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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