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