M ⊂ R n M F : M → R n ,F(x)=(F 1 (x), ··· ,F n (x)) F (x) x C ⊂ R 3 τ τ : C → R 3 C τ(x) C x x ∈ C ✿ ✛ ❍ ❍ ❍ ❍ ❍❨ t τ (x) x C ϕ(t)=(cost, sin t),t ∈ (0, 2π) ϕ  (t)=(−sin t, cos t) S ⊂ R 3 S S N : S → R 3 N(x) ⊥ T x S, ∀x ∈ S S N s x N (x) ❇ ❇ ❇ ❇ ❇▼ ✲✒     S ϕ(φ, θ)=(cosφ sin θ, sin φ sin θ, cos θ), (φ, θ) ∈ (0, 2π) × (0,π). ∂ϕ ∂φ =(−sin φ sin θ,cos φ sin θ, 0), ∂ϕ ∂θ =(−cos φ cos θ, sin φ cos θ, −sin θ) N = ∂ϕ ∂φ × ∂ϕ ∂θ R R 2 V k R (v 1 , ··· ,v k ) (w 1 , ··· ,w k ) V P =(p ij ) k×k w j =  i p ij v i (v 1 , ··· ,v k ) (w 1 , ··· ,w k ) det P>0 (v 1 , ··· ,v k ) (w 1 , ··· ,w k ) det P<0 V (v 1 , ··· ,v k ) [v 1 , ··· ,v k ] −[v 1 , ··· ,v k ] V µ µ =[v 1 , ··· ,v k ] R k R R 2 R 3 ✲ e 1 ✲ e 1 ✻ e 2 ✩ ✛ ✲ e 1 ✻ e 3   ✒ e 2 ✛ R 1 , R 2 , R 3 M ⊂ R n k µ = {µ x : µ x T x M,x ∈ M} a ∈ M (ϕ, U) a [D 1 ϕ(u), ··· ,D k ϕ(u)] = µ ϕ(u) u ∈ U M M M µ M µ µ R 3 N = D 1 ϕ×D 2 ϕ M ∂M M ∂M O M µ (ϕ, U) ∈O i : R k−1 → R k ,i(u 1 , ··· ,u k−1 )=(u 1 , ··· ,u k−1 , 0) {(ϕ ◦ i, i −1 (U)) : (ϕ, U) ∈O,U  H k = ∅} ∂M x ∈ ∂M (ϕ, U) ∈O x  x =[D 1 ϕ(u), ··· ,D k−1 ϕ(u)],x= ϕ(u).  x (ϕ, U) ∈O ∂M  = { x : x = ϕ(u) ∈ ∂M,(ϕ, U) ∈O} ∂M (ϕ, U), (ψ, W) ∈O x ψ = ϕ ◦h det h  > 0 k h h k (w 1 , ··· ,w k−1 , 0) = 0, va h k (w 1 , ··· ,w k−1 ,w k ) > 0khiw k > 0. w =(w 1 , ··· ,w k−1 , 0) h  (w) (D 1 h k (w)=0 ··· D k−1 h k (w)=0 D k h k (w) > 0). det h  (w)=det(h ◦ i)  (w 1 , ··· ,w k−1 )D k h k (w) > 0 det(h ◦ i)  (w 1 , ··· ,w k−1 ) > 0 (h ◦ i)  (w) D 1 ϕ(u), ··· ,D k−1 ϕ(u) D 1 ψ(w), ··· ,D k−1 ψ(w) T x ∂M (x = ψ(w)=ϕ(u)) [D 1 ψ(w), ··· ,D k−1 ψ(w)] = [D 1 ϕ(u), ··· ,D k−1 ϕ(u)].  x µ x M µ ∂M ∂µ x ∈ ∂M (ϕ, U) x M µ µ x =[D 1 ϕ(u), ··· ,D k ϕ(u)] ∂µ x =(−1) k [D 1 ϕ(u), ··· ,D k−1 ϕ(u)]. (−1) k ϕ µ x = ϕ(u) T x ∂M T x M v ∈ T x M \T x ∂M v M v ∈ ϕ  (u)(H k + ) v M ∂M v 1 , ··· ,v k−1 T x ∂M v ∈ T x M M µ =[v 1 , ··· ,v k−1 ,v] ∂µ x =(−1) k [v 1 , ··· ,v k−1 ] s x ✲   ✒ v ✒ ✛ ✠         H k ∂H k = R k−1 × 0 R k−1 k k M R 2 R 3 N ∂M M R 3 ∂M F =(F 1 ,F 2 ,F 3 ) R 3 • v ∈ R 3 x W F (x)(v)=<F(x),v > F (x) v W F = F 1 dx 1 + F 2 dx 2 + F 3 dx 3 C R 3 F C W F C  C W F =  C F 1 dx 1 + F 2 dx 2 + F 3 dx 3 . • v 1 ,v 2 ∈ R 3 x ω F (x)(v 1 ,v 2 )=<F(x),v 1 × v 2 > F (x) ∆S v 1 ,v 2 ω F = F 1 dx 2 ∧ dx 3 + F 2 dx 3 ∧ dx 1 + F 3 dx 1 ∧ dx 2 . S R 3 F S ω F S  S ω F =  S F 1 dx 2 ∧ dx 3 + F 2 dx 3 ∧ dx 1 + F 3 dx 1 ∧ dx 2 U R k ω ∈ Ω k (U) ω = f(u)du 1 ∧···∧du k  U ω =  U f(u)du 1 ∧···∧du k =  U f(u)du 1 ···du k . M k µ R n ω ∈ Ω k (V ) V M ω M  M ω M = ϕ(U) (ϕ, U) µ  M ω =  U ϕ ∗ ω. M O = {(ϕ i ,U i ):i ∈ I} µ Θ={θ i : i ∈ I} M O  M ω =  i∈I  ϕ i (U i ) θ i ω  =  i∈I  U i ϕ ∗ i (θ i ω)  , M ω k =1  M  i F i dx i k =2  M  i<j F ij dx i ∧ dx j µ (ϕ, U) (ψ, W ) µ ψ = ϕ ◦ h h det Jh > 0 ϕ ∗ ω = f(u)du 1 ∧···∧du k h ∗ (f(u)du 1 ∧···∧du k )=h ∗ ϕ ∗ ω =(ϕ ◦ h) ∗ ω = ψ ∗ ω.  U ϕ ∗ ω =  U f =  W f ◦◦h det Jh =  W h ∗ (f(u)du 1 ∧···∧du k )=  W ψ ∗ ω. Θ  = {θ  j : j ∈ J} M  j  M θ  j ω =  j  M (  i θ i )θ  j ω =  i,j  M θ i θ  j ω =  i,j  M θ  j θ i ω =  i  M (  j θ  j )θ i ω  i  M θ i ω. M k µ V  M :Ω k (V ) → R  M ω = −  −M ω −M M −µ  U i ϕ ∗ i h(u 1 , ··· ,u k )=(−u 1 , ··· ,u k ) det h  = −1 (ϕ, U) µ (ϕ ◦h, h −1 (U)) −µ Θ  −M ω =  θ∈Θ  h −1 (U) (ϕ ◦ h) ∗ θω =  θ∈Θ (−  U ϕ ∗ θω)=−  M ω. C ϕ : I → R n  C  i F i dx i =  I  i F i ◦ ϕdϕ i =  I (  i F i ◦ ϕ(t)ϕ  i (t))dt.  x 2 +y 2 =1 ydx − xdy x 2 + y 2 =  2π 0 sin td(cos t) − cos td(sin t) cos 2 t +sin 2 t = −  2π 0 dt = −2π. S  S xdy ∧dz =  [0,2π]×[0,π] cos φ sin θd(sin φ sin θ) ∧ d(cos θ) =  [0,2π]×[0,π] cos φ sin θ(cos φ sin θdφ +sinφ cos θdθ) ∧ d(−sin θdθ) =  [0,2π]×[0,π] −cos 2 φ sin 3 θdφ ∧ dθ =? F =(P, Q,R) C 1 V ⊂ R 3 C ⊂ V T =(cosα, cos β,cos γ)  C Pdx+ Qdy + Rdz =  C <F,T >dl=  C (P cos α + Q cos β + R cos γ)dl. S ⊂ V N = (cos α, cos β,cos γ)  S Pdy∧dz+Qdz∧dx+Rdx∧dy =  S <F,N>dS=  S (P cos α+Q cos β+R cos γ)dS. v ∈ R 3 T v W F (v)=<F,v> W F = Pdx+ Qdy + Rdz. W F (v)=<F,T >v =<F,T>dl(v). C R 3 T  C W F =  C <F,T >dl. v 1 ,v 2 ∈ R 3 N v 1 × v 2 ω F (v 1 ,v 2 )=<F,v 1 × v 2 > ω F = Pdy ∧ dz + Qdz ∧dx + Rdx ∧ dy. ω F (v 1 ,v 2 )=<F,N>v 1 × v 2  =<F,N>dS(v 1 ,v 2 ) S N  S ω F =  <F,N>dS. M k V ⊂ R n ∂M  M dω =  ∂M ω, ∀ω ∈ Ω k−1 (V ). M µ ∂µ ∂M {(ϕ i ,U i ):i ∈ I} µ M U i A i : R k−1 → R k , (u 1 , ··· ,u k−1 )=(u 1 , ··· ,u k−1 , 0) {(ϕ i ◦ , −1 (U i )) : i ∈ I  } I  = {i ∈ I : U i ∩∂H k = ∅} ∂M (−1) k ∂µ {θ i : i ∈ I}  M dω =  M d(  i∈I θ i ω)=  i∈I  ϕ i (U i ∩H k ) dθ i ω.  ∂M ω =  ∂M (  i∈I  θ i ω)=  i∈I   ϕ i (U i ∩∂H k ) θ i ω. ϕ = ϕ i ,U = U i ,A = A i =[α 1 ,β 1 ] ×···×[α k ,β k ] U ∩ ∂H k = ∅ i ∈ I \ I   ϕ(U) dω =0 U ∩ ∂H k = ∅ i ∈ I   ϕ(U∩H k ) dω =(−1) k  ϕ(U∩∂H k ) ω. ϕ ∗ ω = k  j=1 a j (u 1 , ··· ,u k )du 1 ∧···∧  du j ∧···∧du k ∈ Ω k−1 (U) ϕ ∗ ω ∈ Ω k−1 (A) a j (u)=0 u ∈ U (ϕ ◦ ) ∗ ω = a k (u 1 , ··· ,u k−1 , 0)du 1 ∧···∧du k−1 . ϕ ∗ (dω)= k  j=1 da j ∧ du 1 ∧···  du j ···∧du k = k  j=1 (−1) j−1 ∂a j ∂u j du 1 ∧···∧du k .  ϕ(U) dω =  U ϕ ∗ (dω)=  A k  j=1 (−1) j−1 ∂a j ∂u j du 1 ∧···∧du k =  j   l=j [α l ,β l ] (a j (··· ,β j , ···) − a j (··· ,α j , ···))du 1 ···  du j ···du k =0. (u 1 , ··· ,β j , ··· ,u k ) (u 1 , ··· ,α j , ··· ,u k ) ∈ U a j  ϕ(U∩H k ) dω =  U∩H k k  j=1 (−1) j−1 ∂a j ∂u j du 1 ∧···∧du k =  A∩H k k  j=1 (−1) j−1 ∂a j ∂u j du 1 ∧···∧du k =  j (−1) j−1 (  [α 1 ,β 1 ]×···×[0,β k ] ∂a j ∂u j du 1 ∧···∧du k ). j = k,  [α j ,β j ] ∂a j ∂u j du j = a j (u 1 , ··· ,β j , ··· ,u k ) − a j (u 1 , ··· ,α j , ··· ,u k )=0 j = k,  [0,β k ] ∂a k ∂u k du k = a k (u 1 , ··· ,β k ) − a k (u 1 , ··· , 0) = −a k (u 1 , ··· , 0)  ϕ(U∩H k ) dω =(−1) k   j=k [α j ,β j ] a k (u 1 , ··· , 0)du 1 ···du k−1 .  ϕ(U∩∂H k ) ω =  A∩R k−1 ×0 a k (u 1 , ··· , 0)du 1 ···du k−1 . M M R ω(x)=xdx V R n F : V → R C 1 ϕ :[a, b] → V  ϕ([a,b]) dF = F (ϕ(b)) − F (ϕ(a)). D ⊂ R 2 C = ∂D P, Q C 1 D  D ( ∂Q ∂x − ∂P ∂y )dxdy =  C Pdx+ Qdy. S ⊂ R 3 N ∂S = C P, Q,R C 1 S  S ( ∂Q ∂x − ∂P ∂y )dx∧dy+( ∂R ∂y − ∂Q ∂z )dy∧dz+( ∂P ∂z − ∂R ∂x )dz∧dx =  C Pdx+Qdy +Rdz. V ⊂ R 3 ∂V = S P, Q,R C 1 V  V ( ∂P ∂x + ∂Q ∂y + ∂R ∂z )dxdydz =  S Pdy ∧ dz + Qdz ∧dx + Rdx ∧ dy. D C R 2  D dxdy =  C xdy = −  C ydx = 1 2  C (xdy −ydx). V S R 3  V dxdydz =  S xdy ∧dz =  S ydz ∧ dx =  S zdx ∧dy = 1 3 (  S xdy ∧dz +  S ydz ∧ dx +  S zdx ∧dy) U R n ω = n  i=1 a i dx i ∈ Ω 1 (U) ω f ∈ C 1 (U) df = ω ω dω =0 ∂a i ∂x i = ∂a i ∂x j i, j  C ω =0 C ⊂ U R 2 \{0} xdy −ydx x 2 + y 2 2π =0 R n \{0} n  i=1 (−1) i x i x n/2 dx 1 ∧···  dx i ···∧dx n .  dx i dx i R 3 e 1 ,e 2 ,e 3 U R 3 ∇ = ∂ ∂x 1 e 1 + ∂ ∂x 2 e 2 + ∂ ∂x 3 e 3 f : U → R f grad f = ∇f = ∂f ∂x 1 e 1 + ∂f ∂x 2 e 2 + ∂f ∂x 3 e 3 . F = F 1 e 1 + F 2 e 2 + F 3 e 3 U F rot F = ∇×F =         e 1 e 2 e 3 ∂ ∂x 1 ∂ ∂x 2 ∂ ∂x 3 F 1 F 2 F 3         F div F =< ∇,F >= ∂F 1 ∂x 1 + ∂F 2 ∂x 2 + ∂F 3 ∂x 3 . h 1 : X(U) → Ω 1 (U),h 2 (F 1 e 1 + F 2 e 2 + F 3 e 3 )=F 1 dx 1 + F 2 dx 2 + F 3 dx 3 . h 2 : X(U) → Ω 2 (U),h 2 (F 1 e 1 +F 2 e 2 +F 3 e 3 )=F 1 dx 2 ∧dx 3 +F 2 dx 3 ∧dx 1 +F 3 dx 1 ∧dx 2 . h 3 : C ∞ (U) → Ω 3 (U),h 3 (f)=fdx 1 ∧ dx 2 ∧ dx 3 . . F 3 e 3 )=F 1 dx 1 + F 2 dx 2 + F 3 dx 3 . h 2 : X(U) → Ω 2 (U),h 2 (F 1 e 1 +F 2 e 2 +F 3 e 3 )=F 1 dx 2 ∧dx 3 +F 2 dx 3 ∧dx 1 +F 3 dx 1 ∧dx 2 . h 3 : C ∞ (U) → Ω 3 (U),h 3 (f)=fdx 1 ∧ dx 2 ∧ dx 3 . . R 2 R 3 N ∂M M R 3 ∂M F =(F 1 ,F 2 ,F 3 ) R 3 • v ∈ R 3 x W F (x)(v)=<F(x),v > F (x) v W F = F 1 dx 1 + F 2 dx 2 + F 3 dx 3 C R 3 F C W F C  C W F =  C F 1 dx 1 + F 2 dx 2 + F 3 dx 3 . •. =0 R n {0} n  i=1 (−1) i x i x n/2 dx 1 ∧···  dx i ···∧dx n .  dx i dx i R 3 e 1 ,e 2 ,e 3 U R 3 ∇ = ∂ ∂x 1 e 1 + ∂ ∂x 2 e 2 + ∂ ∂x 3 e 3 f : U → R f grad f = ∇f = ∂f ∂x 1 e 1 + ∂f ∂x 2 e 2 + ∂f ∂x 3 e 3 . F = F 1 e 1 + F 2 e 2 + F 3 e 3 U F rot F = ∇×F