39 3 3.1 2 Ax 1,y 1 Bx 2,y 2 x y fx, y z fx, y x 1,y 1, 0 x 1,y 1,fx 1,y 1 x 2,y 2, 0 x 2,y 2,fx 2,y 2 A s I fx, yds lim fx i,y i Δs. 3.1.1 Δs 0 x i,y i N Δs 1 I lim Δx 2 +Δy 2 0 x 1 fx i,y i Δx i 2 +Δy i 2 2 Δyi lim fx i,y i 1+ Δx i Δx 0 Δx i x2 fx, yx 1+y x 2 dx. 3.1.2 y x A B yx 3 4 4 1
: frds frds A B B A r B frds r A frds. 3.1.3 : frds frds. 3.1.4 A B B A : frds + A 1 A 2 frds A 2 A 3 frds. A 1 A 3 3.1.5 : frds A 1 A 2 A 3 A 4 A 1 frds A 1 A 2 A 3 frds. A 1 A 4 A 3 3.1.6 s t frds t rt t fr ds dt. 3.1.7 dt ds 2 dx 2 +dy 2 +dz 2, 3.1.8 ds 2 dx 2 dy 2 dz 2 + + v 2. 3.1.9 dt dt dt dt ds/dt v frds frvtdt. 3.1.10 U fr dr U lim fx i,y i,z i Δr i Δr 0 lim f x x i,y i,z i Δx i + f y x i,y i,z i Δy i + f z x i,y i,z i Δz i Δr 0 fr dr. 3.1.11
: fr dr f x x, y, zdx + f y x, y, zdy + f z x, y, zdz fr ˆtds frt dr dt. dt 3.1.12 r t zx, y z z dr z z dx + dy dz 3.1.13 zr dr dz zr B zr A 3.1.14 φr φr : φ φ φ φr dr dx + dy + z dz dφ φr B φr A. 3.1.15 3.2 2 y fx y 0 : b fxdx lim fx i Δx. 3.2.1 a Δx 0 2 z fx, y z 0 4 : fx, ydxdy lim ΔxΔy 0 fx i,y i ΔxΔy. 3.2.2
2 : b φ2 x fx, ydxdy fx, ydy dx. 3.2.3 a φ 1 x y φx : b φ2 x dxdy fx, y dx dy fx, y. 3.2.4 a φ 1 x a x-y a V 2 dxdy a 2 x 2 y 2 x 2 +y 2 a 2 a a 2 x 2 2 dx a 2 x 2 y 2. 3.2.5 a a 2 x 2 dy y y a 2 x 2 sin θ a V 2 2 2 a a a a a π/2 dx dx π/2 π/2 π/2 dθ a 2 x 2 cos θ a 2 x 2 1 sin 2 θ dθa 2 x 2 cos 2 θ dxa 2 x 2 π 2 4π 3 a3. 3.2.6 2 dxdy x y 0<x<1,0<y<a 1 a dx 1 dy e y log x dx xa 1 0 0 0 log x a 1 a dy dx x y 1 dy 0 0 0 y +1 loga +1. 3.2.7 1 x y u v 1 fxdx fxu dxu du 3.2.8 du
2.1: z fx, y 2 2 u ux v vy fx, ydxdy u2 u 1 du dxu du v2 v 1 dv dyv fxu,yv 3.2.9 dv u ux, y v vx, y x xu, v y yu, v 3.3.1 ΔxΔy ΔuΔv fdudv 2 2 ΔxΔy rδφδr fx, ydxdy fr, φ rdrdφ 3.2.10 fx, ydxdy x, y fxu, v,yu, v dudv 3.2.11 u, v x, y/u, v Jacobian : x, y J u, v u u. 3.2.12 v v
3 2 : fx, y, zdxdydz lim fx i,y i,z i ΔxΔyΔz. 3.2.13 ΔxΔyΔz 0 3 : b φ2 x ψ2 x,y dxdydzfx, y, z dx dy dzfx, y, z. 3.2.14 a φ 1 x ψ 1 x,y z ψx, y y φx x, y, z fx, y, zdxdydz fxu, v, w,yu, v, w,zu, v, w u, v, w dudvdw. 3.2.15 J x, y, z u, v, w u v w u v w z u z v z w 3.2.16 r x, y, z fr fρ, φ, z fr, θ, φ ΔρρΔφΔz ΔrrΔθrsin θδφ fx, y, zdxdydz fρ, φ, zρ dρdφdz fr, θ, φr 2 sin θ drdθdφ. 3.2.17 [ ] 3.3
1 x 2 xy : fx, y, zd lim fx i,y i,z i Δ Δ 0 lim fx i,y i,z i x i,y i ΔxΔy 1 ΔxΔy 0 ˆn ẑ z 2 fx, y, z 1+ + ˆn 1.3.19 z 2 dxdy 3.3.1 Ar ˆnd lim Ax i,y i,z i ˆnΔ Δ 0 lim Ax i,y i,z i x i,y i ˆn ΔxΔy ΔxΔy 0 ˆn ẑ z, z, 1 A x,a y,a z z 2 1+ z 2 + +1 z A x z A y + A z z 2 + z 2 dxdy dxdy. 3.3.2 vr ˆnd 3.3.3 current ρrvr ˆnd 3.3.4 Er Er ˆnd 3.3.5 Br Br ˆnd 3.3.6
3.4 2.3 jδxδyδz ΔV ΔxΔyΔz V Gauß V A dv A ˆnd 3.4.1 2.3.13 j A Ax, y + 12 Δy, z + 12 Δz ˆxΔyΔz +Ax +Δx, y + 12 Δy, z + 12 Δz ˆxΔyΔz + Ax + 1 2 Δx, y, z + 1 2 Δz ŷδzδx + Ax + 1 2 Δx, y +Δy, z + 1 2 Δz ŷδzδx + Ax + 12 Δx, y + 12 Δy, z ẑδxδy + Ax + 12 Δx, y + 12 Δy, z +Δz ẑδxδy Ax x, y, z + A yx, y, z + A zx, y, z ΔxΔyΔz. 3.4.2 z 3.4.1 Ax + A y + A z dxdydz A x ˆx ˆn + A y ŷ ˆn + A z ẑ ˆn d.3.4.3 z V 3 V z ψ 2 x, y z ψ 1 x, y xy ψ2 x,y dxdy dz A z ψ 1 x,y z dxdy A z x, y, ψ 2 x, y A z x, y, ψ 1 x, y A z ẑ ˆnd. 3.4.4 dxdy ẑ ˆnd dxdy ẑ ˆnd [ ]
1. ρ R r 0 r 2. a l r a r l 3.5 2.3 A z ΔxΔy Δ ΔxΔy A tokes A ˆnd A ˆtds 3.5.1 2.3 2.3.32 : Ar dr Ar 0 + 1 2 Δx ˆx Δx ˆx + Ar 0 +Δx ˆx + 1 Δy ŷ Δy ŷ 2 small loop +Ar 0 + 1 2 Δx ˆx +Δyŷ Δx ˆx+Ar 0 + 1 Δy ŷ Δy ŷ 2 Ay A x ΔxΔy A ẑδxδy A ˆnΔ. 3.5.2 3.3.2 A A A ˆnd z A x z A y + A z dxdy [ z Az A y z Ax z z A z Ay + A ] x dxdy. 3.5.3
A x z ψx, y 2.1 φ fx, y ψx, y. dx dy A x x, y, z A xx, y, z dx dy A xx, y, ψx, y z b dx [ A x x, φ 2 x,ψx, φ 2 x + A x x, φ 1 x,ψx, φ 1 x]. a 3.5.4 3.5.1 A x b b A x x, y, zdx A x x, φ 1 x,ψx, φ 1 xdx A x x, φ 2 x,ψx, φ 2 xdx a a 3.5.4 A y A z dx dy z A z x, y, z z A z x, y, z dx dy z,a z x, y dz da z 3.5.5 z,a z z A z z A z x y [ ] v 1 v 2 v 3 3 I 1 v ˆnd, I 2 v ˆnd, I 3 v ˆtds 1 2 1 0, 0, 0 a z a 2 x 2 y 2 2 xy 1 2 z x 2 + y 2 δ 2,δ 0 v 1 ωy, ωx, 0 ωy v 2 x 2 + y, ωx 2 x 2 + y, 0 2 v 3 ωy x 2 + y 2, ωx x 2 + y 2, 0
z 0 2 A dxdy A x A y ux, y vx, y vx, y Ay A x dxdy A x dx + A y dy 3.5.6 ux, y dxdy ux, ydx + vx, ydy 3.5.7
1. 2. fx +Δx, y +Δy n0 Δz 2 f Δx Δy 2 2 f λ 1 Δx 1 2 + λ 2 Δx 2 2 1 Δx n! +Δy n fx, y 2 f 2 f 2 Δx Δy 3. 4. ˆn f, 1, f f 2 f + 2 +1 ɛ ijk ɛ lmk ɛ ijk ɛ klm ɛ ijk ɛ mkl δ il δ jm δ im δ jl. 5. Ar ˆx ŷ ẑ z A x A y A z. 6. φψ φψ + φ ψ, φa φ A + φ A, φa φ A + φ A, A B B A A B, A B B A B A A B + A B, A B B A +A B + B A+A B, φ 0, A 0, A A 2 A. 7. f f r ˆr + 1 f r θ ˆθ + 1 f r sin θ φ ˆφ
8. 9. 10. A [ A 1 r sin θ [ 1 r 2 r 2 sin θa r + sin θ r θ r sin θa θ+ ] φ ra φ. θ sin θa φ 1 r sin θ [ 2 1 f r 2 r 2 sin θ f + sin θ r r θ ] A θ ˆr + φ [ 1 A r r sin θ φ 1 ] r r ra φ ˆθ [ 1 + r r ra θ 1 ] A r ˆφ. r θ sin θ f θ + 1 φ sin θ ] f. φ 11. 12. A 1 ρ f f ρ ˆρ + 1 f ρ φ ˆφ + f z ẑ [ ρ ρa ρ+ A φ φ + ] z ρa z. 13. A 1 A z ρ φ A φ Aρ ˆρ + z z A z 1 ˆφ + ρ ρ ρ ρa φ 1 ρ A ρ ẑ φ 14. 15. 16. 17. 2 f 1 ρ f ρ ρ ρ fx, ydxdy x, y u, v V u u A dv + 1 ρ 2 2 f φ 2 + 2 f z 2 x, y fxu, v,yu, v u, v dudv v v A ˆnd. A ˆnd A ˆtds