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(MATH101)notes.pdf
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PRELIMINARIES

Some Useful Notations . there exists . for all
.
implies

.
equivalent to (i.)
belongs to
{x|q(x)} set speci.cation
A .B A is a subset of B
A B the union of A and B
A B the intersection of A and B

n
Points and Sets in R
What is Rn? Rn is the set {(x1,x2, ..., xn)|xi are real numbers}. An element, or a point, of Rn is an n-tuple (x1,x2, ..., xn).
n
Ris the Cartesian product of n Rs.
2
Example R= RR= {(x, y)| x, y belong to R}
Recall the following way to construct the product of two sets De.nition The Cartesian product of two sets X and Y is the set X Y = {(x, y)|x belongs to X and y belongs to Y }.
n
A rectangular region in Rcan be constructed as the Cartesian product of n intervals.
2
Example [a, b] [c, d]= {(x, y)| a x b & c y d} It is a closed subset of R.
Rectangular (Cartesian) Coordinates
In Analytic Geometry, the location of a point in three-dimensional space can be determined

by its coordinates with respect to a coordinate system.
Usually the system is rectangular; it means that the coordinate axes are perpendicular to each
other.

Arrangement of the axes in 3D is according to the Right-Hand Rule.

Important Open Sets
Corresponding to open neighborhoods N(x0)= {x||x .x0|<}
and deleted open neighborhoods

N(x0)= {x|0 < |x .x0|<}in the one dimensional case, the basic open sets in two dimensions are:
1
(i) An open disk in R2, centered at (x0,y0) and with radius ,isthe set
D(x0,y0)= {(x,y)| (x. x0)2 +(y. y0)2 <}.
(ii) A deleted open disk centered at (x0,y0) with radius is the set
D(x0,y0)= {(x,y)| 0 < (x. x0)2 +(y. y0)2 <}.
Elementary Functions xn,sin x and cos x, ex (exp x)and ln x They are continuous and di.erentiable in their natural domains.
PART I: VECTORS AND VECTOR-VALUED FUNCTIONS

(plus elements of solid &analytic geometry)
Vectors and Scalars
A vector is described by two things: a direction and a magnitude
A vector is denoted by .
PQ or Aor A.
.

Its magnitude (or length) is denoted by |PQ| or |A|.
P is the initial point; Q is the terminal point.
Example 1. Displacement in 2D or 3D
2.
Velocity

3.
Force

Abstract vectors discard the exact locations of the initial/terminal points.
A vector can be concieved as a pointed stick which can be translated freely as long as the direction and length are unchanged.
Ascalar is simply a real number; it is usually denoted by a lower-case alphabet.
Fundamental Properties & Operations of Vectors Displacement vectors (or pointed sticks) can be used to illustrate the basic properties of general vectors.
1.
Equality: A= Bi. they have the same magnitude & direction locations of the initial points are irrelevant

2.
Parallel vectors: A . Bi. they have the same (or opposite) direction

3.
Negative vector: .A

has the same magnitude as A, but opposite direction

4.
Addition: A+ Bde.ned by the triangle rul