Lecture 7 - Vector Valued Functions | Calculus - Engineering Mathematics PDF Download

 Page 1


                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Vector Valued Functions 
 
Paper: Calculus 
Course Developer: Gurudatt Rao Ambedkar 
Department/College: Assistant Professor, Department of 
Mathematics, Acharya Narendra Dev College, University of Delhi 
 
 
 
 
 
 
 
 
 
Page 2


                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Vector Valued Functions 
 
Paper: Calculus 
Course Developer: Gurudatt Rao Ambedkar 
Department/College: Assistant Professor, Department of 
Mathematics, Acharya Narendra Dev College, University of Delhi 
 
 
 
 
 
 
 
 
 
                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
Table of Contents: 
 Chapter : Vector Valued Functions 
 Learning Outcomes 
 Introduction 
 
? 1: Vector Function 
? 2: Limit of a vector function 
? 3: Continuity of a vector function 
? 4: Derivative of a vector function 
? 5: Chain rule for differentiation  
? 6: Space curve 
? 7: Arc length 
? 8: Unit tangent vector of T 
 
 Objective Problems 
 Exercise 
 Summary 
 References 
 
 
 
 
 
 
 
 
Page 3


                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Vector Valued Functions 
 
Paper: Calculus 
Course Developer: Gurudatt Rao Ambedkar 
Department/College: Assistant Professor, Department of 
Mathematics, Acharya Narendra Dev College, University of Delhi 
 
 
 
 
 
 
 
 
 
                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
Table of Contents: 
 Chapter : Vector Valued Functions 
 Learning Outcomes 
 Introduction 
 
? 1: Vector Function 
? 2: Limit of a vector function 
? 3: Continuity of a vector function 
? 4: Derivative of a vector function 
? 5: Chain rule for differentiation  
? 6: Space curve 
? 7: Arc length 
? 8: Unit tangent vector of T 
 
 Objective Problems 
 Exercise 
 Summary 
 References 
 
 
 
 
 
 
 
 
                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
1. Learning outcomes: 
After studying this chapter you should be able to 
? Understand the meaning of the term ‘Vector valued function’ 
? Able to differentiate between vector function and real function 
? How to check the limit point, continuity and differentiability of a vector 
function 
? Understand the concept of space curve, arc length and unit tangent 
vector 
? Understand the relation between position vector and velocity 
 
2. Introduction 
 
We face many problems in our day to day life. These problems are sometime 
become too small and sometime become too serious. Everybody wants a 
better future and mathematics help us to get it. We can model a life 
situation with the help of vector function like the relation between velocity 
and acceleration of a car or bike, the motion of a ceiling fan etc. The 
understanding the concept of vector function helps us to better understand 
the real life problems and to get their solution.  In this chapter we discuss 
about some fundamentals of vector function and will learn to solve the 
problem related to them.  
 
3. Vector Functions: 
A function of one or more than one variable whose range is the set of 
multidimensional or infinite-dimensional vectors is called a vector function or 
vector valued function. 
Let us suppose that i, j and k be the unit vectors along the x-axis, y-axis, 
and z-axis in space. Then a function which depends on a real parameter t 
(time), defined such as 
 ), ( ) ( ) ( ) ( t z t y t x t f k j i ? ? ?      
Page 4


                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Vector Valued Functions 
 
Paper: Calculus 
Course Developer: Gurudatt Rao Ambedkar 
Department/College: Assistant Professor, Department of 
Mathematics, Acharya Narendra Dev College, University of Delhi 
 
 
 
 
 
 
 
 
 
                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
Table of Contents: 
 Chapter : Vector Valued Functions 
 Learning Outcomes 
 Introduction 
 
? 1: Vector Function 
? 2: Limit of a vector function 
? 3: Continuity of a vector function 
? 4: Derivative of a vector function 
? 5: Chain rule for differentiation  
? 6: Space curve 
? 7: Arc length 
? 8: Unit tangent vector of T 
 
 Objective Problems 
 Exercise 
 Summary 
 References 
 
 
 
 
 
 
 
 
                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
1. Learning outcomes: 
After studying this chapter you should be able to 
? Understand the meaning of the term ‘Vector valued function’ 
? Able to differentiate between vector function and real function 
? How to check the limit point, continuity and differentiability of a vector 
function 
? Understand the concept of space curve, arc length and unit tangent 
vector 
? Understand the relation between position vector and velocity 
 
2. Introduction 
 
We face many problems in our day to day life. These problems are sometime 
become too small and sometime become too serious. Everybody wants a 
better future and mathematics help us to get it. We can model a life 
situation with the help of vector function like the relation between velocity 
and acceleration of a car or bike, the motion of a ceiling fan etc. The 
understanding the concept of vector function helps us to better understand 
the real life problems and to get their solution.  In this chapter we discuss 
about some fundamentals of vector function and will learn to solve the 
problem related to them.  
 
3. Vector Functions: 
A function of one or more than one variable whose range is the set of 
multidimensional or infinite-dimensional vectors is called a vector function or 
vector valued function. 
Let us suppose that i, j and k be the unit vectors along the x-axis, y-axis, 
and z-axis in space. Then a function which depends on a real parameter t 
(time), defined such as 
 ), ( ) ( ) ( ) ( t z t y t x t f k j i ? ? ?      
                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
is a vector function or vector-valued function of t
,
where ) (t x , ) (t y , and ) (t z
are the  co-ordinate function (real-valued functions) of the real parameter t . 
The motion of a particle in space or in the plane is usually describe by these 
functions.  
To describe the motion of plane, we simply assume that the coordinates are 
chosen to make the plane of motion the xy -plane.  In this case we only 
consider the motion in the  xy -plane  and assume that the z-coordinate of 
the particle is zero as a special case of motion in space. The vector function 
in two dimensional planes is defined as  
 ), ( ) ( ) ( t y t x t f j i ? ? 
Where i, j are the unit vectors along the X-axis and Y-axis and   
) ( ), ( t y t x are real-valued functions of the real parametert . 
Example 1:  A vector valued function  j i j i
t t
te e y x t P ? ? ? ? ) ( , represent the  
position of a particle in thexy -plane at the time t .How fast is it moving and 
in what direction and  where is the particle at 0 ? t ? 
Solution:    At         , 0 ? t
 
We have 
   
, 0 * 0
, 1
0
0
? ? ?
? ? ?
e te y
e e x
t
t
 
So,  . ) 0 ( i ? P 
 
Page 5


                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Lesson: Vector Valued Functions 
 
Paper: Calculus 
Course Developer: Gurudatt Rao Ambedkar 
Department/College: Assistant Professor, Department of 
Mathematics, Acharya Narendra Dev College, University of Delhi 
 
 
 
 
 
 
 
 
 
                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
Table of Contents: 
 Chapter : Vector Valued Functions 
 Learning Outcomes 
 Introduction 
 
? 1: Vector Function 
? 2: Limit of a vector function 
? 3: Continuity of a vector function 
? 4: Derivative of a vector function 
? 5: Chain rule for differentiation  
? 6: Space curve 
? 7: Arc length 
? 8: Unit tangent vector of T 
 
 Objective Problems 
 Exercise 
 Summary 
 References 
 
 
 
 
 
 
 
 
                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
1. Learning outcomes: 
After studying this chapter you should be able to 
? Understand the meaning of the term ‘Vector valued function’ 
? Able to differentiate between vector function and real function 
? How to check the limit point, continuity and differentiability of a vector 
function 
? Understand the concept of space curve, arc length and unit tangent 
vector 
? Understand the relation between position vector and velocity 
 
2. Introduction 
 
We face many problems in our day to day life. These problems are sometime 
become too small and sometime become too serious. Everybody wants a 
better future and mathematics help us to get it. We can model a life 
situation with the help of vector function like the relation between velocity 
and acceleration of a car or bike, the motion of a ceiling fan etc. The 
understanding the concept of vector function helps us to better understand 
the real life problems and to get their solution.  In this chapter we discuss 
about some fundamentals of vector function and will learn to solve the 
problem related to them.  
 
3. Vector Functions: 
A function of one or more than one variable whose range is the set of 
multidimensional or infinite-dimensional vectors is called a vector function or 
vector valued function. 
Let us suppose that i, j and k be the unit vectors along the x-axis, y-axis, 
and z-axis in space. Then a function which depends on a real parameter t 
(time), defined such as 
 ), ( ) ( ) ( ) ( t z t y t x t f k j i ? ? ?      
                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
is a vector function or vector-valued function of t
,
where ) (t x , ) (t y , and ) (t z
are the  co-ordinate function (real-valued functions) of the real parameter t . 
The motion of a particle in space or in the plane is usually describe by these 
functions.  
To describe the motion of plane, we simply assume that the coordinates are 
chosen to make the plane of motion the xy -plane.  In this case we only 
consider the motion in the  xy -plane  and assume that the z-coordinate of 
the particle is zero as a special case of motion in space. The vector function 
in two dimensional planes is defined as  
 ), ( ) ( ) ( t y t x t f j i ? ? 
Where i, j are the unit vectors along the X-axis and Y-axis and   
) ( ), ( t y t x are real-valued functions of the real parametert . 
Example 1:  A vector valued function  j i j i
t t
te e y x t P ? ? ? ? ) ( , represent the  
position of a particle in thexy -plane at the time t .How fast is it moving and 
in what direction and  where is the particle at 0 ? t ? 
Solution:    At         , 0 ? t
 
We have 
   
, 0 * 0
, 1
0
0
? ? ?
? ? ?
e te y
e e x
t
t
 
So,  . ) 0 ( i ? P 
 
                 Vector Valued Functions  
 
Institute of Lifelong Learning, University of Delhi                                                       
 
 
Figure 1: The position vector and velocity vector at t = 0  
And it is the vector from the origin to the position ) 0 , 1 ( A of the particle at 
time 0 ? t . 
 Next, let us think about the speed and direction of motion.  
Let us think about the point where the particle will reach after t ? time. So x
andy
 
both will increase and P will be changed by the amount 
 ? ? ? ? ? j i P x
         (1)
 
So the speed of the particle will be defined by 
 
t
P
t
P
?
?
?
?
?
 
Then from the equation 1, 
 
t
y
t
x
t
P
?
?
?
?
?
?
?
?
j i
 
Let   
 
, 0 ? ?t
 
 
? ?
t t t
t t
te e e
dt
dy
dt
dx
dt
dP
v
t
y
t
x
t
P
? ? ?
? ? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ? ?
j i
j i
j i
0 0
lim lim