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16.360 Lecture 13
Basic Laws of Vector Algebra
Scalars:
e.g. 2 gallons, $1,000, 35ºC
Vectors:
e.g. velocity: 35mph heading south 3N force toward center
AaA^
A
Aa ^
16.360 Lecture 13
• Cartesian coordinate system
x
y
z
A
xA
yA
zA^^^
zAyAxAA zyx
222zyx AAAA
)cos(AAz
)cos()sin( AAx
)sin()sin( AAz
16.360 Lecture 13
• Vector addition and subtraction
C = B+A = A +B,
|C| = |B+A| = |A| +|B|,
A
B
C
A
B
C
head-to-tail rule parallelogram
rule
A
B
D
D = A - B = -(B –A),
A
B
D’ = (B-A)
)()()()()(^^^^^^^^^
zzxyxxzyxzyx BAzBAyBAxBzByBxAzAyAx
)()()()()(^^^^^^^^^
zzxyxxzyxzyx BAzBAyBAxBzByBxAzAyAx
16.360 Lecture 13
• position and distance
x
y
z
A
xA
yA
zA
B
D = A - B = -(B –A),
)()()()()(^^^^^^^^^
zzxyxxzyxzyx BAzBAyBAxBzByBxAzAyAx
222 )()()(|| zzxyxx BABABAD
16.360 Lecture 13
• Vector multiplication
1. simple product
2. scalar product (dot product)
)()()(^^^
zyx AzAyAxA
)()()(^^^
zyx kAzkAykAxAkB
ABBAAB cos
ABAB )()()(
^^^
zyx AzAyAxA )()()(
^^^
zyx BzByBxB
)]()()([
)]()()([)]()()([^^^
^^^^^^
zzyyxx
zyxzyx
ABzAByABx
AzAyAxBzByBxAB
16.360 Lecture 13
Properties of scalar product (dot product)
CBABCAB
)(
BAAB
a) commutative property
b) Distributitve property
2AAA
16.360 Lecture 13
3. vector product (cross product)
ABABnBA sin^
A
B
ABABnBA sin^
^
n
AB
ABBA
a) anticommutative property
b) Distributitve property
CABACBA
)(
c) 0AA
,^^^
zyx ,^^^
xzy
,^^^
yxz
16.360 Lecture 13
3. vector product (cross product)
)()()(
)()(^^^
^^^^^^
xyyxzxxzyzzy
zyxzyx
BABAzBABAyBABAx
BzByBxAzAyAxBA
zyx
zyx
BBB
AAA
zyx
BA
^^^
16.360 Lecture 13
Example vectors and angles
In Cartesian coordinate, vector A is directed from origin to point P1(2,3,3), and vector B is directed from P1 to pint P2(1,-2,2). Find: (a) Vector A, its magnitude |A|, and unit vector a(b) the angle that A makes with the y-axis(c) Vector B(d) the angle between A and B(e) perpendicular distance from origin to vector B
B
A
332^^^
zyxA
222 332 A
22/)332(^^^^
zyxA
Aa
)cos(^^
AyyAyA
)32()32()21(^^^
12 zyxPPB
)cos( ABBABA
)180sin( 0ABAd
16.360 Lecture 13
4. Scalar and vector triple product
a) scalar triple product
b) vector triple product
)()())()()( BACACB
CCC
BBB
AAA
BACCABCBA
zyx
zyx
zyx
)()()( BACACBCBA
zyx
zyx
zyx
CCC
BBB
AAA
CBA )(
CBACBA
)()(
)()()( BACCABCBA
16.360 Lecture 13
Example vector triple product
2^^^
zyxA ^^
zyB
?)( CBA
?)( CBA
^^^
^^^^^^
223
302
110 zyx
zyx
CCC
BBB
zyx
CB
zyx
zyx
^^^
^^^
062
222
2113)( zyx
zyx
CBA
32^^
zxC
16.360 Lecture 14
• Cartesian coordinate system
x
y
z
dl
xl
yl
zl
^^^^^^
zdzydyxdxzdlydlxdlld zyx
dydzxdldlxsd zyx
^^
dxdzydldlysd zxy
^^
dxdyzdldlzsd yxz
^^
dxdydzdv
16.360 Lecture 14
• Cartesian coordinate system
x
y
z
xs
s
directions of area
16.360 Lecture 14
• Cylindrical coordinate system
z
x
y
),,( 111 zrP
^
r
^
^
z
^^^
zr ^^^
rz ^^^
rz
0^^
rr 1^^
rr
zr AzAArA^^^
222zr AAAA
1R
1
^
1
^
1 zzrrR
16.360 Lecture 14
• the differential areas and volume
^
drdzsd
^
rdzrdsd r
^
zdrrdsd z
z
x
y
dr rd
drdzrddv
16.360 Lecture 14
Example: cylindrical area
z
x
y
5
3
30
60
3/
6/
3
0
60
30
35
ddzrdS
^
rdzrdsd r
16.360 Lecture 14
• Spherical coordinate system
^^^
r^^^
R^^^
R
0^^
rr 1^^
rr
AAARA R
^^^
222 AAAA R
16.360 Lecture 14
• differential volume in Spherical coordinate system
dRRddRR
dldldlRld R
sin^^^
^^^
ddRRdldlRsd R sin2^^
dRdRdldlsd R sin^^
dRdRdldlsd R
^^
ddRdRdldRdldv sin2
16.360 Lecture 14
• Examples
(1) Find the area of the strip
60
30
360
0
2 sin ddRSdS R
(2) A sphere of radius 2cm contains a volume charge density 2cos4v
Find the total charge contained in the sphere
100/2
0
260
0
2180
0
2
100/2
0
2260
0
180
0
2
100/2
0
2260
0
180
0
))(sincos4(
))(sincos4(
)sin(
dddRR
dddRR
RdddRdvQ vv
16.360 Lecture 15
• Cartesian to cylindrical transformation
,22 yxr
),(tan 1
x
y
),cos(rx
),sin(ry
16.360 Lecture 15
• Cartesian to cylindrical transformation),cos(ˆˆ xr ),sin(ˆˆ yr
),sin(ˆˆ x ),cos(ˆˆ y
,cosˆsinˆˆ yx
),cos(ˆ)ˆˆ(ˆˆ axybxaxr
),sin(ˆ)ˆˆ(ˆˆ byybxayr
,sinˆcosˆˆ yxr
,sinˆcosˆˆ rx
,cosˆsinˆˆ ry
16.360 Lecture 15
• Cartesian to cylindrical transformation
),cos(ˆˆ xr ),sin(ˆˆ yr
),sin(ˆˆ x ),cos(ˆˆ y
,sincos yxr AAA
,ˆˆˆ zAyAxAA zyx
,ˆˆˆ zAArAA zr
,cossin yx AAA
,sincos AAA rx
,cossin AAA ry
16.360 Lecture 15
• Cartesian to Spherical transformation
,222 zyxr
),(tan22
1
z
yx
),(tan 1
x
y
,cossin Rx
,sinsin Ry
,cosRz
16.360 Lecture 15
• Cartesian to Spherical transformation
,ˆˆˆ bzarR
,sinˆˆ bzR
,cosˆˆ arR ,sinˆcosˆˆ yxr
,cosˆsinsinˆcossinˆˆ zyxR
,cosˆˆ cr,ˆˆˆ dzcr ,sinˆˆ cr
,sinˆsincosˆcoscosˆˆ zyx
,cosˆsinˆˆ yx
16.360 Lecture 15
• Cartesian to Spherical transformation
,ˆˆˆˆ edcRx
,sinˆˆ bzR
,cossinˆˆ xR
,sinˆcosˆˆ yxr
,sinˆcoscosˆcossinˆˆ Rx
,cosˆˆ cr,ˆˆˆ dzcr ,sinˆˆ cr
,sinˆcosˆˆ Rz
,coscosˆˆ x ,sinˆˆ ex
,cosˆsincosˆsinsinˆˆ Ry
16.360 Lecture 15
• Distance between two points:
,)()()(2/12
122
122
121212 zzyyxxRd
,)()sinsin()coscos(2/12
122
11222
11221212 zzrrrrRd
,)cos(sinsincos[cos2
,)coscos()sinsinsinsin()cossincossin(2/1
121212212
122
2/121122
2111222
21112221212
RRRR
RRRRRRRd
16.360 Lecture 16
Gradient in Cartesian Coordinates
Gradient: differential change of a scalar
,
,)ˆˆˆ(
ˆˆˆ
,),,(
ldT
ldzz
Ty
y
Tx
x
T
ldzz
Tldy
y
Tldx
x
T
dzz
Tdy
y
Tdx
x
TzyxdT
),ˆˆˆ( zz
Ty
y
Tx
x
TT
ldTzyxdTT
),,(
The direction of T is along the maximum increase of T.
16.360 Lecture 16
Example of Gradient in Cartesian Coordinates
Find the directional derivative of ,22 zyxT along the direction 2ˆ3ˆ2ˆ zyx
and evaluate it at (1, -1,2).
),)(ˆˆˆ( 22 zyxz
zy
yx
xT
,ˆlaTdl
dT
16.360 Lecture 16
Gradient operator in cylindrical Coordinates
),sin1
(cos
,
r
T
r
T
x
z
z
T
x
T
x
r
r
T
x
T
,ˆˆˆz
Tz
y
Ty
x
TxT
,sinˆcosˆˆ rx
,cosˆsinˆˆ ry
),cos1
(sin
,
r
T
r
T
y
z
z
T
y
T
y
r
r
T
y
T
z
Tz
T
rr
Tr
r
T
r
T
r
T
r
T
r
T
r
T
r
T
r
Tr
z
Tz
y
Ty
x
TxT
ˆ1ˆˆ
)cossincos1
sin1
cossin(ˆ
)sincos1
sinsincos1
cos(,ˆ
ˆˆˆ
22
22
16.360 Lecture 16
Gradient operator in cylindrical Coordinates
z
x
y
dr rd
zr l
Tz
l
T
l
TrT
ˆˆˆ
z
Tz
T
rr
Tr
l
Tz
l
T
l
TrT
zr
ˆ1ˆˆ
ˆˆˆ
,drlr , rdl ,dzlz
16.360 Lecture 16
Gradient operator in Spherical Coordinates
,ˆˆˆ
l
T
l
T
l
TRT
R
T
R
T
RR
TR
l
T
l
T
l
TRT
R
sin
1ˆ1ˆˆ
ˆˆˆ
,dRlR , Rdl ,sin dRl
16.360 Lecture 16
Properties of the Gradient operator
,)( VUVU
,)( UVVUUV
,)( 1 VnVV nn
16.360 Lecture 17
Flux in Cartesian Coordinates
,sdEfluxTotal
16.360 Lecture 17
Flux in Cartesian Coordinates
,)1(
)ˆ()ˆˆˆ(
,
1
111
zyE
dydzxEzEyEx
dsnEF
x
zyface x
face
,)2(
)ˆ()ˆˆˆ(
,
2
222
zyE
dydzxEzEyEx
dsnEF
x
zyface x
face
,
)]1()2([
)]1()2([21
zyxx
E
zyxx
EE
zyEEFF
x
xx
xx
16.360 Lecture 17
Definition of divergence in Cartesian Coordinates
,
)]1()2([
)]1()2([21
zyxx
E
zyxx
EE
zyEEFF
x
xx
xx
,43 zyxy
EFF y
,65 zyxz
EFF z
,)( zyxz
E
y
E
z
EsdE zyx
S
),(z
E
y
E
x
EEEdiv zyx
16.360 Lecture 17
Properties of divergence
,)( 2121 EEEE
,01 E
If No net flux on any closed surface.
Divergence theorem
,
)(
dxdydzE
zyxz
E
y
E
z
EsdE
v
zyx
S
16.360 Lecture 17
Divergence in Cylindrical Coordinates
z
x
y
dr rd
,)1(
)ˆ()ˆˆˆ(
,
1
11
zrE
zrrEzEEr
dsnEF
r
zface r
rrface
,)2(
)ˆ()ˆˆˆ(
,
1
22
zrE
zrrEzEEr
dsnEF
r
zface r
rrface
,)(
)]1()2([
)]1()2([21
zrrEr
zrr
rErE
zrEEFF
r
rr
rr
16.360 Lecture 17
Divergence in Cylindrical Coordinates
z
x
y
dr rd
,)(
)]1()2([
)]1()2([21
zrrEr
zrr
rErE
zrEEFF
r
rr
rr
,)(
)]1()2([
)]1()2([43
zrE
zrEE
zrEEFF
,)(
)]1()2([
)]1()2([65
zrrEz
zrrz
EE
rrEEFF
z
zz
zz
,
))(
(
vE
zrrz
E
r
E
rr
rEsdE zr
S
16.360 Lecture 17
Divergence in Spherical Coordinates
,sin)(
sin)]1()2([
sin)]1()2([
2
22
221
RERR
RR
ERER
REEFF
R
RR
RR
,sin)1(
sin)ˆ()ˆˆˆ(
,
2
1
11
RE
RRREEER
dsnEF
R
face R
RRface
,sin)2(
sin)ˆ()ˆˆˆ(
,
2
2
22
RE
RRREEER
dsnEF
R
face R
RRface
16.360 Lecture 17
Divergence in Spherical Coordinates
,)(sin
)]1(sin)2([sin
sin)]1()2([43
RRE
RREE
RREEFF
R
,sin)(
sin)]1()2([
sin)]1()2([
2
22
221
RERR
RR
ERER
REEFF
R
RR
RR
,)(
)]1()2([
)]1()2([65
RE
RREE
RREEFF
16.360 Lecture 17
Divergence in Spherical Coordinates
,)(sin43
RREFF R,sin)( 221
RERR
FF R
,)(65
RREFF
vE
RRER
ER
ERRR
RERERERR
sdE
R
R
S
sin)](sin
1)(sin
sin
1)(
1[
,)]()(sin)([
222
2
16.360 Lecture 18
Circulation of a Vector
,ldBnCirculatio
,0
ˆˆˆˆ
ˆˆˆˆ
00
00
dyyBxdxxBx
dyyBxdxxBxnCirculatio
a
d
d
c
c
b
b
a
16.360 Lecture 18
Circulation of a Vector
,2
ˆ 0
r
IB
,
,ˆ2
ˆ
0
2
0
0
I
dr
IldBnCirculatio
16.360 Lecture 18
Curl in Cartesian Coordinates
sBn
sdz
B
z
Bz
x
B
z
By
z
B
y
Bx
xyz
B
z
Bzzxzy
z
B
y
BxldBn
zxzxyz
zxyz
c
)(ˆ
)](ˆ)(ˆ)(ˆ[
)(ˆ)(ˆ
,)(ˆ
)(ˆ
,
2
,2 2
zyz
B
y
Bx
yByBzBzBx
ldBn
yz
zzzyface yy
zyface
)(ˆ)(ˆ)(ˆz
B
z
Bz
x
B
z
By
z
B
y
BxB zxzxyz
16.360 Lecture 18
Vector identities involving the curl
BABA
)(
0)( A
0)( V
Stokes’s theorem
sdBldBnc
)(
16.360 Lecture 18
Curls in Rectangular, Cylindrical and Spherical Coordinates
zyx BBBzyx
zyx
B
ˆˆˆ
zr ArAAzr
zrr
rB
ˆˆˆ1
ARRAA
R
RRR
RB
R sin
ˆsinˆˆ
sin
12
16.360 Lecture 18
Laplacian Operator of a scalar
zyx AzAyAxz
Vz
y
Vy
x
VxV ˆˆˆˆˆˆ
2
2
2
2
2
2
)(
z
V
y
V
x
V
Az
Ay
Ax
V zyx
2
2
2
2
2
22 )(
z
V
y
V
x
VVV
16.360 Lecture 18
Laplacian Operator of a vector
zyx EzEyExE ˆˆˆ
2
2
2
2
2
2
2
2
2
2
2
22 ˆˆˆ)(
z
Ez
y
Ey
x
ExE
zyxE zyx
)()(2 EEE
