曲线坐标:柱坐标 坐标元:r=√x2+y2,b= arctan,z动矢:可,e,囝 er= cos Bex sin e sin be+cos Be 39/384 dede de deg de, de, a Or 00 r 010 微分算符:V=e1x+e ahaz 矢上场:f=fre1+fe+f2E2 标上场的梯度:Vb=6br+er+eaz 矢上场的散度:V·f= 01 rBEO2/·(f,n+fen+E司2) Of 1ofo Of,- 1 der de Ofr 1 ofg Of f Or r ae az f→ 2 Or2 r002 az2 r Or ose
39/384 JJ II J I Back Close ➢❶❿■: ❰❿■ ❿■✄➭ r = p x2 + y2 , θ = arctan y x , z ➘➙➭ ~er, ~eθ , ~ez ~er = cos θ~ex + sin θ~ey ~eθ = − sin θ~ex + cos θ~ey ∂~er ∂r =0 ∂~er ∂θ =~eθ ∂~er ∂z = ∂~eθ ∂r =0 ∂~eθ ∂θ =−~er ∂~eθ ∂z = ∂~ez ∂r = ∂~ez ∂θ = ∂~ez ∂z =0 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) = ∂fr ∂r + 1 r ∂fθ ∂θ + ∂fz ∂z +~eθ 1 r · (fr ∂~er ∂θ +fθ ∂~eθ ∂θ ) = ∂fr ∂r + 1 r ∂fθ ∂θ + ∂fz ∂z + fr r ~f → ∇~ ⇒ ∇2 = ∇ · ~ ∇~ = ∂ 2 ∂r 2 + 1 r ∂ 2 ∂θ2 + ∂ 2 ∂z 2 + 1 r ∂ ∂r
曲线坐标:柱坐标 de deg de, de, de Or 40/384 010 021028210 微分算符:V=e1+6-+2 ra02 az2 r or 标量场的梯度:V=e+1000 中量场的散庭:F=11mo,t Or r ae az r 中量场的旋度:V×f=同+e+2。]×(Fen+fe+f2e) of of of of, of of a0 ra0 az afg Of fo,_Of Of,- Of, afe Or roe ose
40/384 JJ II J I Back Close ➢❶❿■: ❰❿■ ∂~er ∂r =0 ∂~er ∂θ =~eθ ∂~er ∂z = ∂~eθ ∂r =0 ∂~eθ ∂θ =−~er ∂~eθ ∂z = ∂~ez ∂r = ∂~ez ∂θ = ∂~ez ∂z =0 ❻➞➂❰➭∇ = ~er ∂ ∂r +~eθ 1 r ∂ ∂θ +~ez ∂ ∂z ∇2= ∂ 2 ∂r 2 + 1 r ∂ 2 ∂θ2 + ∂ 2 ∂z 2 + 1 r ∂ ∂r ■þ⑤✛❋Ý➭∇φ = ~er ∂φ ∂r +~eθ 1 r ∂φ ∂θ +~ez ∂φ ∂z ➙þ⑤✛ÑÝ➭∇·~f = ∂fr ∂r + 1 r ∂fθ ∂θ + ∂fz ∂z + fr r ➙þ⑤✛❫Ý➭∇ ×~f = [~er ∂ ∂r +~eθ 1 r ∂ ∂θ +~ez ∂ ∂z ] × (fr~er+fθ~eθ+fz~ez) = ~ez ∂fθ ∂r −~eθ ∂fz ∂r −~ez ∂fr r∂θ +~er ∂fz r∂θ +~eθ ∂fr ∂z −~er ∂fθ ∂z + fr r ~eθ × ∂~er ∂θ + fθ r ~eθ × ∂~eθ ∂θ =~ez[ ∂fθ ∂r − ∂fr r∂θ + fθ r ]+~eθ [ ∂fr ∂z − ∂fz ∂r ]+~er[ ∂fz r∂θ− ∂fθ ∂z ]
曲线坐标:矢坐标 坐标元:r=Vx2+y2+z2,b= arctan(√x2+y2/z),p= arctan(y/x) 基矢:E,E,创x,y平面基矢:E cos ex+sin ge, sin gex+cos pey 41/384 ex-cos ep-sin pee ey=sin peet cos oeo e,= sin g er.= 0 ere=0 er ex-cos per e,- cos o sin 0 er. e=sin e. e= sinsin 0 ea·en=cos6 ea=- sin 0 ee·e=0 eoexcos oeo e,=cos o cos 0 eo ey- sin oeo ep -sin o cos 0 SIn eo·ey=Coso e,=0 cos o sin g sin o sin cos 6 cos o cos 0 sin o cos 6 -sin 6 COS 0 cos o sin g cos o cos B -sin o sin o sin g sin o cos 6 cos o COS sin e ose
41/384 JJ II J I Back Close ➢❶❿■: ➙❿■ ✠ ✲ ✻ ✒ ❅ ❅❅❘ z θ φ ~r x y ~ρ ✫✪ ✬✩ ✻✒ ❅ ❅ ❅■❅ ✲ ~er −~e ~ez θ ~eρ θ θ ✫✪ ✬✩❡r ✻✒ ❅ ❅ ❅■❅ ✲ ~eρ ~e ~ey φ φ ~ex φ ~ez ❿■✄➭ r= p x2+y2+z 2 , θ=arctan(p x2+y2/z), φ=arctan(y/x) ➘➙➭ ~er, ~eθ , ~eφ x, y➨→➘➙➭ ~eρ ~eρ= cos φ~ex+sin φ~ey ~eφ= − sin φ~ex+cos φ~ey ~ex= cos φ~eρ−sin φ~eφ ~ey = sin φ~eρ+cos φ~eφ ~er · ~eρ = sin θ ~er · ~ez = cos θ ~er · ~eφ = 0 ~er · ~ex = cos φ~er · ~eρ = cos φ sin θ ~er · ~ey = sin φ~er · ~eρ = sin φ sin θ ~eθ · ~eρ = cos θ ~eθ · ~ez = − sin θ ~eθ · ~eφ = 0 ~eθ · ~ex = cos φ~eθ · ~eρ = cos φ cos θ ~eθ · ~ey = sin φ~eθ · ~eρ = sin φ cos θ ~eφ · ~ex = −sinφ ~eφ · ~ey = cos φ ~eφ · ~ez = 0 ~er ~eθ ~eφ = cos φ sin θ sin φ sin θ cos θ cos φ cos θ sin φ cos θ − sin θ − sin φ cos φ 0 ~ex ~ey ~ez ~ex ~ey ~ez = cos φ sin θ cos φ cos θ − sin φ sin φ sin θ sin φ cos θ cos φ cos θ − sin θ 0 ~er ~eθ ~eφ
曲线坐标:矢坐标 cos o sin g sin o sin g cos 6 cos o cos 6 sin o cos 6 -sin B 42/384 sin g Cos o 0 cos o sin g cos o cos 6 -sin o sin o sin g sin o cos 0 cos o os 6 sin e 0 e Or 9g ee Do cos de e sin de -cos bea 沿基中方向的无穷交线叉:dl1=dr,dl2=rdO,dl3= rsin edo 个分算符:V=ea+nO 10 mersin 0 a0 中量场:f=fer+fE+f do 1 标量场的梯度:;Vo=6ar++e rsin 0 ag ose
42/384 JJ II J I Back Close ➢❶❿■: ➙❿■ ✫✪ ✬✩ ✻✒ ❅ ❅ ❅■❅ ✲ ~er −~e ~ez θ ~eρ θ θ ✫✪ ✬✩❡r ✻✒ ❅ ❅ ❅■❅ ✲ ~eρ ~e ~ey φ φ ~ex φ ~ez ~er ~eθ ~eφ = cos φ sin θ sin φ sin θ cos θ cos φ cos θ sin φ cos θ − sin θ − sin φ cos φ 0 ~ex ~ey ~ez ~ex ~ey ~ez = cos φ sin θ cos φ cos θ − sin φ sin φ sin θ sin φ cos θ cos φ cos θ − sin θ 0 ~er ~eθ ~eφ ∂~er ∂r = ∂~eθ ∂r = ∂~eφ ∂r = 0 ∂~er ∂θ = ~eθ ∂~eθ ∂θ = −~er ∂~eφ ∂θ = 0 ∂~er ∂φ = sin θ~eφ ∂~eθ ∂φ = cos θ~eφ ∂~eφ ∂φ = − sin θ~er − cos θ~eθ ÷➘➙➄➉✛➹→✂❶✄➭ dl1 = dr, dl2 = rdθ, dl3 = r sin θdφ ❻➞➂❰➭∇=~er ∂ ∂r +~eθ 1 r ∂ ∂θ +~eφ 1 r sin θ ∂ ∂φ ➙þ⑤➭~f =fr~er+fθ~eθ+fφ~eφ ■þ⑤✛❋Ý➭ ∇φ = ~er ∂φ ∂r +~eθ 1 r ∂φ ∂θ +~eφ 1 r sin θ ∂φ ∂φ
曲线坐标:矢坐标 OrOr a1 0 0 ao cos eeg ab=-sin 0e -cos eee 43/384 ao 微分算符:D=a010 10 rra0 rsin6矢量场:f=f6,+fen+fe 1 ag 10 标量场的梯度:V=en+e0ra+ Prsin e0 10 矢量场的散度:Vf=0+en0+er3nb(Ee+Ee+fe) of. 1 of 1 of de de rsin e ao a0 a0 rsin 0 ao ap of 1a 1 of 2f fo cos 6 or r ae rsin 0 ao r rsin 8 f→V2= 0210210220cos00 ar2+r2002+r2 sin tr2 sin 00 ose
43/384 JJ II J I Back Close ➢❶❿■: ➙❿■ ∂~er ∂r = ∂~eθ ∂r = ∂~eφ ∂r = 0 ∂~er ∂θ = ~eθ ∂~eθ ∂θ = −~er ∂~eφ ∂θ = 0 ∂~er ∂φ = sin θ~eφ ∂~eθ ∂φ = cos θ~eφ ∂~eφ ∂φ = − sin θ~er − cos θ~eθ ❻➞➂❰➭∇=~er ∂ ∂r +~eθ 1 r ∂ ∂θ +~eφ 1 r sin θ ∂ ∂φ ➙þ⑤➭~f =fr~er+fθ~eθ+fφ~eφ ■þ⑤✛❋Ý➭ ∇φ = ~er ∂φ ∂r +~eθ 1 r ∂φ ∂θ +~eφ 1 r sin θ ∂φ ∂φ ➙þ⑤✛ÑÝ➭ ∇·~f =[~er ∂ ∂r +~eθ 1 r ∂ ∂θ +~eφ 1 r sin θ ∂ ∂φ] · (fr~er+fθ~eθ+fφ~eφ) = ∂fr ∂r + 1 r ∂fθ ∂θ + 1 r sin θ ∂fφ ∂φ +~eθ 1 r ·(fr ∂~er ∂θ +fθ ∂~eθ ∂θ )+~eφ 1 r sin θ ·(fr ∂~er ∂φ +fθ ∂~eθ ∂φ ) = ∂fr ∂r + 1 r ∂fθ ∂θ + 1 r sin θ ∂fφ ∂φ + 2fr r + fθ cos θ r sin θ ~f → ∇~ ⇒ ∇2=∇· ~ ∇~ = ∂ 2 ∂r 2 + 1 r 2 ∂ 2 ∂θ2 + 1 r 2 sin2 θ ∂ 2 ∂φ2 + 2 r ∂ ∂r + cos θ r 2 sin θ ∂ ∂θ