review of vector analysis
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Review of Vector Analysis
Special functions
• Functions
– Determine a functional relationship between two or
more variables
– We have studied many elementary functions such as
polynomials, powers, logarithms, exponentials,
trigonometric and hyperbolic functions.
– Error function
– Gamma function
– Laplace transform
– Fourier transform
2
The error function
• It occurs in the theory of probability, distribution of
residence times, conduction of heat, and diffusion
matter:
dzexx
z
0
22erf
0 x z
erf x
22 ze
z: dummy variable
1erf
Proof in next slide
3
dyedxeIR
yR
x
00
22
x and y are two independent Cartesian coordinates
dydxeIR
yxR
0
)(
0
2 22
drdreI r
R2
1
00
2 2
in polar coordinatesError between the volume determined by x-y and r-
The volume of has a base area which is
less than 1/2R2 and a maximum height of e-R2
22
2
1 ReR
2
4
1
4
12 ReI
4
1, 2 IR
dzexx
z
0
22erf
1erf
4
More about error function
Differentiation of the error function:22
erf xexdx
d
dzexx
z
0
22erf
Integration of the error function:
Cexx
Cdxexxxxdx
x
x
2
2
1erf
2erferf
The above equation is tabulated under the symbol “ ierf x” with
1C
(Therefore, ierf 0 = 0)
Another related function is the complementary error function “erfc x”
dzexxx
z
22
erf1 erfc
5
The Fourier Transformation
The Laplace Transformation
, sxf x s e f x dx
F
0
, sxf x s e f x dx
L
• The Laplace transform is
s
dtett st
F
ff
0
L
The gamma function
dtetn tn
0
1)(
for positive values of n.
t is a dummy variable since the value of the definite integral is independent of t.
(N.B., if n is zero or a negative integer, the gamma function becomes infinite.)
)1()1(
)1(
)(
0
2
0
1
0
1
nn
dtetnet
dtetn
tntn
tn
repeat
)!1(
)1()1)(2)...(2)(1()(
n
nnn
The gamma function is thus a generalized factorial, for positive integer
values of n, the gamma function can be replaced by a factorial.
8
Vector analysis
It is much quicker to manipulate a single symbol than the
corresponding elementary operations on the separate
variables.
This is the original idea of vector.
Any number of variables can be grouped into a single symbol in
two ways:
(1) Matrices
(2) Tensors: array of components that are functions of the
coordinates of a space.The principal difference between tensors and matrices is the
labelling and ordering of the many distinct parts.
Tensors
21 izziyxz Generalized as zm
A tensor of first rank since one suffix m is needed to specify it.
The notation of a tensor can be further generalized by using more than
one subscript, thus zmn is a tensor of second rank (i.e. m, n) .
The symbolism for the general tensor consists of a main symbol such as z
with any number of associated indices. Each index is allowed to take any
integer value up to the chosen dimensions of the system. The number of
indices associated with the tensor is the “rank” of the tensor.
10
Tensors of zero rank (a tensor has no index)
• It consists of one quantity independent of the number of
dimensions of the system.
• The value of this quantity is independent of the
complexity of the system and it possesses magnitude
and is called a “scalar”.
• Examples:
– energy, time, density, mass, specific heat, thermal
conductivity, etc.
– scalar point: temperature, concentration and pressure
which are all signed by a number which may vary with
position but not depend upon direction.
11
Tensors of first rank (a tensor has a single
index)
• The tensor of first rank is alternatively names a “vector”.
• It consists of as many elements as the number of
dimensions of the system. For practical purposes, this
number is three and the tensor has three elements are
normally called components.
• Vectors have both magnitude and direction.
• Examples:
– force, velocity, momentum, angular velocity, etc.
12
Tensors of second rank (a tensor has two
indices)
• It has a magnitude and two directions associated
with it.
• The one tensor of second rank which occurs
frequently in engineering is the stress tensor.
• In three dimensions, the stress tensor consists
of nine quantities which can be arranged in a
matrix form:
333231
232221
131211
TTT
TTT
TTT
Tmn
13
The physical interpretation of the stress tensor
x
zy
pxx
xyxz
zzzyzx
yzyyyx
xzxyxx
mn
p
p
p
T
The first subscript denotes the plane and the second subscript denotes the
direction of the force.
xy is read as “the shear force on the x facing plane acting in the y
direction”.
14
15
Review of Vector Analysis
Vector analysis is a mathematical tool with which
Transport phenomena concepts are most conveniently
expressed and best comprehended.
A quantity is called a scalar if it has only magnitude (e.g.,
mass, temperature, electric potential, population).
A quantity is called a vector if it has both magnitude and
direction (e.g., velocity, force, electric field intensity).
The magnitude of a vector is a scalar written as A or
AA A
16
A unit vector along is defined as a vector whose
magnitude is unity (that is,1) and its direction is along
A
A
A
AeA )e( A 1
Thus
Ae
which completely specifies in terms of A and its direction Ae
A
AeAA
17
A vector in Cartesian (or rectangular) coordinates may
be represented as
or
where AX, Ay, and AZ are called the components of in the
x, y, and z directions, respectively; , , and are unit
vectors in the x, y and z directions, respectively.
zzyyxx eAeAeA )A,A,A( zyx
A
A
xe
ze
ye
kAjAiAA zyx
Suppose a certainvector is given by
The magnitude or absolute value of the vector is
(from the Pythagorean theorem)
zyx e4e3e2V
V
385.5432V 222
V
18
19
The Radius Vector
A point P in Cartesian coordinates may be represented by
specifying (x, y, z). The radius vector (or position vector) of
point P is defined as the directed distance from the origin O
to P; that is,
The unit vector in the direction of r is
zyx ezeyexr
r
r
zyx
ezeyexe zyx
r
222
zkyjxir
20
Vector Algebra
Two vectors and can be added together to give
another vector ; that is ,
Vectors are added by adding their individual components.
Thus, if and
A B
C
BAC
zzyyxx eAeAeA zzyyxx eBeBeBB
zzzyyyxxx e)BA(e)BA(e)BA(C
21
Parallelogram Head to rule tail rule
Vector subtraction is similarly carried out as
zzzyyyxxx e)BA(e)BA(e)BA(D
)B(ABAD
22
The three basic laws of algebra obeyed by any given vector
A, B, and C, are summarized as follows:
Law Addition Multiplication
Commutative
Associative
Distributive
where k and l are scalars
ABBA
C)BA()CB(A
kAAk
A)kl()Al(k
BkAk)BA(k
23
When two vectors and are multiplied, the result is
either a scalar or a vector depending on how they are
multiplied. There are two types of vector multiplication:
1. Scalar (or dot) product:
2.Vector (or cross) product:
The dot product of the two vectors and is defined
geometrically as the product of the magnitude of and the
projection of onto (or vice versa):
where is the smaller angle between and
A
ABcosABBA
BA
B
AB
A
BA
A B
B
B
A B
A
B
24
If and then
which is obtained by multiplying and component by
component
),A,A,A(A ZYX )B,B,B(B ZYX
ZZYYXXBABABABA
A B
ABBA
CABACBA )(
A A A2
A2
eX ex ey ey eZ ez 1
eX ey ey ez eZ ex 01
0
kkjjii
ikkjji
25
The cross product of two vectors and is defined as
where is a unit vector normal to the plane containing
and . The direction of is determined using the right-
hand rule or the right-handed screw rule.
A
A
nABesinABBA
B
B
ne
ne
BA Direction of and using (a) right-hand rule,(b) right-handed
screw rule
ne
26
If and then
zyx
zyx
zyx
BBB
AAA
eee
BA
),A,A,A(A ZYX )B,B,B(B ZYX
zxyyxyzxxzxyzzy e)BABA(e)BABA(e)BABA(
27
Note that the cross product has the following basic
properties:
(i) It is not commutative:
It is anticommutative:
(ii) It is not associative:
(iii) It is distributive:
(iv)
ABBA
ABBA
C)BA()CB(A
CABACBA )(
0AA )0(sin
28
Also note that
which are obtained in cyclic permutation and illustrated
below.
yxz
xzy
zyx
eee
eee
eee
Cross product using cyclic permutation: (a) moving clockwise leads to positive results;
(b) moving counterclockwise leads to negative results
29
Scalar and Vector Fields
A field can be defined as a function that specifies a particular
quantity everywhere in a region (e.g., temperature
distribution in a building), or as a spatial distribution of a
quantity, which may or may not be a function of time.
Scalar quantity scalar function of position scalar field
Vector quantity vector function of position vector field
30
Line Integrals
A line integral of a vector field can be calculated whenever a
path has been specified through the field.
The line integral of the field along the path P is defined asV
2
1
P
PP
dl cos Vdl V
31
The cart is constrained to move along the prescribed path
from points a to b.
32
Example. The vector is given by where Vo
is a constant. Find the line integral
where the path P is the closed path below.
It is convenient to break the path P up into the four parts P1,
P2, P3 , and P4.
dl VIP
V xoeVV
33
For segment P1, Thus
For segment P2, and
xedxdl
o o
1
xx
0x
x
0
ooooxxoxxo
P
xV)0x(Vdx)ee(V)edx()eV(dl V
yedydl
)e (since )()(dl x 0002
y
yy
y
yxo
P
eedyeVVo
V Voe x
34
For segment P3,
left) the to points dl length aldifferenti (the xedxdl
oo
xx
x
xxo
P
xV- )edx()eV(dl Vo
03
0
4
dl VP
field) ive(conservat 00xV0xV I oooo
P P PP 2 3 41
35
Surface Integrals
Surface integration amounts to adding up normal
components of a vector field over a given surface S.
We break the surface S into small surface elements and
assign to each element a vector
is equal to the area of the surface element
is the unit vector normal (perpendicular) to the surface
element
ne dsds
ne
ds
The flux of a vector field A through surface S
36
(If S is a closed surface, is by convention directed
outward)
Then we take the dot product of the vector field at the
position of the surface element with vector . The result is
a differential scalar. The sum of these scalars over all the
surface elements is the surface integral.
is the component of in the direction of (normal
to the surface). Therefore, the surface integral can be
viewed as the flow (or flux) of the vector field through the
surface S
(the net outward flux in the case of a closed surface).
ds
ds
ds
V
cosV
SS
cos ds VdsV
V
37
Example. Let be the radius vector
The surface S is defined by
The normal to the surface is directed in the +z direction
Find
V
dyd
dxd
cz
S
dsV
zyx ezeyexV
38
V is not perpendicular to S, except at one point on the Z axis
Surface S
39
SS
cosdsVdsV
c4d(-d)]-2dc[d
dx)]d(d[cdydxcyx
ccyxdsV
cyx
ccos dxdyds cyxV
2
dx
dx
dscos
222
dx
dx
dy
dy
V
222
S
222
222
40
Introduction to Differential Operators
An operator acts on a vector field at a point to produce
some function of the vector field. It is like a function of a
function.
If O is an operator acting on a function f(x) of the single
variable X , the result is written O[f(x)]; and means that
first f acts on X and then O acts on f.
Example. f(x) = x2 and the operator O is (d/dx+2)
O[f(x)]=d/dx(x2 ) + 2(x2 ) = 2x +2(x2 ) = 2x(1+x)
41
An operator acting on a vector field can produce
either a scalar or a vector.
Example. (the length operator),
Evaluate at the point x=1, y=2, z=-2
Thus, O is a scalar operator acting on a vector field.
Example. , ,
x=1, y=2, z=-2
Thus, O is a vector operator acting on a vector field.
)]z,y,x(V[O
O(A ) A A yx ezey3V
)V(O
scalar32.640zy9VV)V(O 22
A2AAA)A(O yx ezey3V
vectore65.16e49.95
e4e1240)e2e(6
ez2ey6zy9)ezey3()V(O
yx
yxyx
yx22
yx
42
Vector fields are often specified in terms of their rectangular
components:
where , , and are three scalar features functions of
position. Operators can then be specified in terms of ,
, and .
The divergence operator is defined as
zzyyxx ezyxVezyxVezyxVzyxV ),,(),,(),,(),,(
xV yVzV
zyx Vz
Vy
Vx
V
xV
yV zV
43
Example . Evaluate at the
point x=1, y=-1, z=2.zyx
2 e)x2(eyexV V
0Vz
1Vy
x2Vx
x2VyVxV
zyx
zy2
x
31x2V
Clearly the divergence operator is a scalar operator.
Ain Aout
0 A
The flux leaving the one end must exceed the flux entering at the other end.
The tubular element is “divergent” in the direction of flow.
Therefore, the operator is frequently called the “divergence” :
AA divDivergence of a vector
44
The operator is of vector form, a scalar product can be obtained as :
z
A
y
A
x
A
kAjAiAz
ky
jx
iA
zyx
zyx
)(
application
The equation of continuity :
0)()()(
tw
zv
yu
x
where is the density and u is the velocity vector.
0)(
tu
Output - input : the net rate of mass flow from unit volume
A is the net flux of A per unit volume at the point considered, counting
vectors into the volume as negative, and vectors out of the volume as positive.
zzyyxx BABABABA
46
1. - gradient, acts on a scalar to produce a vector
2. - divergence, acts on a vector to produce a scalar
3. - curl, acts on a vector to produce a vector
4. -Laplacian, acts on a scalar to produce a scalar
V
V
V
V2
Hamilton’s operator
Tz
Tk
y
Tj
x
Ti
which defines the operator for determining the complete
vector gradient of a scalar point function.
The operator is pronounced “del” or “nabla”.
The vector T is often written “grad T” for obvious reasons.
can operate upon any scalar quantity and yield a vector
gradient.
47
zk
yj
xi
More about the Hamilton’s operator ...
z
Tk
y
Tj
x
TidrTdr (vector) · (vector)
dzz
Tdy
y
Tdx
x
T
z
Tk
y
Tj
x
TikdzjdyidxTdr
TT ddrdr
dTT
But T is the vector equilvalent
of the generalized gradient48
Laplacian operator
In Cartesian coordinates
In cylindrical coordinates, the Laplacian operator is
In spherical coordinates, the Laplacian operator is
49
Curl
The curl is a vector operation that can be used to determine whether there is
a rotation associated with a vector field.
The curl operation determines both the sense and the
magnitude of the rotation.50
Another form of the vector product :
zyx AAA
zyx
kji
A
zyx
zyx
BBB
AAA
kji
BA
is the “curl” of a vector ; AcurlA
What is its physical meaning?
Assume a two-dimensional fluid element
uv
x
y xx
vv
yy
uu
O A
B
Regarded as the angular velocity of OA, direction : k
Thus, the angular velocity of OA is ; similarly, the angular velocity of OB is x
vk
y
uk
y
u
x
vk
vu
yx
kji
0
0u
51
52
Coordinate Systems
In order to define the position of a point in space, an
appropriate coordinate system is needed. A considerable
amount of work and time may be saved by choosing a
coordinate system that best fits a given problem. A hard
problem in one coordinate system may turn out to be easy
in another system.
We will consider the Cartesian, the circular cylindrical, and
the spherical coordinate systems. All three are orthogonal
(the coordinates are mutually perpendicular).
53
Cartesian coordinates (x,y,z)
The ranges of the coordinate variables are
A vector in Cartesian coordinates can be written as
The intersection of three orthogonal infinite places
(x=const, y= const, and z = const)
defines point P.
z
y
x
zzyyxxzyx eAeAeAor )A,A,A(
A
Constant x, y and z surfaces
54
zyx edzedyedxdl
Differential elements in the right handed Cartesian coordinate system
dxdydzd
55
z
y
x
adxdy
adxdz
adydzdS
56
Cylindrical Coordinates . (r, , z)
- the radial distance from the z – axis
- the azimuthal angle, measured from the x-axis in the xy – plane
- the same as in the Cartesian system.
A vector in cylindrical coordinates can be written as
Cylindrical coordinates amount to a combination of
rectangular coordinates and polar coordinates.
)z,,(
z
20
0
21222 /)A(A
Aor ),,(
z
zzz
AA
eAeAeAAA
57
Positions in the x-y plane are determined by the values of
Relationship between (x,y,z) and )z,,(
and
zz x
ytan yx 122
58
eee
eee
eee
z
z
z
0eeeeee
1eeeeee
z
zz
Point P and unit vectors in the cylindrical coordinate system
59
z and ,
semi-infinite plane with its edge along the z - axis
Constant surfaces
60
Differential elements in cylindrical coordinates
Metric coefficient
zp adzadaddl
dzdddv
61
Spherical coordinates .
- the distance from the origin to the point P
- the angle between the z-axis and the radius
vector of P
- the same as the azimuthal angle in cylindrical coordinates
),,r(
0 r
0
Colatitude( polar angle)
0 2
62
21222 /)A(A
Aor ),,(
AA
eAeAeAAA
r
rrr
eee
eee
eee
r
r
r
0eeeeee
1eeeeee
rr
rr
A vector A in spherical coordinates may be written as
Point P and unit vectors in spherical coordinates
63
cosrz
sinsinry
cossinrx
22
11-22
1222
yx
xcos
x
ytan
z
yxtan zyxr
r
zcos
ztan 11
Relationships between space variables )z,,( and ),,,r(),z,y,x(
64
and ,,rConstant surfaces
65
Differential elements in the spherical coordinate system
adsinrardadrdl r
ddrdsinrdv 2
66
Summary of the Transformation
between Coordinate Systems
67
1.
2.
3.
POINTS TO REMEMBER
68
4.
5.
6.
7.
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