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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