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
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