the mathematics for chemists (i) (fall term, 2004) (fall term, 2005) (fall term, 2006) department of...
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The Mathematics for Chemists (I)
(Fall Term, 2004)
(Fall Term, 2005)
(Fall Term, 2006)
Department of ChemistryNational Sun Yat-sen University
化學數學（一）
Chapter 3 Vector Algebraand Analysis
• Definition• Scalar (dot) product• Vector (cross) product• Scalar and vector fields• Applications
Content covered in the textbook: Chapter 16
Assignment: pp372-374: 15,17,18,24,25,30,32,35,40,41, 43,47,48,55,58,60
Definition (naïve)
a
B
A
a=AB
Vectors are a class of quantities that require both magnitude and direction fortheir specification.
Initial pointInitial point
Terminal point Terminal point
Unit vector: a vector of unit length.
Null vector: a vector of zero length. (its direction is meaningless.)
aa
a
au
||
^
Examples of Vectors
• Position, velocity, angular velocity, acceleration
• Force, torque, momentum, angular momentum
• Electric and magnetic fields, electric and magnetic dipole moments,
Vector Algebra
Equality:
Addition:
Subtraction:
Scalar multiplication:
a b＝
a + b = b + a
( ) a + -b a - b
a
ba+b
a
ba+b
-ba-b
a1.5a
-a-0.5a
a
b
Example
)(2
1)(
2
1
2
1baADOAODOC
1'
2OC OA AC a AB ��������������������������������������������������������
OCbaabaCO )(2
1)(
2
1
Oa
b
A
B D
C C’
Show that the diagonals of a parallelogram bisect each other.
We need to show that the midpoints of OD and AB coincide.
Classroom Exercise
1 1 1( ) ( ) ( )
3 3 3OX OO OA OB 0 a b a + b��������������������������������������������������������
( ) 2OC a + b��������������
OCOX3
2 O
A
B
C
a
b X
Show that the mean of the position vectors of the vertices of a triangle is the position vector of the centroid of the triangle.
'3
2BCBX
B
OA
C
a
b X
C’
C’’''
3
2ACAX
Components and Decomposition
cosaON
( , )x ya aa
x ya a a i j
x
y
i
j
a
axi
ayj
x=(2,3),y=(4.2,-5.6)
θ
a
O N P
Components and Decomposition(in 3D Space)
axi
azk
ayj
( , , )x y z x y za a a a a a a i j k
( , , )x y za a aa
2 2 2x y za a a a
Vector Algebra Restated
Equality:
Addition:
Subtraction:
Scalar multiplication:
( , , )x y za a aa ( , , )x y zb b bb
a = b xx ba yy ba zz ba
( , , )x x y y z za b a b a b a + b
( , , )x y zc ca ca caa
( , , )x x y y z za b a b a b a - b
Example
(2,3,1)a (1, 2,0) b (5,2, 1) c
2 3 ? d a b c
( , , )x y zd d dd
2)51322(32 xxxx cbad
2)2)2(332(32 yyyy cbad
3))1(0312(32 zzzz cbad
(2, 2,3) d
2 2 2 2 2 22 ( 2) 3 17x y zd d d d
The Center of Mass (Gravity)
N
iiN m
Mmmm
M 1221
1)(
1iN1 rrrrR
N
iimM
1
N
iii xm
MX
1
1
N
iii ym
MY
1
1
N
iii zm
MZ
1
1
m1
m4
m2
m3
r1 r2
r3r4
Dipole Moments
r1
r2
r
-q
q
rrrrrμ 2 qqqq )( 121
1 21
( )N
N ii
q q q q
1 2 N iμ r r r r
N
ii
N
i
N
iii Qqqq
11 1
)() Rμ(0)RrRrμ(R ii
N
ii
1
μμμμμ N21
Dependence of reference frame:
Total dipole moment:
If the total charge Q is zero (e.g., in a molecule), then
) μ(R μ(0)
mCD 301033564.31
Electric dipole moments
Symmetry and Dipole Moment
)
( , , )
(0,0,0)
k
k a a a a a a a a a a a a
k
1 2 3 4
1 2 3 4
μ μ μ μ μ
(r r r r
0
The total dipole moment of a tetrahedron:
r1=(a,a,a), r2=(a,-a,-a), r3=(-a,a,-a), r4=(-a,-a,a)
r3r2
r1
r4
Base Vectors
)0,0,1(i )0,1,0(j )1,0,0(k
),,(
),0,0()0,,0()0,0,(
)1,0,0()0,1,0()0,0,1(
zyx
zyx
zyx
zyx
aaa
aaa
aaa
aaa
kjia
axi
azk
ayj
Orthogonal basis:
Nonorthogonal basis:
cbaυ cba
Classroom Exercise
kjia 32 jib 2
?32 bad
ki
jikji
jikjibad
27
)63()264(
)2(3)32(232
Scalar Differentiation of a Vector
)()( ttt aaa
t
ttt
tdt
dtt
)()(lim)(lim
00
aaaa
kjia )()()()( tatatat zyx
kjia
dt
da
dt
da
dt
da
dt
d zyx
A
B
O
a(t)
a(t+Δt)
Δa
dt
da
Parametric Representation of a Curve
kjir )()()()( tztytxt
O
x
y
z
r(t)C
jir )sin()cos()0,sin,cos()( tbtatbtat
tatx cos)( tbty sin)(
0)( tz
a
bt
Position, Velocity, Momentum, AccelerationNewton’s Second Law
Linear momentum:
rr
dt
dkjikji
rυ zyxdt
dz
dt
dy
dt
dx
dt
d
222zyx
)(2
1
2
1 2222zyxmmvT
kji
kji
rυa
2
2
2
2
2
2
2
2
dt
zd
dt
yd
dt
xd
dt
d
dt
d
dt
d
dt
d
dt
d
zyx
dt
dpF
m
x y z
x y z
d dx dy dzm m m m
dt dt dt dt
m m m
p p p
rp υ i j k
i j k
i j k
Speed:
Kinetic energy:
Acceleration:
Velocity:
Newton’s second law:
Classroom Exercise
• Write the expression of momentum in terms of the planar polar coordinates
. . . . .
. . . .
cos sin
cos sin sin cos
( cos sin ) ( sin cos )
r r i r j
v r r i r i r j r j
p mv m r r i m r r j
����������������������������
The Scalar (Dot) Product
cosab bazzyyxx bababa ba
cos2cos))((2)()()( 22222 abbaOBOAOBOAAB
2 2 2 2
2 2 2 2 2 2
2 2
( ) ( ) ( ) ( )
( ) ( ) 2( )
2( )
x x y y z z
x y z x y z x x y y z z
x x y y z z
AB b a b a b a
a a a b b b a b a b a b
a b a b a b
a b
zzyyxx bababaab cos
Proof:
θ a
b
O B
A
ab
AB
Example)1,1,3( a )3,2,1( b
8)3()1(2113 ba
8)1()3(1231 ab abba
cosab
ba
11aa 14bb
154
8cos
86.498702.0
154
8cos 1
0,2
ba 0,
2 ba
0,2
ba
0 bacaba If 0a
cb )( cba
θ a
b
O B
A
θ aO θ a
b
O
b
Classroom exercise
Orthogonal and Coincident
)1,,2( μ )2,2,4( υ
26)2(1)2(420 υμ
aaa 2222 aaa zyx aaa
3
Cartesian Base Vectors
0 ji
0 kj
0 ik
1 ii
1 jj
1 kk
zzyyxx
zzyy
xyzxyxxx
zyxzyx
bababa
baba
babababa
bbbaaa
kkjj
ijkijiii
kjikjiba
)()(
axi
azk
ayj
Orthogonality:
Normalization (unit length):
Force and Work
),,( 121212 zzyyxx 12 rrd
cosFdW dF FFF d
zx x y y z z x yW F d F d F d W W W
110)3(122 dFW
dFdF
d
FFd cos
)1,3,2(),0,1,2( dF
Force and Work: General Case
C
ABW drF(r)
),,( zyx FFFF
),,( dzdydxd rdzFdyFdxF zyx drF
C zyxAB zdFydFdxFW
x
VFx
y
VFy
z
VFz
dVdzz
Vdy
y
Vdx
x
VzdFydFdxF zyx
),,( zyxVV (r)
BA
B
AAB VVdVW
A
B
O
r(t)
r+Δr
Δr
F(r)
F(r+Δr)
Charges in an Electric Field
xEx
yE y
zE z
CCzEyExE zyx Err )()(
qCqqV Err)(
N
iiN qqqqV
121 )()()()( iN21 rrrr
QC
CqqCqqVN
ii
N
iii
N
iii
Eμ
ErEri111
)(
cosEV Eμ
θμ
E
EF q
Magnetic Moment in a Magnetic Field
x
ABx
y
AB y
z
ABz
CCzByBxB zyx Brr )()(
CqqqV mmm Brr)(
N
iimNmmm qqqqV
1,,2,1, )()()()( iN21 rrrr
, , , ,1 1 1
( )N N N
m i m i m i i m ii i i
m
V q q C q q C
Q C
ir B r B
m B
cosmBV Bm
θm
B
BF mq
HOW DO YOU KNOW THEY ARE PARALLEL WITH EACH OTHER?
The Vector (Cross) Product
baυ sinabυ
a
b
θbsinθ
A new vector can be constructed from two given vectors:
Its magnitude:
Its direction:
bbaυ
abaυ
v
a
b
Right-hand rule:
A
BC
=A
xB
Important properties
baab
0aa
Classroom exercise:
v
a
b
-v
Anti-commutative:
,00aa If the cross product of two vectors is a zero vector,they must be parallel or antiparallel to each other.
cbacba )()(
Nonassociative:
(Proof to be given later)
In Cartesian Basis
kji
ikj
jik 0ii 0jj
0kk
axi
azk
ayj
)()()(
)()(
ijikjk
kkjkikkjjj
ijkijiii
kjikjiba
yzxzzyxyzxyx
zzyzxzzyyy
xyzxyxxx
zyxzyx
babababababa
bababababa
babababa
bbbaaa
jiij
kjiba )()()( xyyxzxxzyzzy babababababa
jkkj kiik
zyx
zyx
bbb
aaa
kji
ba
Example
)1,1,3( a )3,2,1( b
1 ( 3) ( 1) 2 ( 1) 1 3 ( 3) 3 2 1 1
8 5 ( 1,8,5)
a b i j k
i j k
5,8,1 baab
90581 222 ba
Classroom Exercise)3,1,8( a )2,5,4( b
zyx
zyx
bbb
aaa
kji
ba
Calculate the cross product of above two vectors using
Application: Moment of Force (Torque)
FrFr sinT
FrT r
F
O
θ
d
A
An Electric Dipole in an Electric Field
r
E
-q
q
EF q1
EF q2
r1
r2O
EμT EμErErr
FrFrT
21
11
qq )(22
μ
ET
A Magnetic Dipole in a Magnetic Field
r
B
-qm
qm
BF mq1
BF mq2
r1
r2O
BmT BmBrBrr
FrFrT
21
11
mm qq )(22
m
BT
Angular Velocity
r
sinr
rωυ
In a plane:
General case:
vω
r
O
θ
rsinθ
O
r
v
ω
Exercisec)a(bc)b(acba )(
zyx
zyx
bbb
aaa
kji
ba
c)a(bc)b(a
k
j
i
kji
cdcba
kji
kji
bad
cdcba
)]()([
)]()([
)]()([
)()()(
)(
zxxzxzzzyy
yzzyzxyyxx
xyyxyzxxzz
zyx
xyyxzxxzyzzy
xyyxzxxzyzzy
zyx
zyx
babacbabac
babacbabac
babacbabac
ccc
babababababa
babababababa
bbb
aaa
Classroom exercise
Exercise
b)c(ac)ba
k
j
i
kji
dacba
kji
kji
cbd
dacba
(
)]()([
)]()([
)]()([
)()()(
)(
yzzyyzxxzx
zyyxxyzzyz
zxxzzxyyxy
xyyxzxxzyzzy
zyx
xyyxzxxzzyzy
zyx
zyx
cbcbacbcba
cbcbacbcba
cbcbacbcba
cbcbcbcbcbcb
aaa
cbcbcbcbbccb
ccc
bbb
b)c(ac)b(acba )(
c)a(bc)bacba
b)c(ac)bacba
()(
()(
2=| | -( )
c (b c) (c b) c
c b b c c
( ) ( ) [( ) ( ) ( ) ]
( ) ( ) ( )
( ) ( )
x y z y z y z z x x z x y y x
x y z y z y z x x z z x y y x
a a a b c c b b c b c b c b c
a b c c b a b c b c a b c b c
a b c i j k i j k
a b c a c b
( ) ( ) [( ) ] [( ]
( ( ) ( )
a b c d a b c d a c)b (b c)a d
a c) b d (b c) a d
Angular Momentum
prl
rωrωl )(2 mmr
2mrI
rωυ r
p
O
θ
d
Arωp m
)( rωrl m
][(
)(
ω)r(rr)ωr
rωrl
m
m
(moment of inertia)
A special case: ω is perpendicular r:
ωωl Imr 2r
ω
Conservation of Angular Momentum
dt
dprFr
pr
prp
r dt
d
dt
d
dt
d)(
dt
dlT
dt
dpF
mB
T
BmT Bmμ
T dt
d
Bμμ
μm
dt
d
For nuclear spins:
NMR measures how fast a nuclear spin precesses.
If T=0, angular momentum is conserved.
Scalar and Vector Fields
),,()( zyxfff r
),,()( zyxυrυυ
The Gradient of a Scalar Field
kjiz
f
y
f
x
ffgrad
kjizyx
kji
kji
z
f
y
f
x
f
fzyx
ffgrad
The gradient of a scalar field is a vector.
Vector differential operator
The Meaning of the Gradient
),,(),,( dzzdyydxxzyx
dzdydxdzdydxd kjir ),,(
)()( rrr fdfdf
dzz
fdy
y
fdx
x
fdf
dzz
fdy
y
fdx
x
fdzdydx
z
f
y
f
x
f
kjikji
rdfdf
f(r+dr)
f(r)
dr
Gradient is a convenient vector expression ofthe derivative of multi-variable functions.
Example: Gradient232 zyzxV
1
x
V zy
V2
zy
z
V62
kji )62(2 zyzV
Example: Gradient as Force
x
VFx
y
VFy
z
VFz
kjiFz
V
y
V
x
VV
Force is the negative of the gradient of the potential energy.
Example: Gradient as Force
r
qqV
0
21
4
30
21
30
21333
0
21
0
21
4
44
111
4
r
zyxr
r
z
r
y
r
xqq
rzryrx
qqV
r
kjikji
kjiF
r
rr ˆ1 2 1 2
3 20 0
ˆ4 4
q q q q
r r
rF r
12
0
ˆ4
0
q
r
E r
E r E rr
qqV
0
12 4
q1
q2
r
q1
rE
The Divergence of a Vector Field
zyxdiv zyx
υυ
2 2 2
2 2 2
Suppose that vector is the gradient field of a scalar field , thenf
f f fdiv f
x y z x y z
f f f
x y z
υ
υ i j k i j k
2Laplacian operator
002
2
2
2
2
22
z
f
y
f
x
ff Laplace’s Equation
Examples: Divergence
1S
2S
2S
0 B 0/ E
AThe divergence is a measure of flux density:the amount of ‘something’ flowing out of a unit volume per second.
The Curl of a Vector Field
zyx
xyzxyz
zyx
yxxzzyrotcurl
kji
kjiυυυ
fgradifcurl υυ 0
wυυ curlifdiv 00)( f
0)( A
( ) ( ) ( )
( ) ( ) ( )
x y z
y yx xz z
zy yz xz zx yx xy
x y z
f f ff f f f
x y z
f ff ff f
y z z x x y
f f f f f fx y z
f f f
i j k i j k
i j k
i j k
i j k 0
0)( f
0)( AClassroom Exercise
, , , , , ,
, ,
( ) [ ]
0
(using ...)
y yx xz z
z yx y zx z yx x zy y xz x yz
x yz x zy
A AA AA A
x y z y z z x x y
A A A A A A
A A
A i j k i j k
Physical Meaning of curl (rot)
( ) ( ) ( )x y z y z z x x yz y x z y x
x y z
i j k
v ω r i j k
2
y yx xz z
x y z
v vv vv vcurl rot
y z z x x y
x y z
v v v
v v v i j k
i j k
ω
x y z
y z x
z x y
v z y
v x z
v y x
The curl of a velocity field is angular velocity field (x2).
1 1 12 2 2
1 12 2
( ), ( ), ( )y yx xz zv vv vv v
x y zy z z x x y
x y z
x y zv v v
i j k
ω v
Example: Curl
A
0 E
The curl of the velocity filed is a measure of the circulation of fluid around the point.
( )
( )
[( ) (( ) ) ( ) ]
( ) ( ) ( )
( ) ( ) ( ) (
y z z y z x x z x y y x
y z z y z x x z x y y x
y y yx x z zz y x
A B A B A B A B A B A Bx y z
A B A B A B A B A B A Bx y z
A A BA A A AB B B
x y z x y z x
A B A B
A B A B
i j k iA jB k
(
) ( ) ( )
) ( )
yx x z zz y x
BB B B BA A A
y z x y z
A B B A
[( ) ( ) ( ) ]
[ ( ) (
( )
( )
)] [ ( ) ( )] [ ( ) )]
( ( ) (
(
)
y z z y z x x z x y y x
x y y x z x x z y z z y x y y x z x x z y z z y
A B A B A B A B A B A Bx y z
A B A B A B A B A B A B A B A B A B A B A B A By z z x x y
i j k i j kA B
B A A )B B A A B
i j k
You may win 5 points for filling in the details here!
Supplementary (Not Required)
Major Theorems in Integrationa
dF(x)dx
b
dx=F(a)-F(b)
LS
V
S
b a
) ( ( , ) ( , ) )
( )
( ( , , ) ( , , ) ( , , ) )
S L
S
L
Q Pdxdy P x y dx Q x y dy
x y
R Q P R Q Pdydz dxdz dxdy
y z z x x y
P x y z dx Q x y z dy R x y z dz
( )
( )
yx z
V
x y z
S
AA Adxdydz
x y z
A dydz A dxdz A dxdy
( )
( )
y yx xz z
S
x y z
L
B BB BB Bdydz dxdz dxdy
y z z x x y
B dx B dy B dz
( )
( ( , , ) ( , , ) ( , , ) )
V
S
P Q Rdxdydz
x y z
P x y z dydz Q x y z dxdz R x y z dxdy
Green:
Gauss:
Fundamental:
Stokes:
Major Theorems in Integration
( )S L
d d B S B l
( )V S
dV d A A S
LS
V
S
adF(x)
dxb
dx=F(a)-F(b) b a
(D ,M)=( , M) M∂M
The general form:
( )
( ) ( ) ( )
y yx xz z
S
x y z x y z
L L
B BB BB Bdydz dxdz dxdy
y z z x x y
B B B dx dy dz B dx B dy B dz
i j k i j k
( )
( )
yx z
V
x y z
S
AA Adxdydz
x y z
A dydz A dxdz A dxdy
(Stokes’s theorem)
Classroom Exercise?• Find out the result of the following
integration by using Gauss’s theorem:
2 2
2
2 2
2
1 1 12
0 0 0
(2 3 ) (3 2 ) ( 5 )
2 2
[(2 3 ) (3 2 ) ( 5 ) )
( ) (2 2 )
(2 2 ) (2
yx z
S S
V V
xz y z x x y z x y yz
AA Az x yz
x y z
xz y z dydz x x y z dxdz x y yz dxdy d
dV z x yz dxdydz
dz dy z x yz dx dz dy zx
A i j k
A
A S
A
1 13 1
0
0 0
1 1 1 12 1 7
0 60 0 0 0
/ 3 2 ) |
(2 1/ 3 2 ) (2 / 3 ) | (2 1/ 3 )
xx
yy
x xyz
dz dy z yz dz zy y y z dz z z
2 2[(2 3 ) (3 2 ) ( 5 ) )
where is the surface of a cube shown in the diagram.S
xz y z dydz x x y z dxdz x y yz dxdy
S
1
1
1
x
y
z
The Maxwell Equations
0 E
0 B0/ E
B Jt
BE
0 B0/ D
0 0 t EB J
0 D E P E0 / H B M B
Related to the light speed c
Electrostatics Electrodynamics
In a medium:
Vector Spaces
)0,,0,0,1( 1e
)0,,0,1,0( 2e
)1,,0,0,0( ne
n321 eeeea nn aaaaaaaa 321321 ),,,,(
abba nnbabababa 332211
223
22
21 naaaa aaa
…
Norm (length):
Inner (scalar) product:
Inner product space
j i
j iijji if 0
if 1ee
n321 eeeeb nn bbbbbbbb 321321 ),,,,(
You as a vector in a high dimensional space
• Name• Gender• Birth date• Birth place• ID• Student ID• Height• Weight • Favorite drink• Favorite music• ….
(1,0,0,0,0,0,0,0, ,0)1e (0,1,0,0,0,0,0,0,0, ,0)2e
(0,0,0,0,0,0,0,0, ,1)ne
3 (0,0,1,0,0,0,0,0, ,0)e 4 (0,0,0,1,0,0,0,0, ,0)e 5 (0,0,0,0,1,0,0,0,..,0)e
…
6 (0,0,0,0,0,1,0,0, ,0)e 7 (0,0,0,0,0,0,1,0, ,0)e 8 (0,0,0,0,0,0,0,1 ,0)e
( , ,890301, ,12345678, 942020001,175,60, , )Wang Male Taipei b Tea ClassicWang Da - Fu
Operations of Vectors in Fortran
• Array • Loop: Dot/cross product• Gradient• Divergence• Curl• General vector spaces• Subroutine
Dot Product of VectorsWrite a program to calculate the dot product of any two vectors.
C Program for calculating dot product of any two vectorsc
program dotpro1 parameter(n1=3) real a1(n1),a2(n1)1 write(6,*)'Input three components of first vector:' read(5,*)(a1(i),i=1,n1) write(6,*)'Input three components of second vector:' read(5,*)(a2(i),i=1,n1) dot=0. DO 10,I = 1,n1 dot=dot+a1(i)*a2(i) 10 CONTINUE print *,'The dot product is ', dot print *,'Next calculation (0/1)?' read(5,*)i
if(i.eq.1) goto 1 stop
end
Cross Product of VectorsWrite a program to calculate the cross product of any two vectors.
C Program for calculating cross product of any two vectorsc
program crossp1 parameter(n1=3) real a1(n1),a2(n1),crossp(n1)1 write(6,*)'Input three components of first vector:' read(5,*)(a1(i),i=1,n1) write(6,*)'Input three components of second vector:' read(5,*)(a2(i),i=1,n1) CROSSP(1)=a1(2)*a2(3)-a1(3)*a2(2) CROSSP(2)=a1(3)*a2(1)-a1(1)*a2(3) CROSSP(3)=a1(1)*a2(2)-a1(2)*a2(1) print *,'The cross product is \n', (crossp(i),i=1,3) print *,'Next calculation (0/1)?' read(5,*)i
if(i.eq.1) goto 1 stop
end
Cross Product of VectorsWrite a program to calculate the following vector operations.
)( cba b)c(ac)bacba ()(
C Program for calculating cross product of any three vectorsprogram crossp2
parameter(n1=3) real a1(n1),a2(n1),a3(n1),crossp(n1)
real tmp(n1)1 write(6,*)'Input three components of first vector:' read(5,*)(a1(i),i=1,n1) write(6,*)'Input three components of second vector:' read(5,*)(a2(i),i=1,n1) write(6,*)'Input three components of third vector:' read(5,*)(a3(i),i=1,n1) tmp1=0.0
tmp2=0.0 DO 10,I = 1,n1 tmp1=tmp1+a1(i)*a2(i)
tmp2=tmp2+a1(i)*a3(i)10 CONTINUE
do 20 i=1,n1 CROSSP(i)=tmp2*a2(i)-tmp1*a3(i)20 continue print *,'The cross product is \n', (crossp(i),i=1,3) print *,'Next calculation (0/1)?' read(5,*)i
if(i.eq.1) goto 1 stop
end
Gradient of a Scalar Field
kji
kji
z
f
y
f
x
f
fzyx
ffgrad
232)sin( zyzeyxxf x
Write a program to calculate the gradient of a scalar function.
C Program for calculating the gradient of any scalar functionprogram grdt1
parameter(n1=3) real grt(n1)1 write(6,*) 'please input the position (x,y,z):' read(5,*)x,y,z dx=0.001 dy=0.001
dz=0.001 x2=x+dx
y2=y+dyz2=z+dz
f1=x*sin(x+y)+2.0*y*z*exp(x)+3.0*z*z f2=x2*sin(x2+y)+2.0*y*z*exp(x2)+3.0*z*z
fx=(f2-f1)/dx f2=x*sin(x+y2)+2.0*y2*z*exp(x)+3.0*z*z
fy=(f2-f1)/dy f2=x*sin(x+y)+2.0*y*z2*exp(x)+3.0*z2*z2
fz=(f2-f1)/dzgrt(1)=fxgrt(2)=fygrt(3)=fz
print *,'The gradient at (', x,y,z, ') is \n', (grt(i),i=1,3) print *,'Calculating the gradient of next point (0/1)?' read(5,*)i
if(i.eq.1) goto 1 stop
end
The Divergence of a Vector Field
z
v
y
v
x
vdiv zyx
vv
kjiv )62(]cos([2 yzezyxyzzyxxx
Write a program to calculate the divergence of a vector field.
C Program for calculating the divergence of any vector fieldprogram divg1
parameter(n1=3) real vecf(n1)1 write(6,*) 'please input the position (x,y,z):' read(5,*)x1,y1,z1 dx=0.0001 dy=0.0001
dz=0.0001 x2=x1+dx
y2=y1+dyz2=z1+dz
vx1=x1*x1vx2=x2*x2divgx=(vx2-vx1)/dx
vy1=x1*cos(x1+x1*y1+x1*y1*z1)vy2=x1*cos(x1+x1*y2+x1*y2*z2)divgy=(vy2-vy1)/dy
vz1=2.*y1+6.*z1+exp(-y1*z1)vz2=2.*y1+6.*z2+exp(-y1*z2)divgz=(vz2-vz1)/dydivg=divx+divy+divz
print *,'The divergence at (', x,y,z, ') is \n', divg print *,'Calculating the gradient of next point (0/1)?' read(5,*)i
if(i.eq.1) goto 1 stop
end
The Curl of a Vector Field
zyx
xyzxyz
zyx
yxxzzycurl
kji
kjiυυ
kjiv )62(]cos([2 yzezyxyzzyxxx
Write a program to calculate the rot of a vector field.
Using Subroutines(Go to Chapter 2)
Cross Product of VectorsWrite a program to calculate the following vector operations bycalling subroutine for calculating cross product of two vectors.
h)(ge)(dcba )(
)( cba
C Program for calculating cross product of any two vectorsc
program crossp1 parameter(n1=3) real a1(n1),a2(n1),crossp(n1)1 write(6,*)'Input three components of first vector:' read(5,*)(a1(i),i=1,n1) write(6,*)'Input three components of second vector:' read(5,*)(a2(i),i=1,n1) CROSSP(1)=a1(2)*a2(3)-a1(3)*a2(2) CROSSP(2)=a1(3)*a2(1)-a1(1)*a2(3) CROSSP(3)=a1(1)*a2(2)-a1(2)*a2(1) print *,'The cross product is \n', (crossp(i),i=1,3) print *,'Next calculation (0/1)?' read(5,*)i
if(i.eq.1) goto 1 stop
end
C Subroutine for calculating cross C product of any two vectorsC
subroutine(a,b,c,n) real a(n),b(n),c(n)
c(1)=a1(2)*a2(3)-a1(3)*a2(2) c(2)=a1(3)*a2(1)-a1(1)*a2(3) c(3)=a1(1)*a2(2)-a1(2)*a2(1)
returnend
C Program for calculating the crossC product of any number of vectors
program crossp parameter(n=3,n1=3) real a1(n),a2(n),a3(n),TMP1(1),TMP2(N) open(20,file=‘crossp.1’,err=9999) read(20,*)(a1(i),i=1,n) read(20,*)(a2(i),i=1,n) read(20,*)(a3(i),i=1,n) close(20) call vcross(a2,a3,tmp1,n) call vcross(a1,tmp1,tmp2,n) open(30,file=‘crossp.2’,err=9999) write(30,*)(tmp2(i),i=1,n) CLOSE(30)9999 STOP END
C Subroutine for calculating cross C product of any two vectorsC
subroutine(a,b,c,n) real a(n),b(n),c(n) c(1)=a1(2)*a2(3)-a1(3)*a2(2) c(2)=a1(3)*a2(1)-a1(1)*a2(3) c(3)=a1(1)*a2(2)-a1(2)*a2(1) return
end
Cross Product of VectorsWrite a program to calculate the following vector operations bycalling subroutine for calculating cross product of two vectors.
h)(ge)(dcba )(
C Program for calculating the crossC product of any number of vectors program crossp6 parameter(n=3) real a1(n),a2(n),a3(n),a4(n),a5(n),a6(n),a7(n) real TMP1(1),TMP2(N),TMP3(N),TMP4(N)
real TMP5(n),TMP6(n) open(20,file=‘crossp6.1’,err=9999) read(20,*)(a1(i),i=1,n) read(20,*)(a2(i),i=1,n) read(20,*)(a3(i),i=1,n) read(20,*)(a4(i),i=1,n) read(20,*)(a5(i),i=1,n) read(20,*)(a6(i),i=1,n)
read(20,*)(a7(i),i=1,n) close(20) call vcross(a6,a7,tmp1,n) call vcross(a5,a6,tmp2,n) call vcross(a3,a4,tmp3,n) call vcross(tmp1,tmp2,tmp4,n)
call vcross(tmp3,tmp4,tmp5,n) call vcross(a1,tmp5,tmp6,n) open(30,file=‘crossp6.2’,err=9999) write(30,*)(tmp6(i),i=1,n) CLOSE(30)9999 STOP END
C Subroutine for calculating cross C product of any two vectorsC
subroutine(a,b,c,n) real a(n),b(n),c(n) c(1)=a1(2)*a2(3)-a1(3)*a2(2) c(2)=a1(3)*a2(1)-a1(1)*a2(3) c(3)=a1(1)*a2(2)-a1(2)*a2(1) return
end
General Vector Spaces
n321 eeeea nn aaaaaaaa 321321 ),,,,(
n321 eeeeb nn bbbbbbbb 321321 ),,,,(
Write a subroutine to calculate the inner product of two vectors in any dimension.
C Program for calculating dot product of any two vectorsc
program dotpro1 parameter(n1=3) real a1(n1),a2(n1)1 write(6,*)'Input three components of first vector:' read(5,*)(a1(i),i=1,n1) write(6,*)'Input three components of second vector:' read(5,*)(a2(i),i=1,n1) dot=0. DO 10,I = 1,n1 dot=dot+a1(i)*a2(i) 10 CONTINUE print *,'The dot product is ', dot print *,'Next calculation (0/1)?' read(5,*)i
if(i.eq.1) goto 1 stop
end
C subroutine for calculating dot product of any two vectorssubroutine vdot(a,b,n)
real a(n),b(n) dot=0. DO 10,I = 1,n1 dot=dot+a(i)*b(i) 10 CONTINUE return
end
C program for calculating the inner product of two vectors program innerp1 parameter(n=10) do 10 i=1,n a1(i)=i*1.2 a2(i)=-2.*+i*i 10 continue call vdot(a1,a2,dot,n) print *,'The inner product of two vectors is \n', dot
stopend
C subroutine for calculating inner product of any two vectorssubroutine vdot(a,b,dot,n)
real a(n),b(n) dot=0. DO 10,I = 1,n1 dot=dot+a(i)*b(i) 10 CONTINUE return
end
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