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???????? ?? ????? ???????? ??? ????? ?????? ??? ????? ??????? ????????? ????? ???????? ?? ????? ???????? ??? ????? ??? ???? ????? ??? ??????? ??????? ???????. ??????? ?? ???????? ?????? ???? ??????????.?????? ??????????????? ????? ??????? ???? ???????? ?????????? (x,y,z) ??????????? (r, f,z) ?????
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1. ????? ?????: ???????? Vector ????? ???????? Vector Analysis:
??? ???????? (?????? ????????)
????? ??????? (??????? ????????)
???? ??????? ????????
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?????? (???????)
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???? ???????
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??????? ???????? ??????
??????? ???????? ???????? (????????)
?????????? ????????
2. ???????? ?? ????? ???????? ??? ????? ?????? ??? ????? ??????? ????????? ????? ???????? ?? ????? ???????? ??? ????? ??? ???? ????? ??? ??????? ??????? ???????. ??????? ?? ???????? ?????? ???? ??????????.
?????? ??????????????? ????? ??????? ???? ???????? ?????????? (x,y,z) ??????????? (r, f,z) ???????? (r,?, f). ????? ???????? ??????? ?????????? (x,y,z) ???? ????? ?????? ?????? ?????? ?????? ?????? ??????? ??????? (ax, ay, az) & (I, j, k) ??????:
??? ax'=I' & ay'=j' & az'=k'
??????? ?????? ?????? ??? ?????? ??????? ?????? ???? ??? ???: ????? ???? ?????? aA ??? ??? ?????? A
?????:
3. ????? ???????? (??? ????????): ???????? ???? ????? ???????
????? ????? ?? ?????? ??????? ???????? ????????
A'+(B'+C') = (A'+B')+C'
K(A'+B')= KA'+KB'
(K1+K2)A' = K1A'+K2A'
A'+B' = B'+A'
4. ????? ??????? (??????) ???????:
A'.B' = AB Cos C A' & B' ?? ??????? ??? ???????? ? ???
A'.B'= Ax Bx + Ay By + Az Bz
??? ?? cos 90 = 0 & cos 0 =1
???? ????? ?????? ??????? ??????? ?????? ???????
A'.(B'+C') = A'.B' + A'.C'
A'.KB = K(A'.B')
5. ????? ??????? (???????) ???????:
A' x B' = (AB sin ?) an'
??? an' ?? ???? ?????? ????????? ??? ????? ?????? ????????? ??? ?? ???? ?????? ????? ???? ?????? ?????? ???? ?????? ??????
A' x B' = - A' x B'
6. ????? ????? ??????? ????? ??? ???????? ?????
A' x B' = (Ax ax' +Ay ay' + Az az') x
(Bx ax' +By ay' + Bz az')
A' x B' = (AyBz AzBy) ax' + (AzBx - AxBz) ay' +
( AxBy AyBx) az'
????? ??? ????? ????? ??? ????? ??????
?? ????? ????????? ??:
i' x j' = - j' x i' & j' x k' = - k'x j' & k' x i' = - i' x k'
7. ????: ??? ???? A' = ax' ay' & B' = 2 ax'+4ay' 3 az'
???? ???? A'.B' & A' x B' ?????? ????? ?????
A'.B'= (2x1)+(4x-1)+(-3x0) = 2-4+0=-2
Cos? = A'.B'/ AB
??
A'xB' =(AyBz AzBy) ax' + (AzBx - AxBz) ay' + ( AxBy AyBx) az'=
ŚA'xB'Ś = (9+9+36)0.5 = 11.6
Sin ? = ŚA'xB'Ś / AB
8. ????? ?????????? ??? ??? ???? ???? P ??????? ????? ???? ???? ???? ???? ????? ???????? ???? ??? ???????????
?????????? (x,y,z)
?????????? ???????? (r,f,z)
??????? (r,?,f)
9.
??????? f ???? ??? ???????? ??? ?? ??????? ????????? ??????? ?? ?????? ??????? (????? ???? ??????????)
?????? ????? r ?????? ??????? ??????
??????? ????????? ???? ??????? ?? ?????? z ?? ??????? ??????? ??? z?
??????? ?????? ???? ??????? ?? ???? ????? ?????? P.
11. ????? ???? ???????? ?????? P ???
(x+dx, y+dy, z+dz) ?? (r+dr, f+df, z+dz) ??(r+dr,?+d?, f+df)
12. ??? ????? ???????? ???? ?????? ????? dv ???:
??????? ??????? ????
dv=dx dy dz dv= r dr df dz dv = r2sin?drd?df
?????? ??????? ???????? ???? ???? ????? ???????? ??????? ??? ar
ds = (rd?) (r sin?df) = r2 sin? d? df
?????? ????? ???????? dl ?? ????? ???? P ????? ???
dl2 = dx2 + dy2 + dz2 ???????
dl2 = dr2 + r2df2 + dz2 ???????
dl2 = dr2 +r2d?2 + r2 sin2? df2 ????
13. ?????: ???????? ???? ???? ?????? A ???? ???? ?? ?????? (2,-4,1) ??? ?????? (0,-2,0) ??????????? ?????????? ???? ???? ???? ???? ?????? ??? ??? ?????? A?
A'=(0-2)ax' + (-2+4) ay' + (0-1)az' =-2 ax' + 2 ay' -az'
A=[(-2)2+(2)2+(-1)2]0.5= 3
aA'= A'/A= (-2/3)ax' + (2/3) ay' (1/3)az'
aA=[(-2/3)2+(2/3)2+ (-1/3)2]0.5=1
14. ???? ??????? ??? (5,3p/2, 0) & (5,p/2,10) ??????????? ??????????
A'= -5 ay' & B'= 5 ay' + 10 az'
??????? ???????? ??? ???????? ?? A'-B' = 10 ay'+ 10az'
ŚA-BŚ= (102+102)0.5
?? ???? ??????? ?????????? ?????????? ?????? ?????? ??? ???? ?????? ???? ??????? ??????? ?????????? ??????????
15. ???? ?? ?? ?? ?????? A & B ???????? ????
A'=4 ax'-2 ay'-az & B'= ax'+ 4 ay' 4 az
?=90 Cos 90 = 0 ŚA'.B'Ś=0
-----------------------------------------------------------------------
??? ??? A'=2ax' +4ay' & B'= 6ax'- 4AZ' ???? ???? ??????? ?????? ?????? ???????? ????? ??????? ?????? ????????
---------------------------------------------------------------------------
??? ??? A'= ax'+ay' & B'=ax' + 2az' & C'=2ax'+az
???? ??????
(A'xB')xC * ??????? ????? A'x(B'xC') & A'.B'xC' ??????? ?? A'xB'.C'
16. ?????? ?????????? ??????? ?????? ????? ?????? ?=?=?? ??? ?????? ??????? ???? ??? ????? a ????? ???? ????? ???? ??=p & ?=0?
???? ????? ???????? ?? ds=r2 sin? d? df ????? ????? ????? ?? A=2p a2 (cos? - cos??)
????? ?=0 [cos0=1] & ??=180 [cos 180= -1] ??? A=4pa2 ??? ????? ????? ???????
17. ???????????????
18. ???????? ??????? ???? ??????? ?????????
?????? ?????? ?????? Divergence and the divergence theorem *?????? Divergence
???? ??????? ??????? ???? ????? ??? ???? ???? ?? ???? ??? ???? ?? ????? ??????
?????? ??? ????? ?????? ??????? ?? ???? ??????
???????? ??? ???? ????.
????? ???? ???? ???? ?????? ???? ??? ????? (0 ? ) ???? ???? ??????? ???? ????? ??? ????? ?? ??? ???????. ??????? ?????? ????? ???? ?????? ????? ???? ????? ?????? ??????? ?? ???????? ???????? ???????? ????? ????????? ????? ???? ?????? ????? ???? ????? ?????? ??????? ?? ???????? ???????? ???????.
19. ????? ????? ??????? ? ??? ?????? ??????? ???? ??? ??? ??????? ???? ????? ??? ????? ????? ????? ??? ????? ?? ? ?????? ???? ????? ????? ??????? D ????? ???? ???????. ???? ???? ????? ????? ??? ?????? ?????? ?? ?????????- ??????? ???????
???? ???? ?????? ?????? A' ?? ?????? P ???????
20. ?????? ??????????? ??????????:
???? ?????? A ?????? ??????? P? ??? ??? ?????? ??? ????? ?????? ?????????? x,y,z
A=Ax ax + Ayay' + Azaz
??? ???? ?? ??????? ???? ??? ????? ?????? ????? ???? ???? ?????? ??? ??????? ???????.
21.
???? ???? ??????? ????? ??????? ??
?????? ?????? ????? ???? ???? ?????? ??? ??????? ??????? ???
??????? ??? ?v=?x?y?z ????? ?v?0
22. ????????
????????
?????
23. ??????? ??? (?)? ???????
24. ????: ???? ?????? ??? v(x,y,z) ? ?? ???? ??? dL'=dx i'+dy j' + dz k
25. ????: ???? '.r' ? & '.z'? ??? ??? r'=xi' + yj'+ zk'