Giáo trình giải tich 3 part 3 ppt

R n rank (DF 1 , ··· ,DF m )(x)=m, ∀x ∈ M M n − m C p k = n − m x =(x  ,y) ∈ R k × R m = R n F =(F 1 , ··· ,F m ) a ∈ M det ∂F ∂y (a) =0. V  a =(a  ,b) M ∩ V  = {(x  ,y) ∈ V  : F(x  ,y)=0} = {(x  ,y) ∈ V  : y = g(x  )}, g C p U a  ϕ : U → R n ϕ(x  )=(x  ,g(x  )) M a  R 3 S 2 F (x, y, z)=x 2 + y 2 + z 2 − 1=0 F  (x, y, z)=(2x, 2y, 2z) =(0, 0, 0) S 2 S 2 2 C 1  F 1 (x, y, z)=x 2 + y 2 + z 2 − 1=0 F 2 (x, y, z)=x + y + z =0 (ψ, W) M x W  ,U  ψ −1 (x),ϕ −1 (x) W  ψ = ϕ ◦ h, h = ϕ −1 ◦ ψ : W  → U  h −1 h = ϕ −1 ◦ψ ψ −1 (ψ(W )∩ϕ(U)) ϕ −1 (ψ(W )∩ ϕ(U)) h C p rank Dϕ = k k Dϕ(u) u U  D(ϕ 1 , ··· ,ϕ k ) D(u 1 , ··· ,u k ) =0 U  x =(x  ,y) ∈ R k × R n−k i : R k → R k × R n−k i(u)=(u, 0) p = R k × R n−k → R k p(x  ,y)=x  Φ(u, y)=(ϕ(u),y) det DΦ= D(ϕ 1 , ··· ,ϕ k ) D(u 1 , ··· ,u k ) =0 Φ −1 ∈ C p h = ϕ −1 ◦ ψ =(Φ◦ i) −1 ◦ ψ = p ◦Φ −1 ◦ ψ. C p h C p  M ⊂ R n k x 0 ∈ M γ :(−, ) → M C 1 M γ(0) = x 0 γ  (0) M x 0 M x 0 M x 0 T x 0 M (ϕ, U) M x 0 = ϕ(u 0 ) T x 0 M = {v ∈ R n : v = t 1 D 1 ϕ(u 0 )+···+ t k D k ϕ(u 0 ),t 1 , ··· ,t k ∈ R} = Dϕ(u 0 ). R n M F 1 = ···= F m =0 x 0 T x 0 M = {v ∈ R n : v ⊥ grad F i (x 0 ),i=1, ···,m}. T x 0 M v ∈ R n : < grad F 1 (x 0 ),v >= ···=< grad F m (x 0 ),v >=0 S 2 C H k = {x =(x 1 , ··· ,x k ) ∈ R k : x k ≥ 0} R k ∂H k = {x ∈ H k : x k =0} = R k−1 × 0 H k H k + = {x ∈ H k : x k > 0} H k M ⊂ R n k C p x ∈ M V ⊂ R n x U ⊂ R k ϕ : U → R n C p ϕ : U ∩H k → M ∩V rank ϕ  (u)=k u ∈ U x = ϕ(u),u∈ U M u ∈ H k + M u ∈ ∂H k ∂M = {x ∈ M : x M} M s ✲ R 1 ✻x U  H ✲ ϕ s x ✲  ✒ M  V         V ⊂ R n C p F 1 , ··· ,F m ,F m+1 : V → R M = {x ∈ V : F 1 (x)=···= F m (x)=0,F m+1 (x) ≥ 0} ∂M = {x ∈ V : F 1 (x)=···= F m (x)=F m+1 (x)=0} rank (DF 1 , ··· ,DF m )(x)=m, ∀x ∈ M rank (DF 1 , ··· ,DF m+1 )(x)= m +1, ∀x ∈ ∂M M n − m C p ∂M  R 3 B x 2 + y 2 + z 2 ≤ 1 R n ∂B x 2 + y 2 + z 2 =1 M k ∂M k − 1 ∂(∂M)=∅ x ∈ ∂M T x ∂M k − 1 T x M i : R k−1 → R k ,i(u 1 , ··· ,u k−1 )=(u 1 , ··· ,u k−1 , 0) (ϕ, U) M x x ∈ ∂M (ϕ ◦i, i −1 (U)) ∂M x x ∂M ∂(∂M)=∅ T x ∂M D 1 ϕ(u), ··· ,D k−1 ϕ(u) k − 1 T x M  F =(F 1 , ··· ,F m ):V → R m C 1 V ⊂ R n M = {x ∈ V : F 1 (x)=···= F m (x)=0} rank F  (x)=m, ∀x ∈ M f : V → R C 1 f| M f F 1 = ···= F m =0 M a ∈ M (ϕ, U) M a a = ϕ(b) f F 1 = ··· = F m =0 a grad f(a) ⊥ T a M λ 1 , ··· ,λ m ∈ R grad f(a)= λ 1 grad F 1 (a)+···+ λ m grad F m (a) f| M a f ◦ϕ b (f ◦ ϕ)  (b)=f  (a)ϕ  (b)=0 < grad f (a),v >=0, ∀v ∈ Imϕ  (b)=T a M grad f(a) ⊥ T a M rank (grad F 1 (a), ··· , grad F m (a)) = m =codimT a M grad f(a) grad F 1 (a), ··· , grad F m (a)  f F 1 = ···= F m =0 L(x, λ)=f(x) − λ 1 F 1 (x) −···−λ m F m (x),x∈ V,λ =(λ 1 , ··· ,λ m ) ∈ R m a λ ∈ R m (a, λ)              ∂L ∂x (x, λ)=0 F 1 (x)=0 F m (x)=0 f(x, y, z)=x + y + z x 2 + y 2 =1,x+ z =1 L(x, y, z, λ 1 ,λ 2 )=x + y + z −λ 1 (x 2 + y 2 − 1) −λ 2 (x + z − 1)                        ∂L ∂x =1− 2λ 1 x −λ 2 =0 ∂L ∂y =1− 2λ 1 y =0 ∂L ∂z =1 −λ 2 =0 x 2 + y 2 − 1=0 x + z − 1=0 (0, ±1, 1) f max f| E =max{f(0, 1, 1) = 1,f(0, −1, 1) = 0} = f(0, 1, 1) = 1 min f| E = min{f(0, 1, 1) = 1,f(0, −1, 1) = 0} = f(0, −1, 1) = 0 f, F 1 , ··· ,F m C 2 grad f(a)= λ 1 grad F 1 (a)+···+ λ m grad F m (a) ∂L ∂x (a, λ)=0 H x L(x, a) L x H x L(a, λ)| T a M f| M a H x L(a, λ)| T a M f| M a H x L(a, λ)| T a M f| M a f| M f◦ϕ f  (a)ϕ  (b)=0 H(f◦ϕ)(a)(h)= Hf(a)(ϕ  (b)h) F i ◦ ϕ =0 H(F i ◦ ϕ)=0 H(F i ◦ ϕ)(b)(h)= HF i (a)(ϕ  (b)(h) H x L(a, λ)| T a M = H(f ◦ ϕ)(b)| T a M  k ∈ N a ∈ R f(x 1 , ··· ,x n )=x k 1 + ···+ x k n x 1 + ···+ x n = an R 3 R 3 < ·, · > T =(x t ,y t ,z t ) T  =  x 2 t + y 2 t + z 2 t u =(x u ,y u ,z u ),v=(x v ,y v ,z v ) (u, v)=uv ⊥  = u × v =      u 2 <u,v> <v,u> v 2      1 2 =  u 2 v 2 −|<u,v>| 2 . v = v  + v ⊥ v  v u v ⊥ ⊥ u v  = αu, < v ⊥ ,u>=0      <u,u> <u,v> <v,u> <v,v>      =      <u,u> <u,v  > + <u,v ⊥ > <v,u> <v,v  > + <v,v ⊥ >      =      <u,u> α<u,u  > <v,u> α<v,u  >      +      <u,u> 0 <v,u> v ⊥  2      = u 2 v ⊥  2  u, v, w ∈ R 3 (u, v, w)= (u, v) w ⊥  = | <u×v, w > | = |det(u, v, w)| =        <u,u> <u,v> <u,w> <v,u> <v,v> <v,w> <w,u> <w,v> <w,w>        1 2 w = w  + w ⊥ w  w u, v ✂ ✂ ✂ ✂ ✂✍ w ✲ u ✟ ✟ ✟✯ v ✻w ✟ ✟ ✟ ✂ ✂ ✂ ✂ ✂ ✟ ✟ ✟ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✂ ✟ ✟ ✟  k R n R n k v 1 , ··· ,v k ∈ R n k V 1 (v 1 )=v 1 ,V k (v 1 , ··· ,v k )=V k−1 (v 1 , ··· ,v k−1 )v ⊥ k  v k = v  k + v ⊥ k v  k v k v 1 , ··· ,v k−1 G(v 1 , ··· ,v k )=(<v i ,v j >) 1≤i,j≤k V k (v 1 , ··· ,v k )=  det G(v 1 , ··· ,v k )  C ⊂ R 3 ϕ : I → R 3 ,ϕ(t)=(x(t),y(t),z(t)) l(C) I I i =[t i ,t i +∆t i ] l(C)=  i l(ϕ(I i )) ∆t i l(ϕ(I i )) ∼ l(ϕ  (t i )∆t i )=ϕ  (t i )∆t i dl = ϕ  (t)dt =  x  2 t + y  2 t + z  2 t dt C l(C)=  C dl =  I  x  2 t + y  2 t + z  2 t dt S ⊂ R 3 ϕ : U → R 3 ,ϕ(u, v)=(x(u, v),y(u, v),z(u, v)) S U U i =[u i ,u i +∆u i ]×[v i ,v i +∆v i ] (S)=  i (ϕ(U i )) ∆u i , ∆v i (ϕ(U i )) ∼ (D 1 ϕ(u i ,v i )∆u i ,D 2 ϕ(u i ,v i )∆v i ) dS = (D 1 ϕ, D 2 ϕ)dudv =  EG − F 2 dudv, E = D 1 ϕ 2 = x  u 2 + y  u 2 + z  u 2 G = D 2 ϕ 2 = x  v 2 + y  v 2 + z  v 2 F = <D 1 ϕ, D 2 ϕ> = x  u x  v + y  u y  v + z  u z  v S (S)=  S dS =  U  EG − F 2 dudv H ϕ : A → R 3 ,ϕ(u, v, w)=(x(u, v, w),y(u, v, w),z(u, v, w)) H dV = (D 1 ϕ, D 2 ϕ, D 3 ϕ)dudvdw = |det Jϕ|dudvdw H V (H)=  H dV =  A |det Jϕ|dudvdw M ⊂ R n k dV : M  x → dV (x)= k T x M. (ϕ, U) M x = ϕ(u 1 , ··· ,u k ) dV (x)(D 1 ϕ(x)∆u 1 , ··· ,D k ϕ(x)∆u k )=V k (D 1 ϕ(x), ··· ,D k ϕ(x))∆u 1 ···∆u k G ϕ =(<D i ϕ, D j ϕ>) 1≤i,j≤k dV =  det G ϕ du 1 ···du k f : M → R k f M  M fdV M = ϕ(U) (ϕ, U )  M fdV =  U f ◦ ϕ  det G ϕ , G ϕ =(<D i ϕ, D j ϕ>) 1≤i,j≤k . k =1  M fdl k =2  M fdS M O = {(ϕ i ,U i ):i ∈ I} M Θ={θ i : i ∈ I} M O i ∈ I θ i : M → [0, 1] θ i = {x ∈ M : θ(x) =0} θ i ⊂ ϕ i (U i ) x ∈ M V x i ∈ I θ i =0 V  i∈I θ i (x)=1, ∀x ∈ M {supp θ i ,i ∈ I} x O M O M k x ∈ M (ϕ x ,U x ) ∈O x B x ⊃ U x ϕ −1 x (x) B x = B(a, r) g x : R k → R g x (u)=      e − 1 r 2 −u−a 2 , u − a≤r 0 ,  u −a >r. g x ∈ C ∞ ˜g x (y)=g x (ϕ −1 x (y)) y ∈ ϕ x (U x ) ˜g x (y)=0 y ∈ ϕ x (U x ) ˜g x M M x 1 , ··· ,x N ∈ M ϕ x 1 (B x 1 ), ···ϕ x N (B x N ) M θ i = ˜g x i ˜g x 1 + ···+˜g x N {θ i : i =1, ···N } M ϕ x (B x ) M  M O = {(ϕ i ,U i ):i ∈ I} Θ={θ i : i ∈ I} M O  M fdV =  i∈I  ϕ i (U i ) θ i fdV (=  i∈I  U i θ i f ◦ ϕ i  det G ϕ i ). M f M ϕ(U)=ψ(W ) ψ = ϕ ◦ h h G ψ (w)= t Jh(w)G ϕ (h(w))Jh(w)  U f ◦ ϕ  det G ϕ =  W f ◦ ϕ ◦h|det Jh|  det G ϕ ◦ h =  W f ◦ ψ  det t JhG ϕ ◦ h det Jh =  W f ◦ ψ  det G ψ . Θ  = {θ  j : j ∈ J} M  j  M θ  j f =  j  M (  i θ i )θ  j f =  i,j  M θ i θ  j f =  i,j  M θ  j θ i f =  i  M (  j θ  j )θ i f.  ϕ : I → R n ,ϕ(t)=(x 1 (t), ··· ,x n (t)) C  C fdl =  I f ◦ ϕ ϕ   =  I f(ϕ(t))  (x  1 ) 2 (t)+···+(x  n ) 2 (t)dt. ϕ : U → R 3 ,ϕ(u, v)=(x(u, v),y(u, v),z(u, v)) S  S fdS =  U f ◦ ϕ  EG − F 2 , E = D 1 ϕ 2 = x  u 2 + y  u 2 + z  u 2 G = D 2 ϕ 2 = x  v 2 + y  v 2 + z  v 2 F = <D 1 ϕ, D 2 ϕ> = x  u x  v + y  u y  v + z  u z  v C x = a cos t, y = a sin t, z = bt, t ∈ [0,h]  C dl =  h 0  a 2 sin 2 t + a 2 cos 2 t + b 2 dt = h  a 2 + b 2 R ϕ(φ, θ)=(R cos φ sin θ,R sin φ sin θ, R cos θ), (φ, θ) ∈ U =(0, 2π) × (0,π) D 1 ϕ(φ, θ)=(−R sin φ sin θ,R cos φ sin θ, 0) D 2 ϕ(φ, θ)=(R cos φ cos θ, R sin φ cos θ, −R sin θ) E = R 2 sin 2 θ, F =0,G= R 2  S dS =  U  EG − F 2 dφdθ =  2π 0  π 0 R 2 sin θdφdθ =4πR 2 R ϕ(r, φ, θ)=(r cos φ sin θ, r sinφ sin θ, r cos θ), (r, φ, θ) ∈ U =(0,R) × (0, 2π) × (0,π) D 1 ϕ(r, φ, θ)=(cosφ sin θ, sinφ sin θ, cos θ) D 2 ϕ(r, φ, θ)=(−r sin φ sin θ, r cos φ sin θ, 0) D 3 ϕ(r, φ, θ)=(r cos φ cos θ, r sinφ cos θ, −r sin θ)  B(0,R) dV =  U det(<D i ϕ, D j ϕ>)drdφdθ =  R 0  2π 0  π 0        10 0 0 r 2 sin 2 θ 0 00r 2        drdφdθ = 4 3 πR 3 . H(f ◦ ϕ)(b)| T a M  k ∈ N a ∈ R f(x 1 , ··· ,x n )=x k 1 + ···+ x k n x 1 + ···+ x n = an R 3 R 3 < ·, · > T =(x t ,y t ,z t ) T  =  x 2 t + y 2 t + z 2 t u =(x u ,y u ,z u ),v=(x v ,y v ,z v ) (u,. ,v k )=(<v i ,v j >) 1≤i,j≤k V k (v 1 , ··· ,v k )=  det G(v 1 , ··· ,v k )  C ⊂ R 3 ϕ : I → R 3 ,ϕ(t)=(x(t),y(t),z(t)) l(C) I I i =[t i ,t i +∆t i ] l(C)=  i l(ϕ(I i )) ∆t i l(ϕ(I i )). =  x  2 t + y  2 t + z  2 t dt C l(C)=  C dl =  I  x  2 t + y  2 t + z  2 t dt S ⊂ R 3 ϕ : U → R 3 ,ϕ(u, v)=(x(u, v),y(u, v),z(u, v)) S U U i =[u i ,u i +∆u i ]×[v i ,v i +∆v i ] (S)=  i (ϕ(U i )) ∆u i ,