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IFTER PARTY

Dear friends of Mathematics of University of Dhaka, All of you will be very glad to know that we are going to arrange an IFTER PARTY on upcoming Thursday. This party is going to be arranged by second year students. Anti-fee is only tk 100/= per persion. So dear friends, Don't miss it.... Hurry up. To contribute contact with us... 

Real analysis introduction, lecture 1, Real Analysis

Course title: Real Analysis I
course code: MTH 201, 4 credits
Course conducted by Sapla shirin(SS)

Date :03.03.2011

Mathematical analysis studies concepts related in some way to real numbers. So we begin our study of real analysis with a discussion of the real number.

AXIOMS OF REAL NUMBERS
We assume there exists a non empty set R of objects called real numbers which satisfy the ten axioms listed bellow-

The axioms fall in a natural way into three groups which we refer to as

1. The field axioms
2. The Order axioms
3. The completeness axioms

THE FIELD AXIOMS
Along with the set of real number we assume the existence of two operations called addition and multiplication such that for every pair of real numbers x and y the sum x+y uniquely determined by x and y satisfying the following axioms

A1: Closure law of addition, for all a,b ϵ R , a + b ϵ R 

A2: Commutative law of addition,  for all a,b ϵ R , a + b = b+a

A3: Associative law of addition, for all a,b,c  ϵ R , a +( b+c)=(a+b)+c

A4: Existence of additive Identity,  for all a ϵ R , there exists 0 ϵ R such that a+0=0+a=a

A5: Existence of additive inverse, for all a ϵ R , there exists -a ϵ R such that a+(-a)=(-a)+a=0

********(To be continued) ********

 

Vector-valued function, Lecture no 2, Calculus II

Date: 14.02.2011

Describing the graph of vector-valued function

Let r(t) be a vector valued fuction which  can be represented as the resultant vector of two vector called r_o and v
so,

r(t) = r_{o + tv...........................................(1)}

_{equation (1) represents, r(t) passes through the point of position vector} r_{o and as the direction or parallel to v.}

_{Example 1: a problem has taken from the book of  Haward Anton  page no: 868}

_{problem no: 13}

_{Describe the graph of the vector valued function given as, r(t)= (3-2t) i + 5j}

_{Solution: Given r(t)=}(3-2t) i + 5j

                                       _{= 3 i + 0. j + ( -2 i + 5 j ) t}

_{According to the above theory we can say that the graph of r is a straight line in two dimensional space passing through the point (3,0)  and parallel to the line -2 i + 5 j. we can show this in figure bellow-}

_{Example 2: Describe the graph of} 

_{r(t)= 3cos t i +2 sin t j - k}

_{Solution: we get x=3cos t,   y= 2sin t  and z= -1}

 _{the relation between x and y} 

 _{the graph of r(t) is an ellipse in the plane z=-1 at the center(0,0,-1)}

_{major axis length is 6, parallel to X axis and minor axis length is 4, parallel to Y axis.}

Example 3: Describe the graph of 

                r(t)= 2ti-3j+(1+3t)k

Solution:  Given,

                r(t)= 2ti-3j+(1+3t)k

                    = 2ti-3j+k+3tk

                    = 0.i-3j+k+(2i+0.j+3k)t

So the graph of r(t) is a straight line in three dimensional space passing through the point (0,-3,1) and as the direction or parallel to the line 2i+0.j+3k.

Example 4: Describe the graph of 

              r(t)= 2cos t i -3 sin t j + k

Solution: Given,

                        r(t)= 2cos t i -3 sin t j + k

we get, x=2cos t,  y= -3sin t  and z=1

the relation between x and y 

the graph of r(t) is an ellipse in the plane z=1, the center is (0,0,1) .

Graph Sketching 

Norm of a vector valued function : Norm of a  vector valued function r(t) is denoted by ІІr(t)ІІ and is defined by 

ІІr(t)ІІ= Sqrt ((x(t))²+ (y(t))²+ (z(t))²)

 Example 1: Show that the graph of  r(t) is a circle where 

            r(t)= sin t i + 2 cos t j +sin t k

Solution: we get  x= sin t,   y=2 cos t        z= sin t

So the  relation,

x²+ y²+ z²=sin²t+4cos²t+3sin²t

                =4(sin²t+cos²t)=4

So we get Z= x   as      z= sin t
Thus the graph of r(t) lies on the sphere  x²+ y²+ z²= 4 and on the plane  Z= x of which center is (0,0,0) and radius is  2. the figure as bellow

 

(rest of the part of lecture 2 will be published soon)