Chapter 2
Review of Vectors
2.1
Definition of a Vector
A vector is a quantity that posses both a magnitude and a direction and obey the parallelogram law of
addition.
2.2
Vector Addition
Figure 2.1: Vector addition
~ =A
~+B
~
C
2.3
Vector Subtraction
~ =A
~−B
~
D
2.4
Properties of Vectors
~ and B
~ are two vectors then:
If s and t are two scalars and A
~
0A
~
(s + t)A
~=A
~ ;
= ~0 ; +A
~ + tA
~
= sA
~ + B)
~
~ + sB
~
s(A
= sA
~ = s(tA)
~ = t(sA)
~
st(A)
3
~ = −A
~
(−1)A
Figure 2.2: Vector subtraction
2.4.1
Explanation
~ is a vector, sA
~ is defined to be the vector having magnitude s times that of A
~ and
If s is a number and A
pointing in the same direction if s > 0 and in the opposite direction if s is negative.
~ is a scalar multiple of A.
~
Any vector sA
2.5
Scalar Product
~·B
~ = |A||
~ B|cosθ
~
A
Ex.1) Work done by a force F~ during an infinitesimal displacement δ~s is
W = F~ · δ~s
Ex.2)
~ = δ F~
−pδ A
~ is infinitesimal aera vector, δ F~ is the corresponding force
where p is pressure at a point, δ A
I
δ F~ = F~
s
where F~ is the total force on a body
F~ · î = Drag
F~ · k̂ = Lif t
2.6
Vector Product
~×B
~ = |A||
~ B|sinθê
~
A
~ and B
~ and | | sign denotes the magnitude of the vector.
where ê is a unit vector perpendicular to both A
Ex.1)
~o
~r × F~ = M
where Mo is moment about o
2.7
Triple Product
~ · (B
~ × C)
~
A
~ × (B
~ × C)
~
A
=
=
~ · (A
~ × B)
~ =B
~ · (C
~ × A)
~
C
~ · C)
~ B
~ − (A
~ · B)
~ C
~
(A
4
2.8
Unit Vector
A vector whose magnitude is 1 is called a unit vector.
~
A
~
|A|
êA =
~ is the magnitude of vector A
~ and êA is a unit vector in the direction of A.
~
where |A|
2.9
Vector Differentiation
~ and B
~ are two vectors and U
~ =A
~+B
~ then:
If A
~
dU
dt
~
dU
dt
~)
d(nU
dt
2.10
=
=
=
~|
d|U
êu
dt
~
~
dA
dB
+
dt
dt
~
dn ~
dU
U +n
dt
dt
Product Rules
~·A
~ = (|A|)2
A
½
1 if i = j,
êi · êj =
0 if i 6= j.
½
0 if i = j,
êi × êj =
1 if i 6= j.
2.11
Components of a Vector
In 3-D a vector has 3 components. These 3 components are independent of each other. Consider three
~ B,
~ and C.
~ In component form, these vectors in general can be written as:
vectors A,
~ = A1 ê1 + A2 ê2 + A3 ê3
A
~ = B1 ê1 + B2 ê2 + B3 ê3
B
~ = C1 ê1 + C2 ê2 + C3 ê3
C
Based on the component form the following relations can be established:
~·B
~ = A 1 B1 + A 2 B2 + A 3 B3
A
¯
¯
¯ ê1 ê2 ê3 ¯
¯
¯
~×B
~ = ¯ A1 A2 A3 ¯
A
¯
¯
¯ B1 B2 B3 ¯
¯
¯
¯ A1 A2 A3 ¯
¯
¯
~ · (B
~ × C)
~ = ¯ B1 B2 B3 ¯
A
¯
¯
¯ C1 C2 C3 ¯
¯
¯
ê1
¯
~
~
~
A1
A × (B × C) = ¯¯
¯ (B2 C3 − B3 C2 )
ê2
A2
(B3 C1 − C3 B1 )
ê3
A3
(B1 C2 − B2 C1 )
¯
¯
¯
¯
¯
¯
Cartesian and cylindrical coordinate systems are two coordinate systems widely used. Their component
forms are discussed next.
5
2.11.1
Cartesian Coordinate System
~ = Vx î + Vy ĵ + Vz k̂
V
where î, ĵ, k̂ are unit vectors, and Vx , Vy , Vz are components. In general the components are independent of
each other.
For example:
Vx 6= f (Vy , Vz )
The position vector in cartesian system is given as:
~r = xî + y ĵ + z k̂
2.11.2
Cylindrical Coordinate System
Figure 2.3: Cylindrical and spherical coordinate system
~ = Vr êr + Vθ êθ + Vz êz
V
2.12
Relation Between Coordinate Systems
2.12.1
General Transformation
(q1 , q2 , q3 ) are the general coordinates of a 3-D coordinate system.
q1 = q1 (x, y, z)
q2 = q2 (x, y, z)
q3 = q3 (x, y, z)
2.12.2
General Inverse Transformation
x = x(q1 , q2 , q3 )
y = y(q1 , q2 , q3 )
z = z(q1 , q2 , q3 )
6
2.12.3
2.12.4
Transformation
1
r
θ
= (x2 + y 2 ) 2
= arctan(y/x)
(0 ≤ r < ∞)
(0 ≤ θ ≤ 2π)
z
=
(−∞ < z < ∞)
z
Inverse transformation
x = rcosθ
y = rsinθ
z
=
z
2.13
A procedure for Evaluating Scalar Factors, Unit Vectors and
Their Derivatives from Inverse Transformation
2.13.1
Scale Factor
Scale factor defines the relationship between coordinates and distance along coordinates.
2.13.2
General Coordinate System
A position vector ~r in cartesian system is given by
~r = xî + y ĵ + z k̂
and using the inverse transformation can also be written as:
~r = x(q1 , q2 , q3 )î + y(q1 , q2 , q3 )ĵ + z(q1 , q2 , q3 )k̂
The variation of the position vector along the coordinate direction defines the following relations:
∂~r
= h1 ê1
∂q1
∂~r
= h2 ê2
∂q2
∂~r
= h3 ê3
∂q3
where h1 , h2 , and h3 are the scale factors and ê1 , ê2 , and ê3 are the unit vectors in the q1 , q2 , and q3
directions respectively.
2.14
Relation Between Unit Vectors and Their Derivatives
2.14.1
Cartesian System
Cartesian unit vectors are fixed in magnitude and directions and hence are constant vectors.
∂ î
∂x
∂ ĵ
∂x
∂ k̂
∂x
=
=
=
∂ î
∂y
∂ ĵ
∂y
∂ k̂
∂y
=
=
=
7
∂ î
∂z
∂ ĵ
∂z
∂ k̂
∂z
=0
=0
=0
Figure 2.4: Unit vectors in clyndrical coordinate system
2.14.2
Cylindrical System
In general the unit vectors are not constant. For example, in the cylindrical coordinate system the unit
vectors êr and êθ vary with the coordinate θ.
In contrast, the êz vector is a fixed vector like the cartesian vectors.
For the cylindrical coordinate system:
∂~r
= hr êr
∂r
∂~r
= hθ êθ
∂θ
∂~r
= hz êz
∂z
From the inverse transformation for the cylindrical system, the position vector in the cylindrical coordinate
system can be written as:
~r
∂~r
∂r
∂~r ∂~r
·
∂r ∂r
∂~r
∂θ
∂~r ∂~r
·
∂θ ∂θ
=
r cos θ î + r sin θ ĵ + z k̂
=
cos θ î + sin θ ĵ = hr êr
=
h2r (êr · êr ) = h2r
=
cos2 θ + sin2 θ = 1 = h2r or hr = 1
=
−r sin θ î + r cos θ ĵ
=
r2 (sin2 θ + cos2 θ) = h2θ or hθ = r
similarly hz = 1.
∂~r
∂r
∂~r
∂θ
=
cos θ î + sin θ ĵ = hr êr = êr
=
−r sin θ î + r cos θ ĵ = hθ êθ = rêθ
8
From the previous equations the unit vectors in cylindrical coordinate system can be written as:
êr
=
cos θ î + sin θ ĵ
êθ
êz
=
=
− sin θ î + cos θ ĵ
k̂
The unit vectors in the cylindrical system can be related to the cartesian unit vectors î, ĵ, k̂ in matrix form
as:



 


cos θ sin θ 0  î 
 î 
 êr 
êθ
= [A]
=  − sin θ cos θ 0 
ĵ
ĵ






0
0
1
êz
k̂
k̂
Matrices relating unit vectors of orthogonal coordinate systems are orthogonal matrices.
The inverse of an orthogonal matrix is its transpose. Hence the following relations can be established.




 î 
 êr 
êθ
= [A]
ĵ




êz
k̂




 î 
 êr 
T
êθ
= [A]
ĵ




ê
k̂
z
In other words the cartesian unit vectors can be written in
relations.
 

cos θ − sin θ
 î 
=  sin θ cos θ
ĵ


0
0
k̂
The derivatives of the unit vectors in cylindrical system are:
∂êr
=0
∂r
∂êr
= − sin θ î + cos θ ĵ = êθ
∂θ
∂êr
=0
∂z
∂êθ
=0
∂r
∂êθ
= − cos θ î − sin θ ĵ = −êr
∂θ
∂êθ
=0
∂z
General
(q1 , q2 , q3 )
(ê1 , ê2 , ê3 )
(δs1 , δs2 , δs3 )
(h1 , h2 , h3 )
Coordinates
Unit vectors
Distance along coordinates dir.
Scale factor
terms of the cylindrical unit vectors using the


0  êr 
êθ
0 


êz
1
Cartesian
(x,y,z)
(î, ĵ, k̂)
(δx, δy, δz)
(1, 1, 1)
Cylindrical
(r, θ, z)
(êr , êθ , êz )
(δr, rδθ, δz)
(1, r, 1)
∂êz
=0
∂r
∂êz
=0
∂θ
∂êz
=0
∂z
Spherical
(R, ϕ, θ)
(êR , êϕ , êθ )
(δR, Rδϕ, R sin ϕδθ)
(1, R, R sin ϕ)
Relatinship between coordinate systems and the coordinate space is unique. For example the relation between
cartesian and cylindrical coordinate system is given by:
Inverse transformation
x = r cos θ
y = r sin θ
z=z
Transformation
p
r = x2 + ¡y 2 ¢
θ = arctan xy
z=z
2.15
Vector Calculus
2.15.1
Del, The Vector Differential Operator: ∇
∇
= ê1
=
∂
∂
∂
+ ê2
+ ê3
∂s1
∂s2
∂s3
or
ê2 ∂
ê3 ∂
ê1 ∂
+
+
h1 ∂q1
h2 ∂q2
h3 ∂q3
9
where ê1 , ê2 , and ê3 are three mutually orthogonal unit vectors. δs1 , δs2 , and δs3 denote infinitesimal
distances along the coordinate axes. Thus, knowing the scale factors the vector differential operator ∇ can
be written for other coordinate systems.
2.15.2
Cartesian
∇ = î
2.15.3
∂
∂
∂
+ ĵ
+ k̂
∂x
∂y
∂z
Cylindrical
∇ = êr
∂
∂
∂
+ êθ
+ êz
∂r
r∂θ
∂z
2.15.4
General Rules of Differentiation
2.16
Scalar and Vector Field
¶
µ
µ
¶
µ
¶
∂ê2
∂
∂φ2
ê1
· (φ2 ê2 ) = (ê1 ) ·
ê2 + (ê1 ) · φ2
∂q1
∂q1
∂q1
where q1 , q2 , and q3 are general coordinates. ê1 , ê2 , and ê3 are general unit vectors.
Note: The order of the dot (cross) product should be preserved.
A scalar (vector) quantity given as a function of coordinate space and time is called a scalar (vector)field.
Pressure p = p(~r, t)
= p(q1 , q2 , q3 , t) general
= p(x, y, z, t) cartesian
= p(r, θ, z, t) cylindrical
2.16.1
Examples
Scalar fields
T = T (~r, t)
ρ = ρ(~r, t)
e = e(~r, t)
(temperature)
(density)
(internal energy)
Velocity vector field
~ =V
~ (~r, t)
V
(velocity vector)
~ = (V1 , V2 , V3 )
V
V1 = V1 (~r, t), V2 = V2 (~r, t), and V3 = V3 (~r, t).
In general a field denotes a region throughout which a quantity is defined as a function of location within
the region and time.
If the quantity is independent of time, the field is steady or stationary.
2.17
Concept of Gradient
Let φ = φ(~r) be a scalar function of position (~r)
Find the spatial variation of φ?
Since φ is a function of a vector ~r there are an infinite number of directions in which to take the increment
∆~r. The total change in φ, dφ whould in general be different in different directions.
Spatial derivative of φ at a point is expressed as derivatives of φ in three independent directions. Gradient
of a scalar is a vector
10
2.17.1
Concept of Gradient
At any point, the gradient of a scalar function(φ) of position is equal in magnitude and direction to the
greatest derivative of φ with respect to distance at that point.
Rate of change of the scalar φ along two paths are of special importance.
1. Path along which the scalar is constant (Isolines)
2. Path along which the rate of change of the scalar is the maximum (Gradient line)
2.17.2
General
∂φ
∂φ
∂φ
+ ê2
+ ê3
∂s1
∂s2
∂s3
ê1 ∂φ
ê2 ∂φ
ê3 ∂φ
∇φ =
+
+
h1 ∂q1
h2 ∂q2
h3 ∂q3
µ
¶ µ
¶
∂φ
∂φ
∂φ ∂φ ∂φ
∂φ
,
,
∇φ =
=
∂s1 ∂s2 ∂s3
h1 ∂q1 h2 ∂q2 h3 ∂q3
∇φ
2.17.3
= ê1
Cartesian
∂φ
∂φ
∂φ
+ ĵ
+ k̂
∂x
∂y
∂z
µ
¶
∂φ ∂φ ∂φ
∇φ =
,
,
∂x ∂y ∂z
∇φ = î
2.17.4
Cylindrical
∂φ êθ ∂φ
∂φ
+
+ êz
∂r
r ∂θ
∂z
µ
¶
∂φ ∂φ ∂φ
∇φ =
,
,
∂r r∂θ ∂z
∇φ = êr
2.18
Concept of Directional Derivative
Let d~r be a small increment in ~r in some direction ê. If φ = φ(q1 , q2 , q3 ) using chain rule we can write:
dφ =
∂φ
∂φ
∂φ
dq1 +
dq2 +
dq3
∂q1
∂q2
∂q3
By multiplying the numerator and denominator by the respective scale factors it can also be written as:
dφ =
=
=
∂φ
∂φ
∂φ
h1 dq1 +
h2 dq2 +
h3 dq3
h1 ∂q1
h2 ∂q2
h3 ∂q3
∂φ
∂φ
∂φ
ds1 +
ds2 +
ds3
∂s1
∂s2
∂s3
µ
¶
∂φ ∂φ ∂φ
,
,
· (ds1 , ds2 , ds3 )
∂s1 ∂s2 ∂s3
³
´
∂φ ∂φ ∂φ
,
,
where ∂s
= ∇φ and (ds1 , ds2 , ds3 ) = d~r.
1 ∂s2 ∂s3
Therefore
dφ ≡ (∇φ · d~r)
Change in a general direction dφ is simply the scalar product of the gradient at that point and d~r in that
general direction. That is,
dφ =
∇φ · d~r
dφ =
∇φ · |d~r|ê = ∇φ · (dr ê)
11
Therefore
dφ
= ∇φ · ê
dr
• Directional derivative of φ(~r) in any chosen direction is equal to the component of the gradient vector
in that direction.
• The greatest rate of change of φ with respect to coordinate space at a point takes place in the direction
of ∇φ and has the magnitude of the vector ∇φ.
2.18.1
Cartesian
d~r = (dx, dy, dz)
Cylindrical
d~r = (dr, rdθ, dz) = |d~r|ê
In component form the position vector can also be written as:
d~r = (dr, rdθ, dz) = |d~r|ê = ds1 ê1 + ds2 ê2 + ds3 ê3
2.19
Divergence of a Vector Field (∇ · V~ = 0)
Divergence of a vector field can be computed using vector algebra. In a general coordinate system this is
tedious and yet possible as is illustrated here.
~
∇·V
=
=
∇
= ê1
~
V
=
∂
∂
∂
+ ê2
+ ê3
∂s1
∂s2
∂s3
V1 ê1 + V2 ê2 + V3 ê3
µ
¶
∂
∂
∂
ê1
+ ê2
+ ê3
· (V1 ê1 + V2 ê2 + V3 ê3 )
∂s1
∂s2
∂s3
¶
µ
∂
∂
∂
· (V1 ê1 + V2 ê2 + V3 ê3 )
+ ê2
+ ê3
ê1
h1 ∂q1
h2 ∂q2
h3 ∂q3
³
´
All the terms have to be expanded using chain rule differentiation. For example, consider ê1 h1∂∂q1 · (V2 ê1 ):
µ
ê1
∂
h1 ∂q1
¶
·
¸
∂
∂V1
V1 ∂ê1
· (V1 ê1 ) = ê1 ·
(V1 ê1 ) = ê1 ·
ê1 +
h1 ∂q1
h1 ∂q1
h1 ∂q1
In specific coordinate systems the simplifications are easier if the derivatives of the unit vectors are known.
The divergence of a vector in cartesian and cylindrical systems are illustrated next.
2.19.1
Cartesian
~
∇·A
=
=
µ
∂
∂
∂
+ ĵ
+ k̂
∂x
∂y
∂z
∂A1
∂A2
∂A3
+
+
∂x
∂y
∂z
î
12
¶ ³
´
· A1 î + A2 ĵ + A3 k̂
2.19.2
Cylindrical
~=
∇·A
µ
¶
∂
∂
∂
+ êθ
+ êz
êr
· (Ar êr + Aθ êθ + Az êz )
∂r
r∂θ
∂z
~
∇·A
=
µ
¶
∂
êr ·
(Ar êr + Aθ êθ + Az êz )
∂r
{z
}
|
(1)
+
µ
¶
∂
êθ ·
(Ar êr + Aθ êθ + Az êz )
r∂θ
{z
}
|
(2)
+
¶
µ
∂
(Ar êr + Aθ êθ + Az êz )
êz ·
∂z
|
{z
}
(3)
Term (1)
êr ·
Term (2)
µ
∂Ar
∂êr
∂Aθ
∂êθ
∂Az
∂êz
êr + Ar
+
êθ + Aθ
+
êz + Az
∂r
∂r
∂r
∂r
∂r
∂r
¶
=
∂Ar
∂r
·
¸
∂êr
∂Aθ
∂êθ
∂Az
∂êz
1 ∂Aθ
Ar
êθ
∂Ar
·
êr + Ar
+
êθ + Aθ
+
êz + Az
+
=
r
∂θ
∂θ
∂θ
∂θ
∂θ
∂θ
r
r ∂θ
Term (3)
êz ·
∂Az
∂
[Ar êr + Aθ êθ + Az êz ] =
∂z
∂z
therefore
~
∇·A
=
=
=
Ar
1 ∂Aθ
∂Az
∂Ar
+
+
+
∂r
r
r ∂θ
∂z
1 ∂
1 ∂
1 ∂
(Ar r) +
(Aθ ) +
(rAz )
r ∂r
r
∂θ
r ∂z
¸
·
∂
∂
1 ∂
(Ar r) +
(Aθ ) +
(rAz )
r ∂r
∂θ
∂z
~ is written as:
For any orthogonal coordinate system the conservation form of divergence of a vector A
·
¸
1
∂
∂
∂
~=
∇·A
(h2 h3 A1 ) +
(h1 h3 A2 ) +
(h1 h2 A3 )
h1 h2 h3 ∂q1
∂q2
∂q3
2.20
Concept of Divergence
2.20.1
Physical Meaning of Divergence of a Vector Field
The divergence of a vector at a point is the net outflow of the vector per unit volume enclosing the point.
~ with components Ax , Ay , and Az at a point in the vector field surrounded by an elemental
Consider A
control volume ∆V– with an elemental surface ∆S.
For convenience the elemental control volume with its center having a vector with components A x , Ay , and
Az is oriented with edges parallel to x, y, and z axes respectively.
~ through any side = Component of A
~ in the direction normal to side × Area of the surface.
Outflow of A
~ in x-direction(Outflow A
~ from the x-faces):
Net outflow of A
¶ µ
¶¸
µ
¶
µ
¶
·µ
∂Ax ∆x
∂Ax
∂Ax
∂Ax ∆x
− Ax −
∆y∆z =
(∆x∆y∆z) =
∆V–
=
Ax +
∂x 2
∂x 2
∂x
∂x
13
Figure 2.5: Elemental control volume
~ in y-direction = ∂Ay ∆V– and
Similarely, net outflow of A
∂y
∂Az
~
∆V–
net outflow of A in z-direction =
∂z
~ from a point is by definition:
Outflow of A
"
#
~ in all directions
∂Ax
Outflow of A
∂Ay
∂Az
~
∇ · A = lim
=
+
+
–→0
∆V
∆V–
∂x
∂y
∂z
2.20.2
Example
~=V
~ , the velocity vector then ∇ · V
~ is the volume flux from a point.
If A
It is a scalar and its magnitude is the rate at which fluid volume is leaving a point per unit volume.
Flux of volume from a point is by definition:
¸
·
V olume outf low/sec − V olume inf low/sec
lim
V olume→0
V olume
~ ) is mass flux.
∇ · (ρV
where ρ is density.
2.20.3
Problem
Show from the physical meaning of divergence that in polar coordinates:
~ =
∇·V
2.21
1 ∂Vθ
∂Vz
1 ∂
(rVr ) +
+
r ∂r
r ∂θ
∂z
Gauss Divergence Theorem
– in space as shown in Figure 2.6.
Consider a finite control volume V
d~s = ên ds
where d~s is the elemental vector area with ên as the unit vector normal to ds being counted positive when
directed outward.
If
If
~ points out A
~ · d~s is positive.
A
~
~
A points in A · d~s is negative.
~ is found for each cube at its center. The Flow of
The region V– is divided into elementary cubes and div A
14
Figure 2.6: A finite control volume
~ through common faces of adjacent cubes cancel because the inflow through one face equals the outflow
A
~ of all the cubes, only faces on the surface enclosing
through the other. If we now sum the net outflow of A
the region will contribute to the summation.
~ over V– is equal to the net
In the limit, therefore as the cubes approach zero volume the integral of div A
outflow through ds, the surface enclosing the region.
Stated in integral notation:
ZZ
ZZZ
~
~ –
(∇ · A)dV
OA · d~s =
–
S
V
Vector Identities Useful for the Manipulation of Conservation Equations:
Gauss Divergence Theorem:
´
³
´
³ ´ ³
~ •∇ ϕ + ϕ ∇•A
~
~ = A
∇• ϕ A
³
´ ³
´
³
´
~A
~ = ϕA
~ •∇ A
~+A
~ ∇•ϕA
~
∇• ϕ A
³ ´ ³
´
³
´
e = B
e •∇ ϕ + ϕ ∇•B
e
∇• ϕ B
³
´ ³
´
³
´
e •A
~ = B
e •∇ •A
~+A
~ • ∇• B
e
∇• B
Z ³
CS
´ Z
~ =
F~ • dA
(∇•F~ ) dV–
CV
Gradient Theorem:
Z ³
CS
´ Z
~
ϕ dA =
∇ϕ dV–
CV
15