divergence given a vector field, the divergence operation tells if there is a source or sink useful...
TRANSCRIPT
divergence •
• given a vector field, the divergence operation tells if there is a source or sink
• useful for relating electric fields to charges
• vector field ==> scalar
AA div
xxo
x
s s
vv
dsA
0
lim
two surfaces s surround the point xovector field enters from left and leaves from rightevaluate integrals at the 2 surfacesTaylor series about the center point xo
v
dsA
2
1 x
x
AxA
xx
ox
xxo
x
2
x
x
AxA xox
x
Ax
ss
xs
sx
xA ox
2 sx
xA ox
2
z
A
y
A
x
A zyx
A
Cartesian coordinates
source
example
Gauss’s law or divergence theorem
convert from volume integral to surface integral
v
dvv AA dsA
vv
dsA
0
limA
dsAAv
dv
xxuA
v
dvA
dsA
221
20 2 21
01 x
curl or rot
• place paddle wheel in a river
• no rotation at the center
• rotation at the edges
The start of Katrina
Hurricane from a satellite.
• the vector un is out of the screen
• right hand rule s is surface
enclosed within loop• closed line integral
AA curlss
dlAu
0
lim n
• four lines x and y surround the point xo, yo
• vector field follows closed loop
• evaluate integrals• Taylor series about
the center point
s0s
limcurl
dlAu
AAn
2
1oo 2
yy,x dlAdlA
4
3oo 2
yy,x dlA
3
2oo y,2
xx dlA
1
4oo y,2
xx dlA
x2
y
y
Ay,xA x
oox
dlA
x2
y
y
Ay,xA x
oox
y2
x
x
Ay,xA y
ooy
y2
x
x
Ay,xA y
ooy
yxy
A
x
Axy
Cartesian coordinates
zyx AAAzyx
zyx uuu
A
right hand rule
Flow velocity
vorticityMagnetic field
current
xyzxyxzyx
zyx uuu
A
zy xyzxyx uuuA x
xuA xz yyzu zyu
s0s
lim
dlAu
An
dlAuuA nns
dlAuuA nnds
dlAuuA nnds
zy xyzxyx uuuA x
xux
zy yyzxz uuuA x
dlA 1
0xdx
1
0ydy1x
0
1xdx
0
1ydy0x 0
2
1
2
1
2
1
2
1
dxdyA dxdyy2
1
repeated vector operations
curl (grad a) = 0div•curl A = 0div (grad a) = Laplacian (a)
zyx uuuz
a
y
a
x
aa
z
a
y
a
x
azyx
a
zyx uuuxux
y
A
x
A
z
x
A
z
A
yz
A
y
A
x
xy
zxyzA
zyx AAAzyx
zyx uuu
A
zyx uuuz
a
y
a
x
aa
zza
y
ya
xxa
a
zyx uuuzyx
2
2
2
2
2
22
z
a
y
a
x
aa
Phasors•v = V cos (t + ø)•v = Re [V ejt]•where V = V ejø
vjdt
dv
vj
1'vdt
LjRI
Vo
o
53j
120j
e10
e5
o120je5 V
o30t4sin5)t(v
oo 9030t4cos5
o173t4cos5.0)t(i
24j6
e5o120j
cb
ay
0.010.01
1 100
1
100
y
d
0.010.01
1 100
1
100
y
a0.01
0.011 100
1
100
y
b
0.010.01
1 100
1
100
y
c
