1 R n (x (k) = (x (k) 1,, x(k) n )) k 1 lim k,l x(k) x (l) = 0 (x (k) ) 1.1. (i) R n U U, r > 0, r () U (ii) R n F F F (iii) R n S S S = { R n ; r > 0

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1 III ϵ-δ ymgmi/teching/set2018.pdf ymgmi/teching/rel2018.pdf n x = (x 1,, x n ) n R n x 0 = (0,, 0) x = (x 1 ) (x n ) 2 r > 0 r () r () r () = {x R n ; x < r}, r () = {x R n ; x r}. 1

2 1 R n (x (k) = (x (k) 1,, x(k) n )) k 1 lim k,l x(k) x (l) = 0 (x (k) ) 1.1. (i) R n U U, r > 0, r () U (ii) R n F F F (iii) R n S S S = { R n ; r > 0, r () S, r () \ S } S = S \ S S S = S S S R n K (finite covering property) {U i } i I K i I U i I F K i F U i 1.4. R n K 3 (i) K (ii) K K (iii) K Proof. (i) (ii): { j } j 1 K R n K K x K x /2 (x) K { x /2 (x)} x K K x 1,..., x N { xj /2(x j )} 1 j N K x 1,..., x N r > 0 2

3 K r K K K (ii) (i): (olzno ) K K {U i } i I U i K K r > 0 K r 1 K U i 0 0 K 0 1/2 U i 1/ K 1/2 k { k } (i) k U i (ii) 1 j k j K k 1 j k j K {c k } j k c j c k 1/2 j {c k } Cuchy c = lim c k K c K c U i i r > 0 r (c) U i k 1/2 k < r k+1 U i k+1 U i K 1.5. (i) (ii) ϵ > 0, δ > 0, x y δ f(x) f(y) ϵ Proof. (i) M = sup{f(k)} M f(k) M f(k) M j f 1 ((, M j ]), j = 1,... K K (ii) K δ > 0 f(x) f() ϵ for x δ () { δ/2()} K K 1,..., N δ = mx{δ 1,..., δ N } x y δ x j δ j /2 j y j x y + x j δ j f(x) f( j ) ϵ, f(y) f( j ) ϵ f(x) f(y) 2ϵ 1.6. X R m Y R n φ : X Y X (x (k) ) k 1 lim k φ(x(k) ) = φ( lim k x(k) ) 1.7. φ : X Y Y U φ 1 (U) X 2 D R 2 D f D D = i D i 3

4 f(x i ) D i i x i D i i D i D i D i 0 x i f f D f(x) dx D D D i f D [, b] = [ 1, b 1 ] [ n, b n ] R n 2.1. f : R = { i } S(, f) = i f i i, S(, f) = i f i i, = mx{d( i )} f i = sup{f(x); x i }, f i = inf{f(x); x i }, d( i ) = b i i lim S(, f) = lim S(, f) 0 0 f f(x) dx f 2.2. f, g (i) α, β αf + βg (αf(x) + βg(x)) dx = α f(x) dx + β (ii) f(x) (iii) f(x) g(x) f(x) dx f(x) dx 4 f(x) dx. g(x) dx. g(x) dx.

5 2.3. = [, b] * 1 Proof. f(x) dx = b1 bn dx 1 dx n f(x 1,..., x n ) 1 n 2.4. R n D 1 D (x) = { 1 x D 0 x D 1 D D D D f 1 D f D f(x) dx 2.5. Proof. n = 2 D δ = S(1 D, ) S(1 D, ) (h + 2δ)(k + 2δ) hk δ 0 0 D 2.6. [0, 1] Dirichlet f f(x) = 1 Q = f D { 1 if x is rtionl number, 0 otherwise 2.7. (i) D D f f(x) dx (1 D f ) (ii) D f(x) dx = f(x) dx D i D i D *1 b dt f(t) 5

6 2.8. R n A 1 A A A = sup{ 1 A (x) dx; } 2.9. α > 0 A = {(x, y); x > 1, 0 < y < 1/x α } { 1 A = α 1 if α > 1, + otherwise 2.10.,, A = A + A. 3 m n M m,n (R) M m,n (R) mn m n A (Hilbert-Schmidt norm) A A 2 = i,j ij 2 A A A Ax A x (x R m ). A = sup{ Ax / x ; 0 x R n } 1. ( ) A A mn A x(t) R n ( b ) b x(t)dt = x i (t)dt b x(t)dt b 1 i n x(t) dt ( < b) 6

7 Proof. b x(t) dt = lim j x(t j )(t j t j 1 ) b x(t)dt lim sup j x(t j ) (t j t j 1 ) = b x(t) dt 3.1. R n D R m φ D (differentible) m n A φ(x) φ() A(x ) φ(x) = φ() + A(x ) + o(x ) lim = 0 x x A φ () φ (differentil) φ φ (derivtive) * (i) φ (φ ()) ij = φ i x j () (ii) φ D φ D Proof. (i) x (ii) U, b U φ(b) φ() = 1 0 φ ( + t(b ))(b )dt φ : D R C r φ r m φ i (i = 1,..., m) C r 3.4 (Chin Rule). D R n E R m φ : D E, ψ : E R l C r ψ φ : D R l C r (ψ φ) () = ψ (φ())φ () S R n Φ : S S 0 < ρ < 1, x, y S, Φ(x) Φ(y) ρ x y *3 (contrction) *2 differentil derivtive Jcobin Jcobi mtrix *3 7

8 3.5. S R n Φ : S S (i) S Φ() = Φ (ii) x S lim k Φ k (x) = Φ k Φ k Proof. Φ k (x) Φ k (y) ρ Φ k 1 (x) Φ k 1 (y) ρ k x y 0 (k ) x S x k = Φ k (x) x k x k+1 ρ k x 0 x 1 x k x l x k x k x l 1 x l ρk ρ l 1 ρ x 0 x 1 (k < l) (x k ) k 0 Cuchy lim k x k x S x S Φ(x ) = Φ(lim x k ) = lim Φ(x k ) = lim x k+1 = x φ R n D C r D (i) φ E = φ(d) R n (ii) φ (iii) φ E φ(x) x D R n C r 4.2. (i) (ii) 4.3. φ : D E T E D S (T f) φ = S(f φ), f C (E) T S 4.4 ( ). D R n φ : D R n D C l (l 1) D φ φ () R n R n n n U (i) V φ(u) 8

9 (ii) φ U ψ (iii) ψ : V U C l Proof. A = φ () φ A 1 φ φ () = I φ (x) det(φ (x)) x r > 0, r () D, x r (), φ (x) φ (x) I 1 2 (1) r () x φ(x) x R n (1) φ(u) φ(v) (u v) 1 0 φ (u + t(v u)) I u v dt 1 u v (2) 2 u v 2 φ(u) φ(v), u, v r (). (3) Φ y : r () x x φ(x) + y R n (y r/2 (φ())) (2) Φ y (u) Φ y (v) 1 2 u v Φ y (u) u + φ() φ(u) + y φ() 1 2 u + y φ() < r 2 + r 2 = r Φ y r () Φ y ( r ()) r () Φ y (x) = x φ(x) = y x r () V = r/2 (φ()), U = {x r (); φ(x) V } φ U V U, V (3) φ : U V ψ Lipshitz ψ C l ψ y 0 V φ(x 0 ) = y 0 x 0 U φ x 0 φ(x) φ(x 0 ) = φ (x 0 )(x x 0 ) + o(x x 0 ). x 0 U r () φ (x 0 ) ψ(y) ψ(y 0 ) = φ (x 0 ) 1 (y y 0 ) φ (x 0 ) 1 (o(ψ(y) ψ(y 0 ))). (3) o(ψ(y) ψ(y 0 )) = o(y y 0 ) φ (x 0 ) 1 (o(ψ(y) ψ(y 0 ))) = o(y y 0 ) ψ y 0 ψ (y 0 ) = φ (x 0 ) 1, φ(x 0 ) = y 0, x 0 U ψ ψ (y) = φ (ψ(y)) 1 *4 ψ C 1 φ (ψ(y)) C l 1 C 1 C 1 ψ (y) = φ (ψ(y)) 1 C 1 ψ C 2 ψ C l *4 φ (x) 1 φ (x) 9

10 4.5. R m, b R n F (x, y) (, b) R m+n N(, b) R n C r F y (, b) Rm R n C r f f() = b, F (x, f(x)) = F (, b) Proof. φ : N(, b) R m+n φ(x, y) = (x, F (x, y)) (, F (, b)) ψ(x, z) = (g(x, z), f(x, z)) g(x, z) = x, F (x, f(x, z)) = z 4.6. n = 1 f(x) = f(x, b) F (x, f(x)) = b ( ) ( ) f F F (x) = (x, f(x)) / (x, f(x)). x i x i y R n U R n V C 1 φ : U V φ 1 φ 1 C 1 φ U x det(φ (x)) R φ (Jcobin) J φ J φ U 5.2. φ : U V f : V R V *5 [f] f(y) dy = f(φ(x)) J φ (x) dx V U n n = 1 n 1 n 5.3. U, U U ψ : U W, W φ (i) φ(x) = φ (ψ (x)), x U, (ii) φ, ψ Proof. det(φ ()) 0 φ 1 (),, φ n () 0 φ i () 0 x 1 x 1 x 1 ψ : U R n ψ(x) = (φ i (x), x 2,, x n ) *5 [f] {x V ; f(x) 0} V 10

11 φ i φ i φ i x 1 x 2 x n det ψ () = = φ i x 1 () 0 U ψ = ψ U W = ψ(u ) W φ = φ ψ 1 (i) (ii) ψ φ φ (φ i (x), x 2,, x n ) = (φ 1 (x), φ 2 (x),, φ n (x)) i 5.4. (ii) Proof. φ : U V φ i (x 1,, x n ) = x 1 U, V U t, V t (t R) U t = {x = (x 2,, x n ) R n 1 ; (t, x 2,, x n ) U} V t = {y = (y 1,, y i 1, y i+1,, y n ); (y 1,, y i 1, t, y i+1,, y n ) V } U t φ t φ t (x 2,, x n ) = (φ 1 (t, x 2,, x n ),, φ i 1 (t, x 2,, x n ), φ i+1 (t, x 2,, x n ),, φ n (t, x 2,, x n )) f t (y ) dy = f t (φ t (x )) det φ t(x ) dx V t U t t R f(y) dy = f(φ(x)) det φ x 1 (x ) dx V U det φ (x) = det φ x 1 (x ) U 5.3 U U 3r () U r > 0 { r ()} U [f φ] = φ 1 ([f]) 1,, N r 1,, r N [f φ] rj ( j ) 1 j N U j = U j, φ j = φ j 5.5. R n 2rj ( j ) h j 0 (j = 1,, N) N h j (x) = 1 j=1 (x [f φ]) 11

12 Proof. g j 1 if x j r j g j (x) = 2 x j /r j if r j x j 2r j 0 otherwise h j (x) = g j(x) j g j(x) 5.5 f(y) dy = f(y)h j (φ 1 (y)) dy = V j V j V j f(y)h j (φ 1 (y)) dy V j = φ(u j ) [h j ] U j 5.4 f(y)h j (φ 1 (y)) dy = f(φ j (z))h j (φ 1 (φ j (z))) det φ j(z) dz V j W j = f(φ j ψ j (x))h j (φ 1 φ j ψ j (x)) det φ j(ψ j (x)) det ψ j(x) dx U j = f(φ(x))h j (x) det φ (x) dx. U j 5.3(i) f(y) dy = f(y)h j (φ 1 (y)) dy = f(φ(x))h j (x) det φ (x) dx V j V j j U j = f(φ(x)) h j (x) det φ (x) dx = f(φ(x)) det φ (x) dx U j U 12

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