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1 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 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 y x A B yx

2 : frds frds A B B A r B frds r A frds : frds frds A B B A : frds + A 1 A 2 frds A 2 A 3 frds. A 1 A : frds A 1 A 2 A 3 A 4 A 1 frds A 1 A 2 A 3 frds. A 1 A 4 A s t frds t rt t fr ds dt dt ds 2 dx 2 +dy 2 +dz 2, ds 2 dx 2 dy 2 dz v dt dt dt dt ds/dt v frds frvtdt 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 : fr dr f x x, y, zdx + f y x, y, zdy + f z x, y, zdz fr ˆtds frt dr dt. dt r t zx, y z z dr z z dx + dy dz zr dr dz zr B zr A φr φr : φ φ φ φr dr dx + dy + z dz dφ φr B φr A y fx y 0 : b fxdx lim fx i Δx a Δx 0 2 z fx, y z 0 4 : fx, ydxdy lim ΔxΔy 0 fx i,y i ΔxΔy

4 2 : b φ2 x fx, ydxdy fx, ydy dx a φ 1 x y φx : b φ2 x dxdy fx, y dx dy fx, y 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 a a 2 x 2 dy y y a 2 x 2 sin θ a V 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 a dxdy x y 0<x<1,0<y<a 1 a dx 1 dy e y log x dx xa log x a 1 a dy dx x y 1 dy y +1 loga x y u v 1 fxdx fxu dxu du du

5 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 dv u ux, y v vx, y x xu, v y yu, v ΔxΔy ΔuΔv fdudv 2 2 ΔxΔy rδφδr fx, ydxdy fr, φ rdrdφ fx, ydxdy x, y fxu, v,yu, v dudv u, v x, y/u, v Jacobian : x, y J u, v u u v v

6 3 2 : fx, y, zdxdydz lim fx i,y i,z i ΔxΔyΔz ΔxΔyΔz 0 3 : b φ2 x ψ2 x,y dxdydzfx, y, z dx dy dzfx, y, z 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 J x, y, z u, v, w u v w u v w z u z v z w 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 [ ] 3.3

7 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 z 2 dxdy 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 z A x z A y + A z z 2 + z 2 dxdy dxdy vr ˆnd current ρrvr ˆnd Er Er ˆnd Br Br ˆnd 3.3.6

8 jδxδyδz ΔV ΔxΔyΔz V Gauß V A dv A ˆnd j A Ax, y + 12 Δy, z + 12 Δz ˆxΔyΔz +Ax +Δx, y + 12 Δy, z + 12 Δz ˆxΔyΔz + Ax Δx, y, z Δz ŷδzδx + Ax Δx, y +Δy, z 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 z Ax + A y + A z dxdydz A x ˆx ˆn + A y ŷ ˆn + A z ẑ ˆn d 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 dxdy ẑ ˆnd dxdy ẑ ˆnd [ ]

9 1. ρ R r 0 r 2. a l r a r l A z ΔxΔy Δ ΔxΔy A tokes A ˆnd A ˆtds : Ar dr Ar Δx ˆx Δx ˆx + Ar 0 +Δx ˆx + 1 Δy ŷ Δy ŷ 2 small loop +Ar Δx ˆx +Δyŷ Δx ˆx+Ar Δy ŷ Δy ŷ 2 Ay A x ΔxΔy A ẑδxδy A ˆnΔ 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

10 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 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 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 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 , 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

11 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 ux, y dxdy ux, ydx + vx, ydy 3.5.7

12 1. 2. fx +Δx, y +Δy n0 Δz 2 f Δx Δy 2 2 f λ 1 Δx λ 2 Δx Δx n! +Δy n fx, y 2 f 2 f 2 Δx Δy ˆn f, 1, f f 2 f 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

13 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 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 f 1 ρ f ρ ρ ρ fx, ydxdy x, y u, v V u u A dv + 1 ρ 2 2 f φ f z 2 x, y fxu, v,yu, v u, v dudv v v A ˆnd. A ˆnd A ˆtds

δ ij δ ij ˆx ˆx ŷ ŷ ẑ ẑ 0, ˆx ŷ ŷ ˆx ẑ, ŷ ẑ ẑ ŷ ẑ, ẑ ˆx ˆx ẑ ŷ, a b a x ˆx + a y ŷ + a z ẑ b x ˆx + b

δ ij δ ij ˆx ˆx ŷ ŷ ẑ ẑ 0, ˆx ŷ ŷ ˆx ẑ, ŷ ẑ ẑ ŷ ẑ, ẑ ˆx ˆx ẑ ŷ, a b a x ˆx + a y ŷ + a z ẑ b x ˆx + b 23 2 2.1 n n r x, y, z ˆx ŷ ẑ 1 a a x ˆx + a y ŷ + a z ẑ 2.1.1 3 a iˆx i. 2.1.2 i1 i j k e x e y e z 3 a b a i b i i 1, 2, 3 x y z ˆx i ˆx j δ ij, 2.1.3 n a b a i b i a i b i a x b x + a y b y + a z b

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