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: Vector

???????? ?? ????? ???????? ??? ????? ?????? ??? ????? ??????? ????????? ????? ???????? ?? ????? ???????? ??? ????? ??? ???? ????? ??? ??????? ??????? ???????. ??????? ?? ???????? ?????? ???? ??????????.?????? ??????????????? ????? ??????? ???? ???????? ?????????? (x,y,z) ??????????? (r, f,z) ?????

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: Vector

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    1. ????? ?????: ???????? Vector ????? ???????? Vector Analysis: ??? ???????? (?????? ????????) ????? ??????? (??????? ????????) ???? ??????? ???????? ??????? ?????? ??????? ?????? (???????) ???????? ????? ????? ???? ??????? ??????? ???????? ??????? ??????? ???????? ?????? ??????? ???????? ???????? (????????) ?????????? ????????

    2. ???????? ?? ????? ???????? ??? ????? ?????? ??? ????? ??????? ????????? ????? ???????? ?? ????? ???????? ??? ????? ??? ???? ????? ??? ??????? ??????? ???????. ??????? ?? ???????? ?????? ???? ??????????. ?????? ??????????????? ????? ??????? ???? ???????? ?????????? (x,y,z) ??????????? (r, f,z) ???????? (r,?, f). ????? ???????? ??????? ?????????? (x,y,z) ???? ????? ?????? ?????? ?????? ?????? ?????? ??????? ??????? (ax, ay, az) & (I, j, k) ??????: ??? ax'=I' & ay'=j' & az'=k' ??????? ?????? ?????? ??? ?????? ??????? ?????? ???? ??? ???: ????? ???? ?????? a’A ??? ??? ?????? ‘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'

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