1) Outline Pros and Cons and Winners and Losers of Trade Using information from
this Economist Article.
After Nafta trade with Mexico increased as did trade with China after they joined the WTO in 200
4
1. BRIEF INTRODUCTION TO VECTORS AND MATRICES
in 3-dimension: Let x =
x1 x2 x3
y1 and y = y2 , y3
the dot product of x and y is, x y = x1 y1 + x2 y2 + x3 y3 Definition 1.3. Matrix product Let A =
2
1. BRIEF INTRODUCTION TO VECTORS AND MATRICES
other elements are 0. So 2 2 and 3 3 zero 1 0 1 0 0 1 and 0 1 0 0
matrices are 0 0 1
A vector is a matrix with one row or one column. In this chapter, a
CHAPTER 1
Brief Introduction to Vectors and Matrices
In this chapter, we will discuss some needed concepts found in introductory course in linear algebra. We will introduce matrix, vector, vector-valu
5. PHASE AND VECTOR FIELD DIAGRAMS
21
font. You can do this for headings etc. Graph Vector field Goal: Familiar your self with an many different kind of vector fields, defined by x = f (t, x), as poss
20 1. EXPLICITLY SOLVABLE FIRST ORDER DIFFERENTIAL EQUATIONS
equation of tangent line y - yj = mij (x - xi ) we get another point (xi + h , yj + mij h ). 2 2 (3) Draw line segment from (xi , yj ) to (
5. PHASE AND VECTOR FIELD DIAGRAMS
19
Figure 12. Vector field for f (t, x) = sin(t) of the dynamics of solutions as shown in the next not so complicated diagram,
Figure 13. Vector field for f (t, x) =
18 1. EXPLICITLY SOLVABLE FIRST ORDER DIFFERENTIAL EQUATIONS
In Example ?, c = 1 is a sink, c = -2, 3 are source. (x) From phase diagram and the sign of derivative of x = dfdt = df (x) dx (x) = f (x)
5. PHASE AND VECTOR FIELD DIAGRAMS
17
Step one: Set f (x) = 0 solve the equation. The solution is called the critical number of the equation x = f (x). Step two: Plot the solution one a horizontal (or
16 1. EXPLICITLY SOLVABLE FIRST ORDER DIFFERENTIAL EQUATIONS
(t,x) 1 is x(t) = t 3 . The reason is that for f (t, x) = x 3 , fx = x 3 which is undefined at (0, 0). So the conditions of the theorem are
4. INITIAL VALUE PROBLEM AND EXISTENCE THEOREM
15
with a(t) = 2t and b(t) = 4t3 . So x(t) = e-t
2
a(t) dt =
2
2t dt = t2 , and
4t3 et dt + C .
Using method of substitution or Mathcad , we find x(t) =
14 1. EXPLICITLY SOLVABLE FIRST ORDER DIFFERENTIAL EQUATIONS
Here is how to get the graph in Mathcad :
0.2 - we first defined a function x(t, C):= 0.02+Ce-0.2t with two arguments t and C by typing x(t
3. LINEAR EQUATIONS AND BERNOULLI EQUATIONS
13
Solution
From Example 3.3, x= be e
a dt
a dt
dt + C
So x= eat beat dt + C = eat
b at e a
,
+C
is the general solution. When C = 0 we have x = a , a stead
12 1. EXPLICITLY SOLVABLE FIRST ORDER DIFFERENTIAL EQUATIONS
Example 3.3. In modelling a population with its births(per unit time) proportional to current population level and the deaths(per unit time
3. LINEAR EQUATIONS AND BERNOULLI EQUATIONS
11
suppose r(t) = cos2 (t), find the general solution and graph the solution for several different value for the constant C in the same coordinate. Explain
1. VECTORS AND MATRICES
5
as Ax = y with A= x= x1 x2 , and y = y1 y2 . a b c d ,
Definition 1.4. A square (ex. 2 2 or 3 3) matrix A is invertible if there is a matrix A-1 such that AA-1 = A-1 A = I. T
6
1. BRIEF INTRODUCTION TO VECTORS AND MATRICES
Example 1.4. Solve the system of equation 3x1 - 4x2 + 5x3 = 2 -2x1 + 5x2 = 7 x - 5x + 8x = -1 1 2 3 The equation can be rewrite Ax = y with 3 -4 5 A = -
1) Outline Pros and Cons and Winners and Losers of Trade Using information from this
Economist Article. W
2) Below the first article is another one from the NYTimes (most of text is below but some
gra
2 1. NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS
while loop: Typically used when unknown number of steps need to be carried out. You can break a while loop with a break statement or a retur
CHAPTER 1
Numerical Methods for Ordinary Differential Equations
In this chapter we discuss numerical method for ODE . We will discuss the two basic methods, Euler's Method and Runge-Kutta Method. 1. N
18
1. BRIEF INTRODUCTION TO VECTORS AND MATRICES
(2) Condition Number In solving Ax = b, one number is very important, it is called the condition number, which can be defined as C(A) = | s , where s i
16
1. BRIEF INTRODUCTION TO VECTORS AND MATRICES
Here are some sets of linearly independent functions that we encounter in solving a system of differential equations, assume k1 , k2 , , kn are differe
3. LINEARLY INDEPENDENCY
15
Example 3.1. For x1
4 8 , we can form a matrix, 0
2 0 = 3 , x2 = 1 , and x3 = 4 -4
2 3 4 A = 0 1 -4 , 4 8 0
apply rref(type rref and in the place holder type A, then p
14
1. BRIEF INTRODUCTION TO VECTORS AND MATRICES
Notice: - Press [Shift][/]to get the derivative operator and press [Ctrl][I] to get the antiderivative operator. - To get dx(t)simplif y you type dx(t)
12
1. BRIEF INTRODUCTION TO VECTORS AND MATRICES
The antiderivative v(t) dt of an vector-valued function v(t) is a vector-valued function whose entries are the antiderivative of corresponding entries
10
1. BRIEF INTRODUCTION TO VECTORS AND MATRICES
each column represents a eigenvector. Since multiplying an eigenvector by a nonzero constant you still get an eigenvector, so we can simplify 1+ 3 1- 3