Functions of two variables
(part I)
Functions of two (or more) variables
In the course of «Calculus I», we studied real-valued functions of one
real variable: y = f(x).
To model complex phenomena occurring in reality, it is necessary to
introduce more sophisticated mathematical tools.
Let us thus begin a new subject in the study of calculus: the study of
vectors and vector functions.
Indeed, in many applications of mathematics to computer science,
physical, engineering and biological sciences, scientists are concerned
with entities that cannot be expressed by a single numerical value and
need more features to be defined: magnitude and direction.
Let’s start studying them in detail…
Vectors
Physical quantities may be subdivided into two groups:
- Scalar
➔
defined by one numerical parameter only (plus,
eventually, an unit of measure)
Examples: temperature, pressure, …
- Vector
➔
defined by a numerical parameter and an oriented
direction
Examples: force, velocity, acceleration,…
Vectors
Directed line segment
It is a line segment in which we know precisely which point is the
initial point and which one is the terminal point.
B
A
Equivalent directed line segments
Two directed line segments are equivalent if there exists a translation in
which one translates into another, i.e. if they have the same modulus
(length) and direction.
B
A
D
C
Vectors
Free vector
A free vector is the set of all directed line segments which are
equivalent to each other.
It is called “free” since it is not located at a given point in the space
(“free to move”).
v
Common direction
Vectors
Notation for vectors
v v v
AB
Notations for the modulus of a vector
v
Unit vector
v
v
AB
v
A vector having unitary modulus v = 1 and representing a given direction
Vectors
Intrinsic representation of a vector
v=vv
VECTOR
MODULUS
v
v
v
UNIT VECTOR
(identifies a direction in the
plane or space)
Vectors
Equal vectors
Vectors having the same modulus and direction
v=u
Null (or zero) vector
Vector having null modulus and undefined direction
Opposite vectors
Two vectors having same modulus and
opposite direction, whose sum is the null vector
−v
v
v=0
v + ( −v ) = 0
Vectors
How to define a direction in the plane ?
We need one angle:

polar angle
y
v

x
Vectors
How to define a direction in the space ?
We need two angles:
 polar angle
 azimuthal angle
z
y
x
Vectors
Angle between directed line segments
It is the smallest angle formed between the two directions.
u

v
0  
Parallel and point
along the same
direction
Parallel but point
along opposite
direction
Vectors
Orthogonal projection
r and a vector v , let us call the orthogonal projection
(or, simply, the component) of v along r the scalar quantity:
Given a direction
vr = v cos 
v
0  
v

vr = v cos 
r
Vectors
Orthogonal projection
Examples
vr = v cos 
v
v
v

vr  0
v

vr  0
r
vr = 0
vr = v
Vectors
Extrinsic representation of vectors (in plane)
v = v1 + v 2 = v1 e1 + v2 e 2 = ( v1 , v2 )
v1 , v2
e2
orthogonal projections
(components) of v
along reference axes
v2
v
v1
e1
Vectors
Extrinsic representation of vectors (in plane)
v = v1 + v 2 = v1 e1 + v2 e 2 = ( v1 , v2 )
Modulus:
v = v12 + v2 2
Unit vector:
v1
v2
v 
v= =
,
2
2
2
2
v  v1 + v2
v
+
v
1
2





Vectors
Example
v = ( 6,3)
(EXTRINSIC)
e2
v2
v

v1
Modulus:
Unit vector:
Angle:
v = 62 + 32 = 36 + 9 = 45 = 9  5 = 3 5
3   2 1  2 5 5
 6
v=
,
,
=
 =  5 , 5 
3 5 3 5   5 5  

v1
6
2 5
cos  = =
=

v 3 5
5 
 v = ( cos  , sin  )
v
3
5  identifies the direction in plane
si n  = 2 =
=
v 3 5
5 
e1
Vectors
e2
v2
v

e1
v1
v = ( 6,3)
(EXTRINSIC)
Modulus:
v=3 5
Angle:
2 5
cos  =
5
(INTRINSIC)
Vectors
The viceversa also holds:
Modulus:
v=3 5
Angle:
cos  =
(INTRINSIC)
2 5
5
2 5
v1 = v cos  = 3 5
=6
5
5
v2 = v sin  = 3 5
=3
5
v = ( 6,3)
(EXTRINSIC)
NOTE:
There exists a one-to-one correspondence between the two
representations!
Vectors
Extrinsic representation of vectors (in space)
v = v1 + v 2 + v3 = v1 e1 + v2 e 2 + v3 e3 = ( v1 , v2 , v3 )
e3
v1 , v2 , v3
orthogonal projections
(components) of v
along reference axes
v3
 /2
v
v1
e1
v2
 /2
 /2
e2
Vectors
Extrinsic representation of vectors (in space)
v = v1 + v 2 + v3 = v1 e1 + v2 e 2 + v3 e3 = ( v1 , v2 , v3 )
Modulus:
v = v12 + v2 2 + v32
Unit vector:
v3
v1
v2
v 
v= =
,
,
2
2
2
2
2
2
2
2
2
v  v1 + v2 + v3
v
+
v
+
v
v
+
v
+
v
1
2
3
1
2
3




