场不:场的微分性质 (f×g ∑x(路∑可x6(g 384 ij, k GilmemGjkI (kaif;+fjaigk) ∑m(-nnk+6n-)g乓+fg ij, k, m=1 ∑n(-0y1m4-1.+11+619 ij, k, m=1 ∑(gO1+fg)+∑(gO可+6g) i,k=1 gV·f-f.Vg+g·Vf+fV·g ose
29/384 JJ II J I Back Close ⑤Ø: ⑤✛❻➞✺➓ ∇ × ( ~f × ~g) = X 3 i,j,k,l=1 ∂i~ei × (fjgkjkl~el) = X 3 i,j,k,l=1 ~ei × ~eljkl∂i(fjgk) = X 3 i,j,k,l,m=1 ilm~emjkl(gk∂ifj + fj∂igk) = X 3 i,j,k,m=1 ~em(−δijδmk + δikδmj)(gk∂ifj + fj∂igk) = X 3 i,j,k,m=1 ~em(−δijδmkgk∂ifj − δijδmkfj∂igk + δikδmjgk∂ifj + δikδmjfj∂igk) = − X 3 i,k=1 (~ekgk∂ifi + fi∂igk~ek) +X 3 i,j=1 (gi∂ifj~ej + ~ejfj∂igi) = −~g∇ · ~f −~f · ∇~g + ~g · ∇~f +~f∇ · ~g
场论:场的微分性质 微分算符对复张场的作用 Vo(u)=∑6ao(u)=∑ eia-oiu=o(u) 30/384 ou u)=∑a(u) 1). Vu V×f(u)=∑a0(u)=∑ of Eikeko diu=Vu×f(u) uI i、j,k=1 ij, k=1 微分算符对R的作用 xe V aa r=∑xV=∑e以以 ose
30/384 JJ II J I Back Close ⑤Ø: ⑤✛❻➞✺➓ ❻➞➂❰é❊Ü⑤✛❾❫ ∇φ(u) = X 3 i=1 ~ei∂iφ(u) = X 3 i=1 ~ei ∂φ ∂u ∂iu= φ 0 (u)∇u ∇ · ~f(u) = X 3 i=1 ∂ifi(u) = X 3 i=1 ∂fi ∂u ∂iu= ~f 0 (u) · ∇u ∇ ×~f(u) = X 3 i,j,k=1 ijk~ek∂ifj(u) = X 3 i,j,k=1 ijk~ek ∂fj ∂u ∂iu= ∇u ×~f 0 (u) ❻➞➂❰éR~ ✛❾❫ ~r = X 3 i=1 xi~ei ∇ = X 3 i=1 ~ei∂i ∂i = ∂ ∂xi ~r 0 = X 3 i=1 x 0 i ~ei ∇0 = X 3 i=1 ~ei∂ 0 i ∂ 0 i = ∂ ∂x 0 i
R=r-r (x1-x1) 俞论:场的微分性质 ax=6;a1x1=6;ax1=a1x=0 31/384 R=∑(x1-x)=3 V×R=∑(x一x=∑=0=-V× k=1 i、jk=1 1 VR=∑6B=2R ∑eAx-x3xy-x 1 R R ei VR R ose
31/384 JJ II J I Back Close ~ ⑤Ø: ⑤✛❻➞✺➓ R = ~r −~r 0 = X 3 i=1 (xi − x 0 i )~ei ∂ixj = δij ∂ 0 ix 0 j = δij ∂ix 0 j = ∂ 0 ixj = 0 ∇ · R~ = X 3 i=1 ∂i(xi − x 0 i )= 3 = −∇0 · R~ ∇ × R~ = X 3 i,j,k=1 ijk∂i(xj − x 0 j )~ek = X 3 i,j,k=1 ijkδij~ek= 0 = −∇0 × R~ ∇R = X 3 i=1 ~ei∂iR = 1 2R X 3 i,j=1 ~ei∂i [(xj − x 0 j )(xj − x 0 j )] = 1 R X 3 i,j=1 ~ei(xj − x 0 j )∂i(xj − x 0 j ) = 1 R X 3 i,j=1 ~ei(xj − x 0 j )δij = 1 R X 3 i=1 ~ei(xi − x 0 i )= R~ R = −∇0R
场论:场的积分性质 梯度的线积分: V∑,画dx∑/,d司 32/384 B B ∑人x=人dx=人da Vo的意义: △1.V 0A或 Vol cos 8 B-A方向导数△p Vo的大小为沿各方向的导数中最大的导数值,方向沿具有最大‖4 导数的方向。任意方向的方向导数即是Va在此方向上的度影 Vo的方向是等值面的简线方向 ose
32/384 JJ II J I Back Close ⑤Ø: ⑤✛➮➞✺➓ ❋Ý✛❶➮➞: Z B A d~l · ∇φ = X 3 i,j=1 Z B A ~eidxi · ∂jφ~ej = X 3 i,j=1 Z B A dxi∂jφ~ei · ~ej = X 3 i,j=1 Z B A dxi∂jφδij = X 3 i=1 Z B A dxi∂iφ = Z B A dφ = φB − φA ✛ ✻✲ ❄ ❅■❅✒ ✠❅❅❘ s ∇φ✛➾➶➭ ∆~l · ∇φ = φB − φA ➼ |∇φ| cos θ = φB − φA |∆~l| ➄➉✓ê ==== ∆φ |∆~l| ∇φ✛➀✂➃φ÷❼➄➉✛✓ê➙⑩➀✛✓ê❾➜➄➉÷ä❦ ⑩➀ ✓ê✛➄➉✧❄➾➄➉✛➄➉✓ê❂➫ ∇φ ✸❞➄➉þ✛Ý❑✧ ∇φ✛➄➉➫✤❾→✛④❶➄➉➞
散度的体积.: 场不:场的积分性质 体积中的体积/drV,f=ds,f体积,表面的面积 33/384 选z方向求证,T表面.为量下两表面SB~zB(x,y)和SA~zA(x,y) 4x, dr a-f(x, y, z) (x ZB(x,y) dxdy dz fs(x,y, z dxdy(f(x, y, ZB(x, y)-f3(x, y, ZA(,y) fs (x, y, ZB(X,y ))-/dS f(x, y, ZA(x, y)) p ds3 f3(x,y,z 公式只适用于体积中无孔洞的情况 ose
33/384 JJ II J I Back Close ⑤Ø: ⑤✛➮➞✺➓ ÑÝ✛◆➮➞: ◆➮ τ ➙✛◆➮➞ Z τ dτ ∇ · ~f = I τ d~S · ~f ◆➮ τ ▲→✛→➮➞ ➚z➄➉➛②,τ▲→➞➃þ❡ü▲→:SB ∼ zB(x, y)ÚSA ∼ zA(x, y) Z τ dτ ∂ ∂z f3(x, y, z) = Z dxdy Z zB(x,y) zA(x,y) dz ∂ ∂z f3(x, y, z) = Z dxdy[f3(x, y, zB(x, y)) − f3(x, y, zA(x, y))] = Z SB dS3f3(x, y, zB(x, y)) − Z SA dS3f3(x, y, zA(x, y)) = I τ dS3 f3(x, y, z) ú➟➄➲❫✉◆➮➙➹➎➱✛➐➵