1 april 14 triple product 6.3 triple products triple scalar product: chapter 6 vector analysis a b c...
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April 14 Triple product
6.3 Triple products
Triple scalar product:
Chapter 6 Vector Analysis
zyx
zyx
zyx
zyx
zyxzyx
CCC
BBB
AAA
CCC
BBBAAA
kji
CBA ),,()(
CBA
CBA
and ,by formed ipedparallelep theof Volume
cossin|)(|
ABC
)()()(
)()()(
ABCCABBCA
BACACBCBA
A
BC
+_
2
Triple vector product:
),0,0(
0
00
),,(A ),0,,( ),0,0,(
thatso system coordinate Choosing
.??)( figure, theFrom
yx
yx
x
zyxyxx
CB
CC
B
AAACCB
kji
CB
CB
CBCBA
B)C(AC)B(A
B)C(AC)B(A
kji
CBA
)0,,()0,0,)((
)0,,(
00
)(
yxxxxyyxx
yxxyxy
yx
zyx
CCBABCACA
CBACBA
CB
AAA
C
B
B×C A
)( CBA x
yz
This is called the “bac-cab” rule.
3
Example p282.1.
Problems 3.12.
)( Torque Frn
)()()( DCBADCBA
4
Read: Chapter 6:3Homework: 6.3.9,12,19.Due: April 25
5
April 18 Gradient
6.4 Differentiation of vectorsDerivative of a vector: . , kji
AkjiA
dt
dA
dt
dA
dt
dA
dt
dAAA zyx
zyx
Derivative of vector products:
dt
d
dt
d
dt
ddt
d
dt
d
dt
ddt
da
dt
daa
dt
d
BAB
ABA
BAB
ABA
AAA
)(
)(
)(
Derivative in polar coordinates:
jie
jie
cos sin
sin cos
r
eee
ee
eA
eeA
ejie
ejie
dt
dA
dt
dA
dt
dA
dt
dA
dt
dA
dt
dA
dt
dA
dt
dA
dt
d
AAdt
d
dt
d
dt
d
dt
ddt
d
dt
d
dt
d
dt
d
rrrr
rrr
rr
r
r
sincos
cossin
x
yree
r
A
6
6.6 Directional derivative; Gradient
Directional derivative: The changing rate of a field along a certain direction.
u
urr
c
zb
ya
xds
dz
zds
dy
yds
dx
xds
zyxd
cszz
bsyy
asxx
s
),,(
derivative lDirectiona
:Line
0
0
0
0
r0
rus
Gradient: kjizyx
Examples p292.1,3.
cos|| u
ds
d
1. The gradient is in the direction along which the field increases the fastest.
2. The gradient is perpendicular to the equipotential surface =constant.
u
u ds
d
0const, ds
d
7
Read: Chapter 6:4-6Homework:6.4.2,8;6.6.1,6,7,9.Due: April 25
8
zyx
kji
April 21 Divergence and curl
6.7 Some other expressions involving
The operator:
Vector function: kjiV ),,(),,(),,(),,( zyxVzyxVzyxVzyx zyx
z
V
y
V
x
V zyx
VDivergence:
Curl: kji
kji
V
y
V
x
V
x
V
z
V
z
V
y
V
VVVzyx
xyzxyz
zyx
Laplacian:2
2
2
2
2
22
zyx
Vector identities involving : p339.
VVV
3
1i i
ii
iii
i x
VV
xV
x
Examples p297.1,2.
9
Physical meaning of divergence:Let V be the flux density (particles across a unit area in a unit time).
dxdyVdxdzVdydzVdxdydz
dydzVdydzVVdxdydzx
V
dxdydzz
V
y
V
x
Vdxdydz
zyx
xxxdxxxx
zyx
V
V
00
V is the net rate of outflow flux per unit volume.
x0 x0+dx
Physical meaning of curl:Set the coordinate system so that z is along direction at the point (x0, y0, z0 ).V
dxVdyV
dxVVdyVV
dxdyy
V
x
Vdxdyd
xy
yxdyyxxydxxy
xyz
0000
)( VnV
V is the total circulation of V per unit area.
10
Read: Chapter 6: 7Homework: 6.7.6,7,9,13,18.Due: May 2
11
April 23 Line integrals
6.8 Line integrals
Line integral:
Circulation:
B
AdW lF
lF d A
B
Examples p300.1,2
Conservative field: A field is said to be conservative if does not depend on the path in the calculation.
B
AdW lF
Theorem: If F and its first partial derivatives are continuous in a simply connected region, then the following five statements are equivalent to each other.
1)
2) does not depend on the path.
3) is an exact differential.
4)
5)
.0 lF d
B
AdW lF
.0 F
dWd rF
.WF
12
00 :)1()5(
0
pathon dependnot does )(0
:)5()4()3()2()1(
lFF
kji
F
FrFrFlF
d
z
W
y
W
x
Wzyx
W
WddWdPWdP
O
Examples p304.3,5.
zxzyxWyxhzxzzxgyxzyfxzyx
yxhdzFzxgdyFzyfdxFW
zyxyx,x,,,
zyx
323232 ),(),(),(
),(),(),(or
),,()0,,()00()000(
Examples p306.6.
13
Read: Chapter 6: 8Homework: 6.8.9,15,17.Due: May 2
14
April 25 Green’s theorem
6.9 Green’s theorem in the plane
Double integral in an area:
d
c Clr
x
xA
d
c
b
a Clu
y
yA
b
a
QdydyyxQyxQ
dxx
yxQdydxdy
x
yxQ
PdxdxyxPyxP
dyy
yxPdxdxdy
y
yxP
r
l
u
l
),(),(
),(),(
),(),(
),(),(
Green’s theorem in the plane:
CA
QdyPdxdxdyy
P
x
Q
A double integral over an area may be evaluated by a line integral along the boundary of the area, and vice versa.
Examples p311.1,2.
15
Examples p312.3: Divergence theorem in two dimension:
AA
C
C xy
A
yx
CA
yx
dsdxdy
dxdy
dyVdxVdxdyy
V
x
V
QdyPdxdxdyy
P
x
Q
PQVV
nVV
jiV
jijiV
)(
Examples p312.4: Stokes’ theorem in two dimension:
AA
C yx
A
xy
CA
yx
ddxdy
dyVdxVdxdyy
V
x
VQdyPdxdxdy
y
P
x
Q
QPVV
rVkV
jijiV
)(
16
Read: Chapter 6:9Homework: 6.9.2,3,8,10.Due: May 2
17
April 28 Gauss’ theorem
6.10 The divergence and the divergence theoremLetbe the density, v be the velocity of water. The water flow in a unit time through a unit area that is perpendicular to v is given by V=v, which is called flux density. The water flow rate through a surface with unit normal n is given by V·n.
Physical meaning of divergence:Let V be the flux density.
dxdyVdxdzVdydzVdxdydz
dydzVdydzVVdxdydzx
V
dxdydzz
V
y
V
x
Vdxdydz
zyx
xxxdxxxx
zyx
V
V
00
V is the net rate of outflow flux per unit volume.
x0 x0+dx
ddd
dnVV
1lim
0
Proof: For a differential cube,
Sum over all differential cubes, at all interior surfaces will cancel, only the contributions from the exterior surfaces remain.
18
Time rate of the increase of mass per unit volume: density source ,
Vt
Equation of continuity (when there is no sources): 0
t
V
The divergence theorem (Gauss’ theorem):(Over a simply connected region.)
dσd nVV
.
surfaces 6
0
0
0
0
0
0
dσd
dσ
dxdyVdxdzVdydzV
dxdydzz
V
y
V
x
Vdxdydz
dzz
zz
dyy
yy
dxx
xx
zyx
nVV
nV
V
dσnV
Examples p319.
A volume integral may be evaluated by a closed surface integral on its boundary, and vice versa.
19
Gauss’ law of electric field:
For a single charge,
For any charge distribution,
From the divergence theorem,
. ,
ddσqdσi
i nDnD
.4
22
qdrr
qdσ
nD
.
DDnD dddσ
20
Read: Chapter 6:10Homework: 6.10.3,6,7,9.Due: May 8
21
April 30 Stokes’ theorem
6.11 The curl and Stokes’ theorem
Physical meaning of curl:Set the coordinate system so that z is along at the point (x0, y0, z0 ).V
dxVdyV
dxVVdyVV
dxdyy
V
x
Vdxdyd
xy
yxdyyxxydxxy
xyz
0000
)( VnV
V is the total circulation of V per unit area.
ddd
drVnV
1lim)(
0
22
Stokes’ theorem:(Over a simply connected region. The surface does not need to be flat.)
Proof: Set the coordinate system so that x is along ndat an arbitrarily chosen point on the surface. Suppose the coordinates of that point is (x0, y0, z0 ).
Sum over all differential squares, at all interior lines will cancel, only the contributions from the exterior lines remain.
)( rVnV ddσ
sides 4
)(0
0
0
0rVVnV ddyVdzVdydz
z
V
y
Vdydzdσ
dzz
zy
dyy
yzyz
x
A surface integral may be evaluated by a closed line integral at its boundary, and vice versa.
rV d
Examples p328.1.
boundary. same thehave and if )()( 2121 σσσVσV dd Corollary:
23
Ampere’s law:
JHnHrH
nJrH
dd
dId
C
C
24
Read: Chapter 6:11Homework: 6.11.6,10,12,14.Due: May 8
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