R  b a f(x)dx =  β α f(ϕ(t))ϕ  (t)dt ϕ (α, β) (a, b) ω = f(x)dx ϕ ∗ ω = f(ϕ(t))ϕ  (t)dt  b a ω =  β α ϕ ∗ ω k V R k V ω : V ×···×V    k → R v 1 , ··· ,v k ∈ V α ∈ R 1 ≤ i<j≤ k ω(v 1 , ··· ,v i + v  i , ··· ,v k )=ω(v 1 , ··· ,v i , ··· ,v k )+ω(v 1 , ··· ,v  i , ··· ,v k ). ω(v 1 , ··· ,αv i , ··· ,v k )=αω(v 1 , ··· ,v i , ··· ,v k ). ω(v 1 , ··· ,v i , ··· ,v j , ··· ,v k )= − ω(v 1 , ··· ,v j , ··· ,v i , ··· ,v k ). ω ω(v 1 , ··· ,v i ··· ,v j , ··· ,v k )=0 v i = v j i = j ω(v σ(1) , ··· ,v σ(k) )=(σ)ω(v 1 , ··· ,v k ) σ {1, ··· ,k} (σ)  i<j (σ(j) −σ(j)) ⇒ v i = v j 2ω(v 1 , ··· ,v i ··· ,v i , ··· ,v k )=0 ⇒ v i = v j = v + w ω(v 1 , ··· ,v,··· ,w,··· ,v k )+ω(v 1 , ··· ,w,··· ,v,··· ,v k )=0. ⇒ −1 k ⇒ σ i j F R 3 W F (v)=<F,v>, v∈ R 3 1 R 3 F v ω F (v 1 ,v 2 )=<F,v 1 ×v 2 >, v 1 ,v 2 ∈ R 3 2 R 3 F v 1 ,v 2 n R n det(v 1 , ··· ,v n ) v 1 , ··· ,v n ∈ R n Λ k (V ) Λ k (V ) k V (ω + γ)(v 1 , ··· ,v k )=ω(v 1 , ··· ,v k )+γ(v 1 , ··· ,v k ) (αω)(v 1 , ··· ,v k )=αω(v 1 , ··· ,v k ) ω, γ ∈ Λ k (V ),α∈ R (Λ k (V ), +, ·) R Λ 1 (V ) V Λ 1 (V )=V ∗ = L(V, R) ϕ 1 ,ϕ 2 ∈ V ∗ 2 ϕ 1 ∧ ϕ 2 : V × V → R, (ϕ 1 ∧ ϕ 2 )(v 1 ,v 2 )=ϕ 1 (v 1 )ϕ 2 (v 2 ) − ϕ 2 (v 1 )ϕ 1 (v 2 ) = det  ϕ 1 (v 1 ) ϕ 1 (v 2 ) ϕ 2 (v 1 ) ϕ 2 (v 2 )  R 2 ϕ(v 1 ),ϕ(v 2 ) ϕ =(ϕ 1 ,ϕ 2 ):V → R 2 ϕ 1 , ··· ,ϕ k ∈ V ∗ k ϕ 1 ∧···∧ϕ k ∈ Λ k (V ) ϕ 1 ∧···∧ϕ k (v 1 , ··· ,v k )=  σ (σ)ϕ σ(1) (v 1 ) ···ϕ σ(k) (v k )=det(ϕ i (v j )),v 1 , ··· ,v k ∈ V, ϕ 1 ∧···∧ϕ k =  σ (σ)ϕ σ(1) ⊗···⊗ϕ σ(k) ϕ 1 , ··· ,ϕ k ,ϕ  i ∈ Λ 1 (V ),α,β∈ R i =1, ··· ,k ϕ 1 ∧···∧(αϕ i +βϕ  i )∧···∧ϕ k = αϕ 1 ∧···∧ϕ i ∧···∧ϕ k +βϕ 1 ∧···∧ϕ  i ∧···∧ϕ k . ϕ σ(1) ∧···∧ϕ σ(k) = (σ)ϕ 1 ∧···∧ϕ k , σ k V R ϕ 1 , ··· ,ϕ n V ∗ Λ k (V ) {ϕ i 1 ∧···∧ϕ i k , 1 ≤ i 1 < ···<i k ≤ n} ω ∈ Λ k (V ) ω =  1≤i 1 <···<i k ≤n a i 1 ···i k ϕ i 1 ∧···∧ϕ i k dim Λ k (V )=C k n = n! (n − k)!k! {ϕ 1 , ··· ,ϕ n } {e 1 , ··· ,e n } ϕ i (e j )=δ ij ω ∈ Λ k (V ) v 1 , ··· ,v k ∈ V v 1 =  i 1 ϕ i 1 (v 1 )e i 1 , ··· ,v k =  i k ϕ i k (v k )e i k , ω(v 1 , ··· ,v k )=ω(  i 1 ϕ i 1 (v 1 )e i 1 , ··· ,  i k ϕ i k (v k )e i k ) =  i 1 ,··· ,i k ϕ i 1 (v 1 ) ···ϕ i k (v k )ω(e i 1 , ··· ,e i k ) =  i 1 <···<i k  σ ϕ i σ(1) (v 1 ) ···ϕ i σ(k) (v k )(σ)ω(e i 1 , ··· ,e i k ) =  i 1 <···<i k ω(e i 1 , ··· ,e i k )ϕ i 1 ∧···∧ϕ i k (v 1 , ··· ,v k ) {ϕ i 1 ∧···∧ϕ i k , 1 ≤ i 1 < ···<i k ≤ n} ϕ i 1 ∧···∧ϕ i k (e j 1 , ··· ,e j k )=  1 (i 1 , ··· ,i k )=(j 1 , ··· ,j k ) 0 (i 1 , ··· ,i k ) =(j 1 , ··· ,j k ) ω =  i 1 <···<i k a i 1 ·i k ϕ i 1 ∧···∧ϕ i k =0, ω(e i 1 , ··· ,e i k )=a i 1 ···i k =0 Λ k (V )=0 k>n Λ n (V ) C n n =1 ω ∈ Λ n (V ) ω = aϕ 1 ∧···∧ϕ n a ∈ R U R n k k U ω : U → Λ k (R n ). ω C p C p Ω k p (U) k C p U Ω k (U)=Ω k ∞ (U) Ω k p (U) U ⊂ R 3 F : U → R 3 F W F : U → Λ 1 (R 3 ),W F (x, y, z)(v)=<F(x, y, z),v > ω F : U → Λ 2 (R 3 ),ω(x, y, z)(v 1 ,v 2 )=<F(x, y, z),v 1 × v 2 > f : U → R C p+1 x ∈ U f  (x):R n → R f 1 df : U → Λ 1 (R n ),x→ df (x)=f  (x). ix i : R n → R, (x 1 , ··· ,x n ) → x i dx i (x)(v)=x  i (x)v = v i ,v=(v 1 , ··· ,v n ) ∈ R n . df (x)(v)=f  (x)v = ∂f ∂x 1 (x)v 1 + ···+ ∂f ∂x n (x)v n = ∂f ∂x 1 (x)dx 1 (x)(v)+···+ ∂f ∂x n (x)dx n (x)(v). df = n  i=1 ∂f ∂x i dx i 1 ϕ 1 , ··· ,ϕ k ∈ Ω 1 (U) (ϕ 1 ∧···∧ϕ k )(x)=ϕ 1 (x) ∧···∧ϕ k (x),x∈ U, k U 1 dx 1 , ··· ,dx n Ω 1 (U) k U ω =  1≤i 1 <···<i k ≤n a i 1 ···i k dx i 1 ∧···∧dx i k , a i 1 ···i k U C p ω C p U ⊂ R 3 (x, y, z) 0 f : U → R 1 Pdx+ Qdy + Rdz 2 Adx ∧ dy + Bdy ∧ dz + Cdz ∧dx 3 fdx∧ dy ∧dz U ⊂ R 3 F : U → R 3 F =(P, Q, R) W F = Pdx+ Qdy + Rdz ω F = Pdy ∧ dz + Qdz ∧ dx + Rdx ∧dy U, V R m , R n ϕ : U → V, u =(u 1 , ··· ,u m ) → x =(ϕ 1 (u), ··· ,ϕ n (u)) ϕ ∗ :Ω k (V ) → Ω k (U),ω→ ϕ ∗ ω ω =  1≤i 1 <···<i k ≤n a i 1 ···i k (x)dx i 1 ∧···∧dx i k , ϕ ∗ ω(u)=  1≤i 1 <···<i k ≤n a i 1 ···i k (ϕ(u))dϕ i 1 ∧···∧dϕ i k . ϕ : R → R 2 ,ϕ(t)=(x =cost, y =sint) ω(x, y)=xdy −ydx ϕ ∗ ω(t)=costd(sin t) − sin td(cos t)=dt ϕ : R 2 → R 2 ,ϕ(r, θ)=(x = r cos θ, y = r sin θ) ω(x, y)=dx ∧ dy ϕ ∗ ω(r, θ)=d(r cos θ) ∧ d(r sin θ) =(cosθdr − r sin θdθ) ∧(sin θdr + r cos θdθ) = rdr ∧ dθ (do dr ∧dr = dθ ∧dθ =0,dθ∧dr = −dr ∧ dθ). ϕ ∗ (ω 1 + ω 2 )=ϕ ∗ (ω 1 )+ϕ ∗ (ω 2 ),ω 1 ,ω 2 ∈ Ω k (V ) ϕ ∗ (γ 1 ∧···∧γ k )=ϕ ∗ (γ 1 ) ∧···∧ϕ ∗ (γ k ),γ 1 , ··· ,γ k ∈ Ω 1 (V ) ϕ ∗ (dx i )=dϕ i = m  j=1 ∂ϕ i ∂u j du j ϕ : R n → R n ϕ ∗ (f(x)dx 1 ∧···∧dx n )=f(ϕ(u)) det ϕ  (u)du 1 ∧···∧du n . ϕ ∗ ω(u)(v 1 , ··· ,v k )=ω(ϕ(u))(ϕ  (u)v 1 , ··· ,ϕ  (u)v k ). k ∈ N d :Ω k (U) → Ω k+1 (U), d(  1≤i 1 <···<i k ≤n a i 1 ···i k dx i 1 ∧···∧dx i k )=  1≤i 1 <···<i k ≤n da i 1 ···i k ∧ dx i 1 ∧···∧dx i k . n =2 (x, y) d (Pdx+ Qdy)=dP ∧ dx + dQ ∧dy =  ∂P ∂x dx + ∂P ∂y dy  ∧ dx +  ∂Q ∂x dx + ∂Q ∂y dy  ∧ dy =  ∂Q ∂x − ∂P ∂y  dx ∧ dy dx ∧dx = dy ∧dy =0,dy∧ dx = −dx ∧ dy R 3 ω(x, y, z)=sinxydx + e x 2 +y dy +arctgxdz dω =(d sin xy) ∧ dx + d(e x 2 +y ) ∧ dy + d(arctgx) ∧dz =(y cos xydx + x cos xydy) ∧ dx +(2xe x 2 +y dx + e x 2 +y dy) ∧dy + 1 1+x 2 dx ∧ dz =(2xe x 2 +y − x cos xy)dx ∧dy − 1 1+x 2 dz ∧ dx. d (P (x, y, z)dx + Q(x, y, z)dy + R(x, y, z)dz) d (P (x, y, z)dx ∧ dz + Q(x, y, z)dz ∧ dx + Q(x, y, z)dx ∧dy) . ω ∈ Ω k (R n ) k ≥ n dω =0 d(ω 1 + ω 2 )=dω 1 + dω 2 , ∀ω 1 ,ω 2 ∈ Ω k (U) d(γ 1 ∧ γ 2 )=dγ 1 ∧ γ 2 − γ 1 ∧ dγ 2 , ∀γ 1 ,γ 2 ∈ Ω 1 (U). d(dω)=0 d ◦d =0 d(ϕ ∗ ω)=ϕ ∗ (dω) dϕ ∗ = ϕ ∗ d γ 1 = adx i ,γ 2 = bdx j d(γ 1 ∧ γ 2 )=d(adx i ∧ bdx j )=d(abdx i ∧ dx j ) = d(ab) ∧ dx i ∧ dx j =(bda + adb) ∧dx i ∧ dx j = bda ∧dx i ∧ dx j + adb ∧ dx i ∧ dx j =(da ∧dx i ) ∧ bdx j − adx i ∧ db ∧dx j = dγ 1 ∧ γ 2 − γ 1 ∧ γ 2 . dx I = dx i 1 ∧···∧dx i k I =(i 1 , ··· ,i k ) k {1, ···n} ω = a I dx I d(dω)=d(da I ∧ dx I )=d   i ∂a I ∂x i dx i ∧ dx I  =  i d  ∂a I ∂x i  ∧ dx i ∧ dx I =  i    j ∂ 2 a I ∂x j ∂x i dx j   ∧ dx i ∧ dx I = −  i  j ∂ 2 a I ∂x i ∂x j dx i ∧ dx j ∧ dx I ( dx i ∧ dx j = −dx j ∧ dx i ) = −d(dω)( i, j) 2d(dω)=0 ω = a I dx I ∈ Ω k (V ) d(ϕ ∗ ω)=d(a I ◦ ϕdϕ I )=d(a I ◦ ϕ) ∧ dϕ I . ϕ ∗ (dω)=ϕ ∗ (da I ∧ dx I )=ϕ ∗ (da I ) ∧ ϕ ∗ (dy I )=ϕ ∗ (da I ) ∧ dϕ I . d(a I ◦ ϕ)=ϕ ∗ (da I ) ϕ ∗ (da I )=ϕ ∗    j ∂a I ∂x j dx j   =  j ∂a I ◦ ϕ ∂x j dϕ j =  j ∂a I ◦ ϕ ∂x j (  i ∂ϕ j ∂u i du i )=d(a I ◦ϕ). d ω ∈ Ω k (U) ω U dω =0 U ω U η ∈ Ω k−1 (U) ω = dη ω ω d(dη)=0 ω(x, y)= ydx − xdy x 2 + y 2 ∈ Ω 1 (R 2 \ 0) ω dω = x 2 − y 2 (x 2 + y 2 ) 2 dy ∧dx − y 2 − x 2 (x 2 + y 2 ) 2 dx ∧ dy =0 ω f ∈ Ω 0 (R 2 \ 0) ω = df ϕ(t) = (sin t, cos t) ϕ ∗ ω = ϕ ∗ (df )=d(ϕ ∗ f)=d(f ◦ ϕ)=(f ◦ϕ)  dt. ϕ ∗ ω = cos td(sin t) −sin td(cos t) sin 2 t +cos 2 t = dt (f ◦ ϕ)  (t) ≡ 1 f ◦ ϕ(t)=t+ f ◦ ϕ 2π ω = a 1 dx 1 + ···+ a n dx n ∈ Ω 1 (U) f ∈ Ω 0 (U) df = ω f ω f ∂f ∂x 1 = a 1 , ··· , ∂f ∂x n = a n . ω dω =0 a 1 , ··· ,a n ∂a j ∂x i = ∂a i ∂x j i, j =1, ··· ,n. U U U R n x 0 ∈ U C 1 h : U × [0, 1] → U, (x, t) → h(x, t) h(x, 0) = x 0 h(x, 1) = x, ∀x ∈ U U ∀x, y ∈ U [x, y]={x + t(y − x):t ∈ [0, 1]}⊂U R n U ∃x 0 ∈ U : ∀x ∈ U, [x 0 ,x] ⊂ U h(x, t)=x 0 + t(x − x 0 ) U R n U U ω ∈ Ω k (U),dω =0 ⇔∃η ∈ Ω k−1 (U),ω= dη. J t : U → U × [0, 1],J t (x)=(x, t) k =1, 2, ··· K :Ω k (U × [0, 1]) → Ω k−1 (U) ∗ Kd + dK = J ∗ 1 − J ∗ 0 Ω k (U × [0, 1]) a(x, t)dx I b(x, t)dt ∧dx J , I =(i 1 , ··· ,i k ),J =(j 1 , ··· ,j k−1 ). K K(a(x, t)dx I )=0 K(b(x, t)dt ∧dx J )=   1 0 b(x, t)dt  dx J ∗ (Kd + dK)(adx I )=K(da ∧ dx I )+d(0) = (  1 0 ∂a ∂t dt)dx I =(a(x, 1) − a(x, 0)dx I =(J ∗ 1 − J ∗ 0 )(adx I ). ∗ (Kd + dK)(bdt ∧ dx J )=K(db ∧ dt ∧ dx J )+d((  1 0 bdt) ∧ dx J ) = K(  i ∂b ∂x i dx i ∧ dt ∧ dx J )+d((  1 0 bdt) ∧ dx J ) = −  1 0 (  i ∂b ∂x i )dt ∧ dx i ∧ dx J + d((  1 0 bdt) ∧ dx J ) = −d((  1 0 bdt) ∧ dx J )+d((  1 0 bdt) ∧ dx J )=0. (J ∗ 1 − J ∗ 0 )(bdt ∧ dx J )=b(x, 1)d(1) ∧dx J − b(x, 0)d(0) ∧dx J =0. h : U ×[0, 1] → U x 0 ω ∈ Ω k (U) dω =0 η = Kh ∗ ω (k −1) dη = ω ∗ (Kd + dK)h ∗ ω =(J ∗ 1 − J ∗ 0 )h ∗ ω. ⇔ Kdh ∗ ω + dKh ∗ ω =(h ◦ J 1 ) ∗ ω −(h ◦J 0 ) ∗ ω. ⇔ Kh ∗ dω + dKh ∗ ω =(id U ) ∗ ω −(x 0 ) ∗ ω. ⇔ 0+dKh ∗ ω = ω +0. η = Kh ∗ ω U ω 1 ,ω 2 ∈ Ω k (U) dω 1 = dω 2 η ∈ Ω k−1 dη = ω 1 − ω 2 R 2 \0 η η dη = ω η = Kh ∗ ω ω =(x 2 − 2yz)dx +(y 2 − 2zx)dy +(z 2 − 2xy)dz ∈ Ω 1 (R 3 ) dω =0 f df = ω R 3 0 h(x, y, z, t)=(tx, ty, tz) h ∗ ω = t 2 (x 2 − 2yz)(xdt + tdx)+t 2 (y 2 − 2zx)(ydt + tdy)+t 2 (z 2 − 2xy)(zdt + tdz). Kh ∗ ω =  1 0 t 2 (x 2 − 2yz)xdt +  1 0 t 2 (y 2 − 2zx)ydt +  1 0 t 2 (z 2 − 2xy)zdt. f = Kh ∗ ω = 1 3 (x 3 + y 3 + z 3 −6xyz) ω df = ω f df = ω (1) ∂f ∂x = x 2 − 2yz (2) ∂f ∂y = y 2 − 2zx (3) ∂f ∂z = z 2 − 2xy f (1) f = x 3 3 − 2xyz + ϕ(y, z) (2) ∂ϕ ∂y = y 2 ϕ = y 3 3 + ψ(z) (3) ∂ψ ∂z = z 2 ψ = z 3 3 + f = 1 3 (x 3 + y 3 + z 3 ) − 2xyz+ . Kh ∗ ω = 1 3 (x 3 + y 3 + z 3 −6xyz) ω df = ω f df = ω (1) ∂f ∂x = x 2 − 2yz (2) ∂f ∂y = y 2 − 2zx (3) ∂f ∂z = z 2 − 2xy f (1) f = x 3 3 − 2xyz + ϕ(y, z) (2) ∂ϕ ∂y = y 2 ϕ = y 3 3 + ψ(z) (3) ∂ψ ∂z =. f = x 3 3 − 2xyz + ϕ(y, z) (2) ∂ϕ ∂y = y 2 ϕ = y 3 3 + ψ(z) (3) ∂ψ ∂z = z 2 ψ = z 3 3 + f = 1 3 (x 3 + y 3 + z 3 ) − 2xyz+ . ··· ,w,··· ,v,··· ,v k )=0. ⇒ −1 k ⇒ σ i j F R 3 W F (v)=<F,v>, v∈ R 3 1 R 3 F v ω F (v 1 ,v 2 )=<F,v 1 ×v 2 >, v 1 ,v 2 ∈ R 3 2 R 3 F v 1 ,v 2 n R n det(v 1 , ··· ,v n ) v 1 , ···