Vector valued function, Lecture 1. calculus II

Date: 13.02.2011

Sunday
Course Title MTH 202
Course name: Calculus II
Course conducted by Chandra Nath Poddar(CNP)

Chapter:   Vector-Valued Functions


* VECTOR-VALUED FUNCTION
Definition:  Let D be the set of real numbers. A vector valued function r with domain D is a correspondence that assigns to each number t in D exactly one vector r(t) in

V_3.

_{If D is a set of real numbers then r is a vector valued function with domain D iff these are scalar function( real valued function) f,g and h such that}

_{r(t)= f(t) i +g(t) j +h(t) k ,   t} ϵD

Explain with an example: Let us consider the equation bellow

r(t)= (t+1) i +(t²-4) j +t² k  ,  t ϵ R

where r(t) is a vector valued function and Domain of  r(t) is the set of real numbers R 
now if t=1 then 
r(1)= 2 i -3 j + k
 that refers a vector whose position  P(2,-3,1) in three dimensional space.

 here r(t) is drawn in a 3D space in-terms of r(t)'s x,y,z coordinates as a function of t.

that is 

x=f(t)=t+1

y=g(t)=t²-1 

z=h(t)= t²

^{these functions are scalar functions that is real valued function.}

 ^{* Parametric equation and curve: Consider the vector valued function  given bellow}

r(t)= f(t) i +g(t) j +h(t) k , where f, g & h are continuous functions on an interval I

then

• the end points of r(t) determine a space curve C
• The graph of C consists of all points of the form P(f,g,h) in an XYZ   coordinate system that represents to the ordered triple
• The equations x=f(t), y=g(t), z=h(t) are parametric equations of parameter t. where t is a real number

 To be continued

Introduction

All the lectures of  Real Analysis I will be available here. Anyone may collect his/her missed class from here. Thank u.

Introduction

All the lectures of  Linear algebra II will be available here. Anyone may collect his/her missed class from here. Thank u.

Introduction

class lectures of course no. MTH 202 calculus will be published here

kolom haraise

amader Einstain khato Hasnaner ekti kolom haraia gese. jodi kono suridoyban bakti paia thaken, tobe doya kore ferot diben. kolomer soke hasnan pagol para

Introduction

2nd year hons math lab class will start today at 1.30 pm.

Solution of equation in one variable, Lecture 1. Numerical analysis

Date:14/02/2011
Monday
Numerical Analysis
3 credits
course conducted by Mr. Babul Hasan(BH)

Numerical Analysis

Definition: Numerical Analysis involves the study of methods of computing numeric data. It produces a sequences of approximations to many problems. So it is the questions of rate of accuracy.

We seek results in numerical form. To provide efficient numerical methods for obtaining numerical results to the mathematical problems we proceed as follows:

Step 1: We first start with an initial approximation

Step 2: Compute

Step 3: After some iterations we get the desired results. Since the data used are only approximate, being correct to some desired decimal places, the computation result always have errors.

By using it we will solve –

1.    1.   Non Linear equations

2.    2.   Interpolation/ Extrapolation

3.   3.    Numerical Differentiation/ Integration

4.    4.   System  of  linear equations

5.   5.    Differential equations

1.      Non Linear Equation

To find the root of f(x) =0------------------------------------------------------- (1)

If f(x) is a polynomial of degree 2 or 3 then we can solve it easily by exact methods, But f(x) is a polynomial of higher degree or it contains transcendental functions (e.g. 1+cos x, cos x+ tan x, e^x+cos x etc) recourse must be taken to find the root of f(x)=0 by approximation methods.

The methods are :

i.       i) Bisection method

ii.    ii)  Iteration method

iii.    iii) False position method

iv.   iv) Newton-Raphson’s method

v.    v)   Secant method and any others

i) Bisection Method:

If f(x) is differentiable and continuous on (a,b) and has opposite sign at a & b then there must be at least one point c for which f(c)=0

Suppose f(a) is positive & f(b) is negetive. Let the initial approximation be x_o

X_o=(a+b)/2

Then there may be three cases:

Case one:  f(x_o)=0 , Then stop and x_o  is the desired  root. [x_o=c]

Case two:  f(x_o)>0 ,  Then replace ‘a’ by x_o and find the new approximation  x₁=( x_o+b)/2

Case three:  f(x_o)<0,  Then  replace ‘b’ by x_o and find new approximation    x₁=( x_o+a)/2

Then repeat the process until the desired root is obtained .

Problem: Find the real root of  x³-x-1=0 by Bisection method. Correct to 2 decimal places.

Solution:  f(1)= -ve

                 f(2)= +ve

so root lies between 1 and 2.

Let, x_o= (1+2)/2,   then   x_o=1.5

f(x_o)= f(1.5) = (1.5)³-1.5-1

            =0.875

A root lies between 1 and 1.5

Let  x₁ = (1.5+1)/2 = 1.25,  then  x₁=1.25

f(x₁)= f(1.25) =-0.29

 So a root  lies between 1.25 and 1.5

Let,  x₂=(1.25+1.5)/2 =  1.375,  then  x₂=1.375

f(x₂)= f(1.375) = 0.22

Ordinary Differential equations and their solutions, Lecture 1, O.D.E

Date: 12.02.2011
Course conducted by SJ(Sohana Jahan)
Course Title MTH 203
Course Name: Ordinary Differential Equations

Chapter one:
Differential Equations & their Solutions
In order to know what is Ordinary differential equation we must know Differential equation first.

Definition: An equation involving derivatives of one or more dependent variables with respect to one or more independent variable is called a differential equation.


example:

Note: It is to be noted that, in the above equations two types of derivative operator has been used. It should be minded that d/dx( read d dx of something like y,z etc) is used when one or more dependent variable is differentiate with respect to one independent variable.

other hand,  δ/δx (read δ(del) δx(del x)  of something like y,z etc) is used when dependent variable /variables are differentiate with respect to more than one independent variable .

Ordinary Differential Equation:A differential equation involving ordinary derivatives  of one or more dependent variables with respect to a single independent variable is called an ordinary differential equation.
Example:

Classification of differential equations:

we can classify differential equations in the following three  ways

1) Depending on Independent variable

2) Differential equations are linear of non linear

3) According to the order of the de



1. Depending on Independent variable:  Depending on independent variable we can classify differential equation in two ways

a. Ordinary Differential equations: If the independent variable is single by which respect we differentiate one or more dependent variable then these type of differential equations are ordinary diff. equation

Definition & example: Its definition and example are given before

b. Partial Differential equations: If the independent variable is more than one by which respect we differentiate one or more dependent variable then these type of differential equations are partial diff. equation.

Definition: A differential equation involving partial derivatives of one or more dependent variable with respect to more than one independent variable is called a partial diff. equation.

Example:

2.Differential equations are linear o non linear:

a. Linear ordinary equation(defn): A linear ordinary diff. equation of order n in the dependent variable y and the independent variable x, is an equation that is in, or can be expressed in the form

Note: In the above expression it is to be noted that
 i) the dependent variable y and its various derivatives occur in the first power.
 ii) no products of y and/or any of its derivatives are present
 iii) No transcendental functions of y and/or derivatives occur.

Example:

b. Non linear ordinary differential equation: A non linear ordinary  differential equation is an ordinary differential equation that is not linear.

Example:

3. According to Order of the differential equation:

Definition: The order of the highest ordered derivative involved in a differential equation is called the order of the differential equation.
Example:

 In above picture no. 2 is an ordinary diff. equation of second order.

Class Routine

Department of  Mathematics, University of  Dhaka 

Class Routine

Second Year Honors , Session:2009-2010

Day/ Time	8.00-8.50	9.00-9.50	10.00-10.50	11.00-11.50	12.00-12.50	1.00-1.30	1.30-2.20	2.30-4.30
Saturday	No class	Statistics	No class	2HA(201) SS-306 2HB(203) SJ-308	2HA(203) SJ-306 2HB(201) SS-308	Break	No class	No class
Sunday	No class	2HA(204) SN-306 2HB(202) CNP-308	2HA(202) CNP-306 2HB(204) SN-308	No class	No class	Break	2HA(206) MATH LAB-AKH+SKB till 4.30pm
Monday	No class	No class	2HA(202) CNP-306 2HB(205) BH-308	No class	2HA(205) BH-306 2HB(202) CNP-308	Break	No class	No class
Tuesday		2HA(LAB)MMH+ LKS up to 10.50am		2HB(LAB) GS+SR up to 12.50		Break	2HB(206) MATH LAB AKH+SKB
Wednesday	2HA(201) SS-306	No class	No class	Statistics	2HB(201) SS-308	Break	No class	No class
Thursday	2HB(205) BH-308	2HA(204) SN-306 2HB(202) CNP-308	2HA(202) CNP-306 2HB(204) SN-308	2HA(201) SS-306 2HB(203) SJ-308	2HA(203) SJ-306 2HB(201) SS-308	Break	No class	No class

working with maths

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Notice

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Lectures

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Ordinary Differential Equations I

Lectures of Ordinary Differential Equations I will be published here. If anyone miss the class, then he/she can get that lecture from this page. Thanks every body