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
Gauss’s law or divergence theorem
convert from volume integral to surface integral
v
dvv AA dsA
vv
dsA
0
limA
• 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
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
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