEGRAL
A partition P of the interval [a, b] is a nite set of points cfw_x0 , x1 , ., xn satisfying a = x0 < x1 < . <
xn = b. Consider a function f dened and bounded on [a
CALCULUS HANDOUT 3 - LIMITS, CONTINUITY, DIFFERENTIABILITY
LIMITS
Suppose that f (x) is a function dened on an open interval containing the point a (except possibly not
at a itself). We say that L is the limit of the function f as x approaches a if for an
CALCULUS HANDOUT 5 - POWER SERIES. TAYLOR POLYNOMIALS
POWER SERIES
an xn is called power series.
A series of functions of the form
n=0
The Abel-Cauchy-Hadamard theorem: the set of convergence of a power series:
1
,
if = 0
we have:
Considering = lim n |an
CALCULUS HANDOUT 4 - SEQUENCES OF FUNCTIONS, SERIES OF FUNCTIONS
SEQUENCES OF FUNCTIONS
A sequence of real valued functions dened on A R is a function F : N cfw_f | f : A R.
We write F (n) = fn and the sequence of functions is denoted by (fn ).
An element
CALCULUS HANDOUT 2 - SERIES
An innite series is an expression of the form
an = a1 + a2 + . + an + .
n=1
where (an ) is a sequence of real numbers. The number an is called the n-th term of the series.
The n-th partial sum sn of the series
an is the sum of
CALCULUS HANDOUT 1 - SEQUENCES
A sequence of real numbers is a function n an whose domain is the set of positive integers N and
whose values belong to the set of real numbers R. Usual notation: (an ).
A sequence (an ) is increasing if an an+1 for all n N.
Probability & Statistics
LAB 5. Statistics with Maple. CLT. Condence Intervals. Hypothesis testing.
The tools available in the Statistics package of Maple can greatly facilitate various tasks such as
calculations with abstract random variables and symboli
Probability & Statistics
LAB 3. Simulating discrete random variables.
If we want to simulate a discrete random variable X having the probability mass function
P (X = xi ) = pi
i = 1, 2, 3.
where
pi = 1
i
we consider a uniformly distributed random variabl
Probability & Statistics
HANDOUT 4. Introduction to Statistics
Bibliography: Michael Baron - Probability and Statistics for Computer Scientists - chapter 8
WHAT IS STATISTICS? universal language of sciences
STATISTICS = science of collecting, classifying,
Probability & Statistics
LAB 2. Random Number Generators.
To generate a sequence of numbers, the idea is to start with some initial values x1 , x2 , ., xk , called seeds,
and use a recurrence formula such as:
xn = g(xn1 , xn2 , ., xnk )
n>k
To insure that
Probability & Statistics
LAB 4. Continuous random variables.
Part I. Some experiments.
Remember from Lab 3 that generating a uniformly distributed random variable U over the interval (0, 1)
is equivalent to generating a random number in the interval (0, 1
Probability & Statistics
LAB 1. Experiments and simulations.
In probability theory, we will be particularly interested in repeating a chance experiment a large number
of times. For example, considering the experiment of rolling a die, we want to estimate