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Vektor
. Divergence Theorem. Further Applications. Divergence Theorem. Further Applications
Ex. 1Ex. 1) Divergence indep. of coordinates. Invariance of divergence
- Use mean value theorem:
Contoh Contoh ) Physical interpretation of the divergence Steady flow of an incompressible fluid with =1
Total mass of fluid that flows around S from T per time:
Average flow out of T:
For a steady incompressible flow,
Contoh Contoh Heat equation: also, see Transport Phenomena by Bird et al., (2002) Fundamentals of Momentum, Heat, and Mass Transfer by Welty et al. (1976, 2000)
)T(V)z,y,x(fdxdydz)z,y,x(f 000
T
S
dAnv
)T(S
0)T(d111
)T(ST
000
dAnF)T(V
1lim)z,y,x(F
dAnF)T(V
1dVF
)T(V
1)z,y,x(Ff
0v0dAnvS
S
dAnvV
1
Potential Theory. Harmonic FunctionsPotential Theory. Harmonic Functions
Potential theory, it solution a harmonic function
Ex. 4)
Theorem 1:Theorem 1: f(x,y,z): a harmonic function in D. Above form in Ex. 4 is zero.
Ex. 5Ex. 5) Green’s first formula:
Green’s second formula:
Ex. 6Ex. 6)
0z
f
y
f
x
ff
2
2
2
2
2
22
SST
2
T
dAn
fdAnfdVfdVf
SSTT
dAn
gfdAngfdVgfgfdV)gf(
ST
22 dAn
fg
n
gfdVfggf
Sin0f.),funcharmonic:f(0f2
0dVfdVffT
2
T
Theorem 2:Theorem 2:
From the Ex. 6, f is identically zero in T.
UniquenessUniqueness
Teorema StokeTeorema Stoke
- Vector form of Green theorem:
Theorem 1Theorem 1: Stokes’s TheoremStokes’s Theorem (Surface integrals Line integrals) S: piecewise smooth oriented surface in space C: boundary of S F(x,y,z) which has continuous first partial derivatives in D
ContohContoh
vectorgenttanunit,
ds
dr'rds'rFdAnF
CS
)jFiFF(rdFdxdykF 21C
R
C
321
R
312
231
123 dzFdyFdxFdudvN
y
F
x
FN
x
F
z
FN
z
F
y
F
)yx(1)y,x(fz:S,kxjziyF 22
dsssinsd'rFjssiniscosr)1(2
0
2
C
RS
vu
dxdy)1y2x2(dAnF
kjy2ix2rrN,kjiF)2(
Polar coord.
R
S
dudv)v,u(N))v,u(r(F
dAnF
Contoh Contoh Green’s theorem
C
3
R
23
13
C
2
R
32
12
C
1
R
31
21
dzFdudvNx
FN
y
F,dyFdudvN
x
FN
z
F
dxFdudvNy
FN
z
F
kjfifrrrrN
kfjyix)y,x(r)v,u(r),y,x(fZ
yxyxvu
*S
1
*C
1
*S
1y
1 dxdyy
FdxFdxdy)1(
y
F)f(
z
F
Green’s theorem with F2=0
y
f
z
)z,y,x(F
y
)z,y,x(F
y
))y,x(f,y,x(F 111
y
F
x
FkFnF
planexyinS,jFiFF
12
21
C21
R
12 dyFdxFdxdyy
F
x
F
ContohContoh
ContohContoh Physical interpretation of the curl. Circulation
F=v, the circulation of the flow around C:
Stokes’s Theorem Applied to Path IndependenceStokes’s Theorem Applied to Path Independence
for path independence
)4(28ds'rF28y
F
x
FnF
kzyjxziyF,3z,4yx:C?,ds'rF
C
12
3322
C
C
**
SC
ds'rFA
1)P(nFA)P(nFdAnFds'rF
C
ds'rv
C0r
ds'rvA
1lim)P(nv
0dAnFds'rFdzFdyFdxFSCC
321 0F
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