场论:场的积分性质 dTV ds 34/384 divo- f dso dTV xf ds xf Vf的意义:dsf定义为:f从流出的通上 △r.f=ddsf V·f=lim 单位体积从场点流出的通量 V·f=0无净通上外溢,无源;V·f≠0有净通上外溢,有源! 两个等上正负点边荷源的边场线模拟 ose
34/384 JJ II J I Back Close ⑤Ø: ⑤✛➮➞✺➓ Z τ dτ∇ = I τ d~S → Z τ dτ∇φ = I τ d~Sφ Z τ dτ∇ ×~f = I τ d~S ×~f ∇ · ~f✛➾➶➭ I τ d~S · ~f ➼➶➃➭ ~f❧τ✻Ñ✛Ïþ ∆τ ∇ · ~f = I ∆τ d~S · ~f ∇ · ~f = lim ∆τ→0 H ∆τ d~S · ~f ∆τ = ü➔◆➮❧⑤✿✻Ñ✛Ïþ ∇ · ~f = 0➹➚Ïþ✠➘➜➹✌➯ ∇ · ~f 6= 0❦➚Ïþ✠➘➜❦✌! ✛ ✻✲ ❄ ❅■❅✒ ✠❅❅❘ s ❅❅❘✲ ❄✛✠ ✒ ✻❅■❅ s ü❻✤þ✔❑✿❃Ö✌✛❃⑤❶✜❬
俞论:场的积分性质 旋度的面译,:曲面s中的根/ds可xf=pd.f 曲面S边界的曲线积分 x方向,曲面方程:z=g(x,y),法线n=n1e1+n2e2+na33 384 yeuxs dz-gldx-gldy=0 dr-dxe,+ dyes dze3 i∝-gx6x-g36y+62 i=c(-gex-g3y+62)=→c2(1+(gx)2+(gy2 1+gx+g ose
35/384 JJ II J I Back Close ⑤Ø: ⑤✛➮➞✺➓ ❫Ý✛→➮➞: ➢→ S ➙✛→➮➞Z S d~S · ∇×~f = I S d~l · ~f ➢→ S ❃✳✛➢❶➮➞ x➄➉,➢→➄➜:z = g(x, y), ④❶:~n = n1~e1 + n2~e2 + n3~e3 dz − g 0 xdx − g 0 ydy = 0 d~r = dx~e1 + dy~e2 + dz~e3 ~n · d~r = 0 ⇒ ~n ∝ −g 0 x ~ex − g 0 y ~ey + ~ez ~n = c(−g 0 x ~ex − g 0 y ~ey + ~ez) ~n·~n=1 ====⇒ c 2 (1 + (g 0 x ) 2 + (g 0 y ) 2 ) = 1 nx = − g 0 q x 1 + g 0 x 2 + g 0 y 2 ny = − g 0 q y 1 + g 0 x 2 + g 0 y 2 nz = 1 q 1 + g 0 x 2 + g 0 y 2
旋度的面积分:/d.×f=中d.直场不:场的积分性质 x方向,曲面方程:z=g(x,y),简线:i=n1e1+n2e2+nge3 n 36/384 1+g2+ 1+g(2+ 曲面边界在xy平面投影区域曲线边界分为y=yB(x),y=yA(x dS203f-dS3a2f1) dS3(glOf +02f1) Z=g(xy) dS3,f(x,y,g(x,y)) dx dy f(x, y, g(x, y)). dx f(x, yB(x),g(x, yB(x)))-f(,yA(x), g(x, yA(x))l dxf1(x,y,g(x,y)曲面外法线按右手关系规定xy平面的曲线的正向为逆时 公式只适用于曲面中无孔洞的情况! ose
36/384 JJ II J I Back Close ❫Ý✛→➮➞ ⑤Ø: ⑤✛➮➞✺➓ : Z S d~S · ∇ ×~f = I S d~l · ~f x➄➉,➢→➄➜:z = g(x, y), ④❶:~n = n1~e1 + n2~e2 + n3~e3 nx,y = − g 0 q x,y 1 + g 0 x 2 + g 0 y 2 ; nz = 1 q 1 + g 0 x 2 + g 0 y 2 dS2 dS3 = n2dS n3dS = −g 0 y ➢→❃✳✸ xy ➨→Ý❑➠➁➢❶❃✳➞➃:y = yB(x), y = yA(x) Z S (dS2∂3f1 − dS3∂2f1) = − Z S dS3 (g 0 y∂3f1 + ∂2f1) = − Z S dS3 d dyf1(x, y, g(x, y)) = − Z dx Z yB(x) yA(x) dy d dyf1(x, y, g(x, y)) = − Z dx [f1(x, yB(x), g(x, yB(x))) − f1(x, yA(x), g(x, yA(x)))] = I S dx f1(x, y, g(x, y)) ➢→✠④❶❯♠➹✬❳✺➼xy➨→✛➢❶✛✔➉➃❴➒✂ ú➟➄➲❫✉➢→➙➹➎➱✛➐➵!
俞论:场的积分性质 ds. vxf/ dsx Vf d dIf dS×V 37/384 ds×Vp= dsxV)×f=ddxf V×f的意义: △§.V×f=ddi.f=f绕S环绕的环量 V×fcos= Im as dI.f △S Ⅳ×f大小为俞点任最大两位面积环量值,方向沿 具有最大环量的曲面方向.任意方向曲面的环量 是V×f在曲面方向上投影。 V×f=0无净环量,无旋;V×f≠0净环量,有旋 ose
37/384 JJ II J I Back Close ⑤Ø: ⑤✛➮➞✺➓ Z S d~S · ∇ ×~f = Z S d~S × ∇ · ~f = I S d~l · ~f ⇒ Z S d~S × ∇ = I S d~l Z S d~S × ∇φ = I S d~l φ Z S (d~S × ∇) ×~f = I S d~l ×~f ∇ ×~f✛➾➶➭ ∆~S · ∇ ×~f = I ∆S d~l · ~f ≡ ~f✼S❶✼✛❶þ |∇ ×~f| cos θ = lim ∆S→0 H ∆S d~l · ~f ∆S ∇ ×~f ➀✂➃⑤✿❄⑩➀ü➔→➮❶þ❾➜➄➉÷ ä❦⑩➀❶þ✛➢→➄➉➞❄➾➄➉➢→✛❶þ ➫∇ ×~f✸➢→➄➉þÝ❑✧ ∇ ×~f = 0➹➚❶þ,➹❫; ∇ ×~f 6= 0➚❶þ,❦❫✧ ❜✻ ✐✍✌✻✻ ✎☞ ✖✕✻ ✗✔ ✣✢ ✤✜
曲线坐标:柱坐标 坐标 +y2,M= arctan,z基矢 COS er.esino 38/384 sin e COS e=0 e=0 en·d,=0 1 e= cos 0e +sin ge sin 0e cos be cos 0e. -sin Beg e= sin Be + cos Bee de, a 0 oa der de 0 Or az or a0 a 沿基矢方向的无穷小线元:dl1=dr,dl2=rdb,dl3=dz 微分算符:V=e7 矢量场:f=f1er+f6e+f2e2 r ahaz 标量场的梯度:√o=价+enrD5 010 矢量场的散度 r a6 +62-]·(fen+fen+fe2) ose
38/384 JJ II J I Back Close ➢❶❿■: ❰❿■ ❿■✄➭ r = p x2 + y2 , θ = arctan y x , z ➘➙➭ ~er, ~eθ , ~ez ✫✪ ✬✩❡r ✻✒ ❅ ❅ ❅■❅ ✲ ~er ~e ~ey θ θ ~ex θ ~ez ~er · ~ex = cos θ ~er · ~ey = sin θ ~er · ~ez = 0 ~eθ · ~ex = − sin θ ~eθ · ~ey = cos θ ~eθ · ~ez = 0 ~ez · ~ex = 0 ~ez · ~ey = 0 ~ez · ~ez = 1 ~er = cos θ~ex + sin θ~ey ~eθ = − sin θ~ex + cos θ~ey ~ex = cos θ~er − sin θ~eθ ~ey = sin θ~er + cos θ~eθ ∂~er ∂r =0 ∂~er ∂θ =~eθ ∂~er ∂z = ∂~eθ ∂r =0 ∂~eθ ∂θ =−~er ∂~eθ ∂z = ∂~ez ∂r = ∂~ez ∂θ = ∂~ez ∂z =0 ÷➘➙➄➉✛➹→✂❶✄➭ dl1 = dr, dl2 = rdθ, dl3 = dz ❻➞➂❰➭∇ = ~er ∂ ∂r +~eθ 1 r ∂ ∂θ +~ez ∂ ∂z ➙þ⑤➭~f = fr~er+fθ~eθ+fz~ez ■þ⑤✛❋Ý➭ ∇φ = ~er ∂φ ∂r +~eθ 1 r ∂φ ∂θ +~ez ∂φ ∂z ➙þ⑤✛ÑÝ➭ ∇ · ~f = [~er ∂ ∂r +~eθ 1 r ∂ ∂θ +~ez ∂ ∂z ] · (fr~er+fθ~eθ+fz~ez)