i Newton Leibniz Cuchy, Riemnn Weierstrss 2 Grssmnn Poincré Crtn Stokes Newton- Leibniz Tylor Riemnn Riemnn Riemnn Kummer Euler-Mclurin St

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2 i Newton Leibniz Cuchy, Riemnn Weierstrss 2 Grssmnn Poincré Crtn Stokes Newton- Leibniz Tylor Riemnn Riemnn Riemnn Kummer Euler-Mclurin Stirling Fourier Fourier

3 ii Prsevl Lgrnge Jcobin Riemnn Green Guss Stokes Riemnn-Stieltjes Riemnn-Stieltjes Riemnn Green Guss Stokes Gmm Stirling Fourier Plncherel Fourier

4 i Tylor Lgrnge Riemnn Green Guss iii

5 iv Stokes : Riemnn-Stieltjes Riemnn-Stieltjes Stokes Bet Gmm Stirling Fourier

6 ,.,.,.,,. 12.1,..,.,,. R n,. u, u u/ u. u, v u/ u = v/ v ( ). D R n, f : D R D. x D, R n u, lim t [f(x + tu) f(x )]/t, f x u, f u. f u f u. x, ϕ(t) = f(x + tu) t =, f u., u = e i = (,,, 1,,, ) ( i 1 ), f u i. f x i, f f x i f xi. f xi = f ( x i, f yix i = f ) y i x i, ( f xi y i = f ) x i y i, 2.. : 2 f x 2 i = x i ( f x i ), 2 f = ( f ),. x i y i x i y i,,., f(x, y) = xy 1, 2.. f x (x, y) = f x = x (xy) = y, f y = x, f yx = 1, f xy = 1, f xx = f yy = f(x, y) = x 2 + y 2 + xy (x, y ) = (1, 2). 1

7 2. f x (x, y) = x x2 + y + y, 2 f y (x, y) = y x2 + y + x. 2 f 5 (1, 2) = x 5 + 2, f y (1, 2) = (x, y, z ) R 3,., f(x, y, z) = [ (x x ) 2 + (y y ) 2 + (z z ) 2] 1 2. r = [ (x x ) 2 + (y y ) 2 + (z z ) 2] 1 2, f x = 1 r 2 r x = 1 x x r 2 = x x r r 3. f y = y y r 3, f z = z z r 3.,.,, ( )., f : R n R, f... f x = f y =. f(x, y) f(, ) = [f(x, y) f(, y)] + [f(, y) f(, )]. x y, f(x, y) f(, ). f(x, y) f(, ) = f f (ξ, y)(x ) + (, ζ)(y ) =, x y,,, x 2 y f(x, y) = x 4, (x, y) (, ), + y2, (x, y) = (, ).

8 f, f (, ). u = (u 1, u 2 ), u 2, f 1 t 3 u 2 (, ) = lim 1u 2 u t t t 4 u t2 u 2 2 = u2 1 u 2, f (, ). f (, ) ( ).,., ( ). D R n, f : D R, x D. L : R n R, x f(x) f(x ) = L(x x ) + o( x x ), (x x ) f x, L f x, df(x ). f. ( )., f x, f x., ( ). f x, f x,. u., f u (x ) = df(x )(u). (12.1) f(x + tu) f(x ) = L(tu) + o( tu ) = tl(u) + o( t ), (t ). lim[f(x + tu) f(x )]/t = L(u), (12.1). t, f x, f xi (x ). f(x ) = ( f x1 (x ),, f xn (x ) ), (12.2) f x. u = n u i e i = (u 1,, u n ), (12.1) f u (x ) = L ( n ) u i e i = R n,, n u i L(e i ) = n u i f xi (x ). f u (x ) = f(x ) u. (12.3)

9 σ : (, b) R n, σ(t) = ( x 1 (t),, x n (t) ), t (, b). σ(t) t, f x = σ(t ), f σ t, ( f σ ) (t ) = f(x ) σ (t ), (12.4) σ (t ) = ( x 1(t ),, x n(t ) ).., C, t σ(t) σ(t ) C t t. f x = σ(t ) f σ(t) f σ(t ) = f(x ) [σ(t) σ(t ) ] + o ( σ(t) σ(t ) ) = f(x ) [σ(t) σ(t ) ] + o( t t ). t t t t (12.4)., ( ). D R n, f : D R D. x, y D, ξ D, f(x) f(y) = f(ξ) (x y), ξ = θx + (1 θ)y, θ (, 1).. σ(t) = tx + (1 t)y, D t [, 1] σ(t) D. ϕ(t) = f σ(t) θ (, 1), ϕ(1) ϕ() = ϕ (θ). (12.4) ϕ(1) ϕ() = f(ξ) σ (θ) = f(ξ) (x y), ξ = σ(θ) = θx + (1 θ)y. f(x) = ϕ(1), f(y) = ϕ() f x. x, f x.

10 , x = (x, y ) , (x, y) (x, y ), f(x, y) f(x, y ) = f x (ξ, y)(x x ) + f = f (x, y ). [ f x (x, y ) + o(1) y (, ζ)(y y ) ] (x x ) + [ f y (x, y ) + o(1) ] (y y ) = f x (x, y )(x x ) + f y (x, y )(y y ) + o ( (x x, y y ) ), ( ). f : D R, (x, y ) D. f xy f yx (x, y ), f xy (x, y ) = f yx (x, y ).. k, h, θ 1, θ 2 (, 1), ϕ(y) = f(x + h, y) f(x, y), ψ(x) = f(x, y + k) f(x, y ). ϕ(y + k) ϕ(y ) = ϕ y (y + θ 1 k)k, θ 3, θ 4 (, 1), = [ f y (x + h, y + θ 1 k) f y (x, y + θ 1 k) ] k = f xy (x + θ 2 h, y + θ 1 k)kh. ψ(x + h) ψ(x ) = f yx (x + θ 3 h, y + θ 4 k)hk., ϕ(y + k) ϕ(y ) = ψ(x + h) ψ(x ), f xy (x + θ 2 h, y + θ 1 k) = f yx (x + θ 3 h, y + θ 4 k). k, h, f xy, f yx (x, y ).. f : D R. f k, f C k, f C k (D). k 2, C k. k 1, f C k, f, f C (D).

11 xy x2 y 2 f(x, y) = x 2, (x, y) (, ), + y2, (x, y) = (, ). f xy (, ) = 1, f yx (, ) = 1. (, ) f, : u, v, u = v, f u = f v. 2. : (1) f(x, y) = x + y + x 2 + y 2, f x(3, 4), f y(, 1), (2) f(x, y, z) = (cos x/ sin y)e z, (π, π 2, ln 3), (3) f(x, y) = sin(x 2 y), (1, 1). 3. : (1) xy + x y ; x2 (2) tn y ; (3) cos(x2 + y 2 ); (4) ln(x + y x ); 2 (5) x 2 y 3/2 ; (6) e xy+yz+zx ; (7) rctn y x ; (8) xy ; (9) ln(x 1 + x x n ). 4. : (1) x 2 y 3 ; (2) ln(xy); (3) rcsin(x x 2 n); (4) e xy ; (5) tn(rctn x + rctn y); (6) e x2 +xyz. 5. f, f. 6. D R n. f D, f. 7. f : R n R. t f(tx) = t m f(x), f m. : f m mf(x). 8. u = e x cos y, v = e x sin y, u x = v y, u y = v x. n x i f xi (x) = 9. = 2 x y 2, R 2 Lplce, u, v u = u xx + u yy =, v = v xx + v yy =.

12 r = [(x x ) 2 + (y y ) 2 + (z z ) 2 ] 1 2, r 1 =,, R 3 Lplce, u = u xx + u yy + u zz. 11. ( ) f(x, y) x, y,, f ( ), f. 12. ( ), , f xy f yx (x, y ), σ : [α, β] R n, σ R n. σ(t) = (x 1 (t),, x n (t)), t [α, β]. t. x i (t) (1 i n) t = t, σ t, σ (t ) = dσ dt (t ) = dσ = (x dt t=t 1(t ),, x n(t )), σ (t ) σ t. z σ(t) σ(t ) σ(t) σ(t ) 12.1 σ (t ). t t, γ t (t) = σ(t ) + (t t ) σ(t ) σ(t ) t t, t R.

13 8 γ t σ(t ), σ(t ) ( σ ). t t, γ t, γ t (t) = σ(t ) + (t t )σ (t ). γ t σ t., γ t σ(t )., :, t, σ. σ (t ) t. t,,. σ, σ. σ(t ) = σ(t ), t σ(t) σ(t) = o( t t ), (t t ) σ σ t., σ γ t t. : x 1 x 1 (t ) x 1 (t ) = x 2 x 2 (t ) x 2 (t ) = = x n x n (t ) x, n(t ),. σ(t ) (n = 2 ), ( x σ(t ) ) σ (t ) = f, σ(t) = (t, f(t)), (t, f(t)) σ (t ) = (1, f (t )), σ t x t 1 = y f(t ) f, (t ) y = f(t ) + f (t )(x t ),. t >, σ(t) = ( cos t, sin t, t).. t = t, σ (t ) = ( sin t, cos t, 1), x cos t = y sin t = z t, sin t cos t 1 x sin t + y cos t + z t =.. D R 2, ϕ : D R 3. ϕ R 3 ϕ(u, v) = ( x(u, v), y(u, v), z(u, v) ), (u, v) D. t

14 ϕ (u, v ), (u, v ), ϕ(u, v) = ϕ(u, v ) + (u u )ϕ u (u, v ) + (v v )ϕ v (u, v ) + o ( (u u, v v ) ), ϕ u = ( x u, y u, z u ), ϕv = ( x v, y v, z v ). ϕu (u, v ) ϕ(u, v ) (u ) u = u, ϕ u (u, v ) ϕ(u, v) (v ) v = v. ϕ (u, v) = ϕ(u, v ) + (u u )ϕ u (u, v ) + (v v )ϕ v (u, v ), (u, v) R 2. ϕ u (u, v ) ϕ v (u, v ), ϕ ϕ(u, v ), ϕ (u, v )...,.,,.. N = ϕ u (u, v ) ϕ v (u, v ). R 3 N,. n = N/ N. ( (x, y, z) ϕ(u, v ) ) N =, r v n r u 12.3 x x(u, v ) y y(u, v ) z z(u, v ) x u (u, v ) y u (u, v ) z u (u, v ) x v (u, v ) y v (u, v ) z v (u, v ) = S 2 = {(x, y, z) R 3 x 2 + y 2 + z 2 = 1}. n. x = sin θ cos ϕ, y = sin θ sin ϕ, z = cos θ, θ π, ϕ 2π N = (cos θ cos ϕ, cos θ sin ϕ, sin θ) ( sin θ sin ϕ, sin θ cos ϕ, ) = sin θ (x, y, z), n = (x, y, z). (x, y, z ) (x x )x + (y y )y + (z z )z =.

15 1, D R m, ϕ : D R n (m < n). ϕ R n. ϕ(u) = ( x 1 (u 1,, u m ),, x n (u 1,, u m ) ), u = (u 1,, u m ) D. ϕ u, u m ϕ(u) = ϕ(u ) + (u i u i )ϕ ui (u ) + o ( u u ), ϕ ui = ( x 1 u i,, x n u i ), i = 1,, m. ϕ (u) = ϕ(u ) + m (u i u i )ϕ ui (u ), u R m. {ϕ ui (u )} m, ϕ ϕ(u ) m, ϕ u..,. m = n 1, ϕ R n. R n ( ), N = ϕ u1 ϕ un 1, N. ( x ϕ(u ) ) N =. N,. {f i } m {u j } m j=1, (f 1,, f m ) ( (u 1,, u m ) = det fi. (12.5) u j )m m N (N 1,, N n ), N i = ( 1) i 1 (x 1,, x i 1, x i+1, x n ). (u 1,, u n 1 ), x 1 x 1 (u ) x n x n (u ) x 1 u 1 (u x ) n u 1 (u )..... x 1 u n 1 (u x ) n u n 1 (u ) : (1) σ(t) = ( cos t sin t, b sin 2 t, c cos t), t = π 4, (2) σ(t) = (t, t 2, t 3 ), t = t, (3) σ(t) = ( cos t, sin t), t = t. =.

16 : (1) r(u, v) = (u, cos v, sin v), (u, v) = (u, v ), (2) z = x 2 + y 2, (x, y, z) = (1, 2, 5), (3) r(u, v) = ( sin u cos v, b sin u sin v, c cos u), (u, v) = (u, v ). 3. D R n, f : D R. f grph(f) R n+1, grph(f) = {(x, f(x)) x D} R n+1,. 4. x 2 + y 2 + z 2 + w 2 = ( ). D R n, f : D R m, x D. L : R n R m, x f(x) f(x ) = L(x x ) + o( x x ), (x x ) f x, L f x, df(x ). f(x) i (1 i m) f i (x)., f,.,, L m n, Jf(x ). Jf(x ) f x Jcobi, Jcobi. Jf(x ) i f i x f i (x ), ( Jf(x fi ) ) = (x ) x. (12.6) j m n. D R n, x, y D., f f i ξ i D, f i (x) f i (y) = f i (ξ i ) (x y). ξ i f : R R 2, f(t) = (t 2, t 3 ). x = 1, y =, ξ 1 = 1/2, ξ 2 = ±1/ 3, ξ 1 ξ 2., f(x) f(y) = Jf(ξ)(x y) ξ.,

17 ( ). D R n, f : D R m D. x, y D, ξ D, f(x) f(y) Jf(ξ) x y.. f., f(x) f(y). R m u = (u 1,, u m ), g = u f = m u i f i, g D., ξ D, g(x) g(y) = g(ξ) (x y). g(ξ) = m u i f i (ξ). Cuchy-Schwrz g(ξ) m u i f i (ξ) g(x) g(y) = u [f(x) f(y)] ( m u f i (ξ) 2) 1/2 = Jf(ξ). u [f(x) f(y)] g(ξ) x y Jf(ξ) x y. u = [f(x) f(y)]/ f(x) f(y) ( ). D R n R m, f : D R m g : R l, f(d). f x D, g y = f(x ), h = g f x, Jh(x ) = Jg(y ) Jf(x ). (12.7). f x, x f(x) f(x ) = Jf(x )(x x ) + o ( x x ). (12.8) C, f(x) f(x ) C x x. f(x) f(x )., g y,, x x g(y) g(y ) = Jg(y )(y y ) + o ( y y ). (12.9)

18 y = f(x) h(x) h(x ) = Jg(y ) [ f(x) f(x ) ] + o ( f(x) f(x ) ) = [ Jg(y ) Jf(x ) ] (x x ) + Jg(y )o ( x x ) + o ( x x ) = [ Jg(y ) Jf(x ) ] (x x ) + o ( x x ), h x, (12.7). (12.7) h i x j (x ) = m k=1. g i y k (y ) f k x j (x ), i = 1,, l, j = 1,, n. (12.1) f(x, y), ϕ(x), u = f(x, ϕ(x)) x.., u x = f x (x, ϕ(x)) x x + f y (x, ϕ(x)) ϕ(x) = f x (x, ϕ(x)) + f y (x, ϕ(x)) ϕ(x) u = f(x, y), x = r cos θ, y = r sin θ,., ( u x ) 2 + ( u y ) 2 ( u ) 2 1 ( u ) 2. = + r r 2 θ u r = u x x r + u y y r = u u cos θ + x y sin θ u θ = u x x θ + u y y u u = r sin θ + r θ x y cos θ ( u ) 2 1 ( u ) 2 ( u + = r r 2 θ x = ( u x u ) 2 ( cos θ + y sin θ + ) 2 ( u ) 2. + y u u ) 2 sin θ + x y cos θ z = f(u, v, w), v = ϕ(u, s), s = ψ(u, w), z u, z w.., z = f(u, v, w) = f(u, ϕ(u, s), w) = f(u, ϕ(u, ψ(u, w)), w).

19 14, z u = f u + f v v u = f u + f ( ϕ v u + ϕ s s ) u = f u + f v ϕ u + f v ϕ s ψ u z w = f w + f v v w = f w + f v = f w + f v ϕ s ψ w. ( ϕ s s w, ( )..., ( ) ( ).,,., ( ), ; ( ), D R n, f : D R D. D f, x f x f(x). f f. R n {e i } n, f(x) = ( f x 1,, f x n ) (x) = n ) f x i (x)e i. (12.11) R n ( R n), R n R. ( R n ) n, {e j } n j=1, e j : R n R, e j( n ) λ i e i = λ j, λ 1,, λ n R D R n. D x ( R n) ω(x), ω D 1., ω ω(x) = ω j (x) D. n ω j (x)e j, x D, (12.12) j=1, f D, x D df(x) ( R n), 1, df, f.

20 df(x) = n j=1 f x j (x)e j., (12.1), (12.3) df(x)(e j ) = f(x) e j = f x j (x),., f x i, dx i = e i. : df = n j=1 f x j dx j. (12.13), φ D, ω 1, φω 1, x φ(x)ω(x)., f, g, d(f + g) = df + dg, d(fg) = gdf + fdg. g, d(f/g) = (gdf fdg)/g 2..., h = ln., 12.3 dh = d ln dh i = m k=1 g i y k (f)df k, (12.14) z 2 x 2, dh h. + y2 z 2 x 2 + y 2 = d( ln z 2 ln(x 2 + y 2 ) ) = 2 z dz 1 x 2 + y 2 d(x2 + y 2 ) = 2x x 2 + y 2 dx h x = 2x x 2 + y 2, 1. : 2y x 2 + y 2 dy + 2 z dz. h y = 2y x 2 + y 2, h z = 2 z. (1) f(x, y) = xy ; (2) f(x, y) = xy ; (3) f(x, y) = x cos y; xy, (x, y) (, ), (4) f(x, y) = x2 +y2, (x, y) = (, ). x 2 y x (5) f(x, y) = 2 +y, (x, y) (, ), 2, (x, y) = (, ).

21 16 2. : (1) f(x, y) = (xy 2 3x 2, 3x 5y 2 ), (x, y) = (1, 1); (2) f(x, y, z) = (xyz 2 4y 2, 3xy 2 y 2 z), (x, y) = (1, 2, 3); (3) f(r, θ) = (r cos θ, r sin θ), (r, θ) = (r, θ ); (4) f(x, y) = (sin x + cos y, cos(x + y)), (x, y) = (, ). 3. f g Jcobi : (1) f(x, y) = (xy, x 2 y), g(s, t) = (s + t, s 2 t 2 ), (s, t) = (2, 1); (2) f(x, y) = (e x+2y, sin(y + 2x)), g(u, v, w) = (u + 2v 2 + 3w 3, 2v u 2 ), (u, v, w) = (1, 1, 1); (3) f(x, y, z) = (x + y + z, xy, x 2 + y 2 + z 2 ), g(u, v, w) = (e v2 +w 2, sin(uw), uv), (u, v, w) = (2, 1, 3). 4. : (1) z = h(u, x, y), y = g(u, v, x), x = f(u, v), z u, z v ; (2) z = f(u, x, y), x = g(v, w), y = h(u, v), z u, z v, z w. 5. : (1) z = x y ; (2) z = xy x 2 + 2y 2 ; (3) z = x2 y 2 + 3xy 3 2y 4 ; (4) z = x y + y ; x (5) z = cos(x + ln y); (6) z = rctn(x + y); (7) z = x y x + y ; (8) z = ln(x4 y 3 ); (9) z = e x+2y + sin(y + 2x). 6. D R n, f D. df =, f. 7., u(x, y) u = 2 u x u y 2 =, v(x, y) = u(x2 y 2, 2xy) x y w(x, y) = u( x 2 +y, 2 x 2 +y ). 2 8., u(x, t) = 1 2 (x b) 2 πt e 4 2 t u t = 2 2 u x f,,. 1. ( ) f, f x (x, y ), f y (x, y) (x, y ), f (x, y ). 11. ( ) f x f y (x, y ), f xy (x, y ) = f yx (x, y ).

22 12.4 Tylor Tylor,,, f(x, y, z) = x2 y 3 z, x = (1, 1, 1), h = (.3,.2,.6), = f(1.3,.98, 1.6) = f(x + h) f(x ) + Jf(x ) h = 1 + (2, 1.3 2, 1 3 ).2 = ,,.,.. α i Z + (1 i n), α = (α 1,, α n ),. α = n α i, α! = α 1! α 2! α n!. x = (x 1,, x n ) R n, x α = x α 1 1 xα 2 2 xαn n. f, α : D α f(x ) = α 1 x 1 α f α (x ). n x n (Tylor ). D R n, f C m+1 (D), D. x D, θ (, 1) f(x) = m k= α =k D α f() (x ) α + α! α =m+1 D α f( + θ(x )) (x ) α. α!. ϕ(t) = f( + t(x )), t [, 1]. ϕ m + 1, Tylor, θ (, 1), ϕ(1) = ϕ() + ϕ () + 1 2! ϕ () m! ϕ(m) () + 1 (m + 1)! ϕ(m+1) (θ). (12.15)

23 18 ϕ (k) (t) = α =k k! α! Dα f( + t(x ))(x ) α., t =, ϕ (k) () = α =k k! α! Dα f()(x ) α, (12.15). R m = f(x) m k= α =k D α f() (x ) α. α! Tylor, f C m+1 (D) R m = α =m+1 D α f( + θ(x )) (x ) α. (Lgrnge ) α! , x, R m = O ( x m+1).. δ >, B δ () D. M > f C m+1 (D), Bδ (), D α f(x) M, x B δ (), α m + 1. R m M M α =m+1 α =m+1 1 α! (x )α 1 α! x α = C x m+1... (1) f C m (D), Tylor f(x) = = m 1 k= α =k m k= α =k D α f() (x ) α + α! α =m D α f() (x ) α + R m, α! D α f( + θ(x )) (x ) α α! R m = α =m 1 [ D α f( + θ(x )) D α f() ] (x ) α. α!

24 12.4 Tylor 19 : R m = o ( x m), (x ). (Peno ) (2) Tylor f(x) = f() + f() (x ) (x ) 2 f()(x ) +, 2 f = [ 2 f x i x j ] n n, f Hessin, Hess(f). f = tr 2 f, Lplce. (3) Tylor f. Tylor, ( ). D R n, f : D R. x y D, f(tx + (1 t)y) tf(x) + (1 t)f(y), t (, 1), f D. <, f D. (i) f f(y) f(x) + f(x) (y x), x, y D. (12.16) (ii) f C 2 (D), f 2 f ( ).. (i) = x, y D, t (, 1), t [ f(y) f(x) ] f(x + t(y x)) f(x) t, t + (12.16). = f(x) t(y x) + o ( t y x ), = x, y D, t (, 1), z = tx + (1 t)y, f(x) f(z) + f(z) (x z), f(y) f(z) + f(z) (y z). tf(x) + (1 t)f(y) f(z) + f(z) [t(x z) + (1 t)(y z) ] = f(z).

25 2 (ii) = f C 2 (D), Tylor, x, y D, ξ D f(y) = f(x) + f(x) (y x) (y x) 2 f(ξ)(y x). 2 f (12.16), (i) f. = ( ). 2 f(x), R n h, h 2 f(x)h <. Tylor, ε, f(x + εh) = f(x) + f(x) εh ε2 h 2 f(x)h + o ( εh 2) = f(x) + f(x) εh ε2[ h 2 f(x)h + o(1) ]. ε,, (12.16)., Tylor ( ). D R n, f : D R, x D. δ >, f(x ) f(x) ( f(x ) f(x)), x D B δ (x ) \ {x }, x f ( ), f(x ) f ( ). ( ) > ( < ), x, f(x ) ( ). x f, x D, f x, f(x ) =.. u R n, ϕ(t) = f(x + tu). x, t x + tu D. t = ϕ. (12.4) Fermt = ϕ () = f(x ) u, u f(x ) =..,, D R n, f C 2 (D), x f. (i) x f ( ), 2 f(x ) ( ) ; (ii) 2 f(x ) ( ), x f ( ) ; (iii) 2 f(x ), x f , f(x, y) = x 4 + y 4 (x + y) 2.

26 12.4 Tylor 21.. = f x = 4x3 2x 2y, = f y = 4y3 2x 2y. m = (, ), m 1 = (1, 1), m 2 = ( 1, 1)., Hessin : ( ) 12x f(x, y) =. 2 12y 2 2 m 1, m 2, 2 f, m 1, m 2. f(m 1 ) = f(m 2 ) = 2. m 2 f, m. : < x < 1, y = x, x, f(x, x) = 2x 4 4x 2 <, f(x, x) = 2x 4 >, m f(x, y) = x 2 + y 3 3y.. (, 1), (, 1). Hessin : ( ) ( f(, 1) =, 2 f(, 1) = 6 6 ). (, 1), (, 1), f(, 1) = f(x, y) = (y x 2 )(y 2x 2 ).. (, ). (, ), (, ), f f(x, y) = x 2 + y 2 (1 + x) 3.. (, ) f,, f( 2, 3) = 5 < f(, ) =, (, ). : f C 2 (R n ), 2 f I n, I n n, f.. Tylor, f f(x) = f() + f() x x 2 f(ξ)x.

27 22 f(x) f() + f() x x 2, x R n., x, f(x), f. x. x, f(x ) =, f Tylor f(x) = f(x ) (x x ) 2 f(ζ)(x x ), f(x) f(x ) x x 2, x ( ). (x 1, y 1 ),, (x n, y n ) R 2 n, y = x + b, F (, b) = n (x i + b y i ) 2.. F (, b) (, b). {x i }., F (, b) = c 1( n n x i y i c = n n x 2 i ( n n n x i x i ) 2 = 1 2 y i, n y x 2 i 12.5 n n n y i x i y i ), x i (x i x j ) 2 >. (, b) i j F (, b), F,., x y 1 n n x i y i n n n n x 2 i x i y i x i =., f(x, y) = x 2 + y 2 2x y D, D = {(x, y) R 2 x, y, 2x + y 4}.. f D. f D (1, 1/2), f(1, 1/2) = 5/4. : (1) y =, x 2: f(x, ) = x 2 2x, (1, ), f(1, ) = 1. (2) x =, y 4: f(, y) = y 2 y, (, 1/2), f(, 1/2) = 1/4; x

28 12.4 Tylor 23 (3) x, y, 2x + y = 4: f(x, y) = 5x 2 16x + 12, (8/5, 4/5), f(8/5, 4/5) = 4/5., : f(, ) =, f(2, ) =, f(, 4) = 12, f 12, 5/ : (1) , (2) k, x = (x 1, x 2,, x n ) R n, (x 1 + x x n ) k = α =k k! α! xα. 3. f C k, ϕ(t) = f( + t(x )), ϕ (k) (t) = α =k 4. Tylor : k! α! Dα f( + t(x ))(x ) α. (1) f(x, y) = 2x 2 xy y 2 6x 3y + 5, (x, y) = (1, 2); (2) f(x, y, z) = x 3 + y 3 + z 3 3xyz, (x, y, z) = (1, 1, 1); (3) f(x, y) = (1 + x) m (1 + y) n, m, n, (x, y) = (, ); (4) f(x, y) = e x+y, (x, y) = (, ). 5. f m,, f(x) = m k= α =k c α = 1 α! Dα f(), α m. 6. Tylor : (1) f(x, y) = x y, (1, 1) ; c α (x ) α + o ( x m), (2) f(x, y) = cos x cos y, (, ) ; (3) f(x, y) = sin(x 2 + y 2 ), (, ) ; (4) f(x, y) = rctn y 1+x 2, (, ). 7. x 1, x k R n. ϕ(x) = k x x i, ψ(x) = k x x i 2. ϕ, ψ, ψ. 8. f D. : f [ f(x) f(y) ] (x y), x, y D.

29 24 9.,, : (1) f(x, y) = y 2( sin x x/2 ) ; (2) f(x, y) = cos(x + y) + sin(x y); (3) f(x, y) = y x ; (4) f(x, y) = x/y xy; (5) f(x, y) = (x 2 + y 2 )e x2 y 2 ; (6) f(x, y) = ye x2. 1. : (1) f(x, y) = 4(x y) x 2 y 2 ; (2) f(x, y) = x 2 + (y 1) 2 ; (3) f(x, y) = x 2 + xy + y 2 + x y + 1; (4) f(x, y) = x 3 + y 3 3xy. 11. y = x 2 x y 2 =. 12. ( ) < < b, n 1. (, b) n x 1, x 2,, x n,. u = x 1 x 2 x n ( + x 1 )(x 1 + x 2 ) (x n 1 + x n )(x n + b) 13. ( ) D R n, f C 2 (D). f, f D, f :,, A n, A < 1, I n A. u R n, (I n A)u =, u = Au A u, A < 1 u =. A.. v R n, (I n A)x = v, x = Ax + v. ϕ(x) = Ax + v, x, y R n ϕ(x) ϕ(y) = A(x y) A x y, ϕ R n., ϕ, (I n A).,.,.,

30 ,,.. C k C k ( ). D R n, f : D R n C k (k 1), x D. det Jf(x ), x U D y = f(x ) V R n, f U : U V, C k.., x =, y =. L f x =, L, L 1 f x. L 1 f, f., Jf(x ) = I n. x =, f : f(x) = x + g(x), Jg() =. g(x) = f(x) x C k., δ > Jg(x) 1 2, x B δ() D. g(x 1 ) g(x 2 ) 1 2 x 1 x 2, x 1, x 2 B δ (). y B δ/2 (), B δ () ϕ(x) = y g(x). x B δ () f(x) = y, x = y g(x). (12.17) ϕ(x) y + g(x) < δ x δ. (12.18) 2 ϕ ( B δ () ) B δ (). x 1, x 2 B δ () ϕ(x 1 ) ϕ(x 2 ) = g(x 2 ) g(x 1 ) 1 2 x 1 x 2., (12.17) B δ (), x y. (12.18) x y B δ (). U = f 1( B δ/2 () ) B δ (), V = B δ/2 (), f U : U V, h(y) = x y y g(h(y)) = h(y). (12.19)

31 26 (1) h : V U : y 1, y 2 V h(y 1 ) h(y 2 ) y 1 y 2 + g(h(y 1 )) g(h(y 2 )) y 1 y h(y 1) h(y 2 ), h(y 1 ) h(y 2 ) 2 y 1 y 2, y 1, y 2 V. (2) h : V U : y V, y V, h(y) h(y ) = (y y ) [ g(h(y)) g(h(y )) ] = (y y ) Jg(h(y )) (h(y) h(y )) + o ( h(y) h(y ) ). Jf = I n + Jg (1), Jf(h(y )) (h(y) h(y )) = (y y ) + o ( y y ) ), h(y) h(y ) = [ Jf(h(y )) ] 1 (y y ) + o ( y y ). h y. (3) h : V U C k : (2) Jh(y) = [ Jf(h(y)) ] 1, y V. (12.2) f C k Jf C k 1. (2) (12.2) Jh, h C 1. Jf C k 1, h C 1 (12.2) Jh C 1 h C 2., h C k.,, f : R 2 R 2, f(x, y) = (e x cos y, e x sin y). det Jf(x, y) = e x cos y e x sin y e x sin y e x cos y = e2x, f. f,., Jcobi, f : R 2 R 2, f(x, y) = (x 3, y 3 )., f,. f (, ), Jf (, ).

32 f : R 2 R C k (k 1), f y (x, y ), (x, y ) f(x, y) = f(x, y ).., (x, y )., (x, y )., F : R 2 R 2, F (x, y) = (x, f(x, y)). (x, y ), det JF (x, y ) = 1 f x (x, y ) f y (x, y ) = f y (x, y )., (x, y ) F. x x, F 1( x, f(x, y ) ) = ( ϕ(x), ψ(x) ), ϕ(x), ψ(x) C k. F ( ) ( ϕ(x), f(ϕ(x), ψ(x)) = x, f(x, y ) ), ϕ(x) = x, f(x, ψ(x)) = f(x, y ). x f f (x, ψ(x)) + x y (x, ψ(x)) ψ (x) =, [ f ] 1 ψ (x) = y (x, ψ(x)) f (x, ψ(x)). x y = ψ(x) f(x, y) = f(x, y )., ( ) ( ). W R n+m, W (x, y), x = (x 1,, x n ), y = (y 1,, y m ). f : W R m C k, f(x, y) = (f 1 (x, y), f 2 (x, y),, f m (x, y)). ( fi ) (x, y ) W, det J y f(x, y ), J y f(x, y) = y j (x, y). m m x V R n C k ψ : V R m, (1) y = ψ(x ), f(x, ψ(x)) = f(x, y ), x V. (2) Jψ(x) = [ J y f(x, ψ(x)) ] ( ) 1 Jx f(x, ψ(x)), J x f(x, y) = fi x j (x, y) m n.

33 28. F : W R n+m F (x, y) = (x, f(x, y)), (x, y ) F, ,. z y,. y = ψ(x) f(x, y) = f(x, y ) ( ) x 2 + 2y 2 + 3z 2 + xy z 9 =, x = 1, y = 2, z = 1 z x, 2 z x y.. F (x, y, z) = x 2 + 2y 2 + 3z 2 + xy z 9, F (1, 2, 1) =, F z (1, 2, 1) = (6z 1) z=1 = 5, z x, y, z = z(x, y) , (1, 2, 1), (z x, z y ) = F 1 z (F x, F y ) = 1 5 (2x + y, 4y + x) (x,y,z)=(1, 2,1) = (, 7/5). z y = Fz 1 F y = 1 (4y + x), 6z 1 2 z x y = 6 (6z 1) 2 z x(4y + x) 1 6z 1, 2 z x y (1, 2, 1) = (x, y) (r, θ) y (x, y) x = r cos θ, y = r sin θ. r r θ x, y.., r r x x y θ θ = r y x y r ( cos θ r sin θ = sin θ r cos θ ) 1 = 1 r θ x x θ y θ ( ) r cos θ r sin θ, sin θ cos θ r r = cos θ, x θ = sin θ, y x = sin θ r, θ y = cos θ. r

34 F : R 3 R C k (k 1). c R, S = F 1 (c) = {(x, y, z) R 3 F (x, y, z) = c}, S F. (x, y, z) S F (x, y, z), S F. (x, y, z ) S, F z (x, y, z )., (x, y, z ), S z = z(x, y) : (x, y, z ) F (x, y, z(x, y)) = F (x, y, z ) = c. N = ( 1,, z x (x, y ) ) (, 1, z y (x, y ) ) = ( z x (x, y ), z y (x, y ), 1 ) = 1 F z ( Fx (x, y, z ), F y (x, y, z ), F z (x, y, z ) ). F (x, y, z ) F (x, y, z ). S (x, y, z ) S (x, y, z ) F (x, y, z ) (x x, y y, z z ) =. x x F x (x, y, z ) = y y F y (x, y, z ) = z z F z (x, y, z ). F, G : R 3 R C k (k 1), c R. l = F 1 (c) G 1 (c) = {(x, y, z) R 3 F (x, y, z) = G(x, y, z) = c}. (x, y, z) l, (F, G) (y, z) = det F y G y F z G z, l F, G. m = (x, y, z ) l,, m l y = y(x), z = z(x)

35 3 l 12.7, F (x, y(x), z(x)) = G(x, y(x), z(x)) = c, ( y (x) ) ( F y F z ) 1 ( F x ) z (x) = G y G z G x, { Fx + F y y + F z z =, G x + G y y + G z z =. (1, y, z ) F G. m l x x F y F z G y G z = y y F z F x G z G x = z z F x F y G x G y, x x y y z z F x F y F z G x G y G z =.., W R n, Φ : W R m C k (k 1), n > m. c R m, Φ (level set) Σ = Φ 1 (c) = {x W Φ(x) = c}. Σ x, JΦ(x) m, Σ n m., x Σ,, x S n m., Φ = (ϕ 1,, ϕ m ). (ϕ 1,, ϕ m ) (x j1,, x jm ) (x ).

36 , x Σ Ψ(x 1,, x j1,, x jm,, x n ) = (x 1,, x j1,, x jm,, x n ), x ji = ψ i (x 1,, x j1,, x jm,, x n ) (1 i m) C k, Φ Ψ(x 1,, x j1,, x jm,, x n ) = Φ(x 1,, ψ 1,, ψ m,, x n ) = c. x x i JΦ(x )Ψ xi (x ) =, i j 1,, j m. S x ϕ 1 (x ),, ϕ m (x ) A n, A < 1. (I n A) 1 ( 1 A ) A n,, ɛ(a) >, B < ɛ(a), A + B. 3. D R n, f : D R n C k (k 1). det Jf D, f(d) R n. 4. f : R n R n, Jf(x) q < 1, x R n. f. 5., y x : (1) f(x, y) = x 2 y 2, (x, y) = (, ); (2) f(x, y) = sin[π(x + y)] 1, (x, y) = (1/4, 1/4); (3) f(x, y) = xy + ln(xy) 1, (x, y) = (1, 1); (4) f(x, y) = x 5 + y 5 + xy 3, (x, y) = (1, 1). 8. : (1) y z xz 3 z 1 =, y (1, 2, 1); (2) x 2 + 2y 2 + 3z 2 + xy z = 9, 2 z x (1, 2, 1) xv 4y + 2e u + 3 =, 2x z 6u + v cos u = x = 1, y = 1, z = 1, u =, v = 1 u = u(x, y, z), v = v(x, y, z). u, v Jcobi.

37 32 1. : (1) x 3 + 2xy 2 7z 3 + 3y + 1 =, (x, y, z) = (1, 1, 1); (2) (x 2 + y 2 ) 2 + x 2 y 2 + 7xy + 3x + z 4 z = 14, (x, y, z) = (1, 1, 1); (3) sin 2 x + cos(y + z) = 3 4, (x, y, z) = (π/6, π/3, ); (4) x 2 + y 2 = z 2 + sin z, (x, y, z) = (,, ). 11. : (1) x 2 + y 2 + z 2 = 3x, 2x 3y + 5z = 4, (x, y, z) = (1, 1, 1); (2) x 2 + z 2 = 1, y 2 + z 2 = 1, (x, y, z) = (1, 1, 3); (3) 3x 2 + 3y 2 z 2 = 32, x 2 + y 2 = z 2, (x, y, z) = ( 7, 3, 4); (4) x 3 y + y 3 x = 3 x 2 y 2, (x, y) = (1, 1). 12. D R n, f : D R C k (k 1). f f ( ) f 1 (c), f f 1 (c). 13. U, V R n, f : U V, det Jf(x), x U. f,. 14. ( ) f : R n R n, f(x) f(y) x y, x, y R n. f, Lgrnge : A, B, σ C 1,. A, B, σ C, C A B CA + CB. C σ,, C A, B, CA + CB = AB. c > AB, {P R 2 P A + P B = c}, A, B. c, σ. σ C, C., P A + P B C, σ C., U R n, f : U R, Φ : U R m (n > m) C k (k 1). Σ = Φ 1 () = {x U Φ(x) = }, f Σ, Φ(x) =.

38 12.6 Lgrnge 33 x Σ, JΦ(x) m, Σ m., ( ) Σ R n n m, Σ x { ϕ 1 (x),, ϕ m (x)}, ϕ i Φ i. x Σ f, f(x ) Σ x., γ(t) Σ, γ() = x. t = f γ(t), : ( ) () f γ =, f(x ) γ () =. γ () Σ x. γ f(x ) Σ x., λ 1,, λ m R, {λ i } m Lgrnge. f(x ) = m λ i ϕ i (x ). (12.21) (Lgrnge ). f, Φ, x Σ f. JΦ(x ) m, λ R m,. f(x ) λ JΦ(x ) =. (12.22) (ϕ 1,, ϕ m ) (x 1,, x m ) (x )., z = (x m+1,, x n) V C k ψ : V R m y = ψ(z ), Φ ( ψ(z), z ) =, z V. y = (x 1,, x n), y = (x 1,, x m ), z = (x m+1,, x n ), x = (y, z) U. x = (y, z ), Jψ(z ) = (J y Φ) 1 (x ) J z Φ(x ). (12.23) x f, z f ( ψ(z), z ) ( ), z, (12.23), J y f(x ) Jψ(z ) + J z f(x ) =. (12.24) J z f(x ) = J y f(x ) (J y Φ) 1 (x ) J z Φ(x ). (12.25)

39 34 λ = J y f(x ) (J y Φ) 1 (x ), J y f(x ) = λ J y Φ(x ). (12.26) (12.25) λ J z f(x ) = λ J z Φ(x ). (12.27) (12.26) (12.27) (12.22)., (12.22) : x, (x, λ) m F (x, λ) = f(x) λ i ϕ i (x). y (x 1) 2 + y 2 = 1 (, 1),. 1. (x 1) 2 + y 2 1 = d = x 2 + (y 1) 2. 1 x : F (x, y, λ) = x 2 + (y 1) 2 λ[(x 1) 2 + y 2 1], x λ(x 1) = F x = F y = F λ = y 1 λy = (x 1) 2 + y 2 1 = x = = 12.8 λ 1 λ y = 1 1 λ λ = 1 ± 2 d 2 = λ 2, d = λ, d,, 2 + 1, 2 1.? V.,, x, y, z, A = 2(xy + yz + zx), V = xyz. xyz V = A. z y x 12.9

40 12.6 Lgrnge 35 F (x, y, λ) = 2(xy + yz + zx) λ(xyz V ), : F x = F y = F z = F λ = = 2(y + z) λyz = 2(z + x) λzx = 2(x + y) λxy = xyz V = = x = y = z = V 1 3. x, y z, A +, A, 6V α i >, x i >, i = 1,, n. x α 1 1 xαn n x 1 = x 2 = = x n.. ( ) α1+ +α α1 x α n x n n, α α n f(x 1,, x n ) = ln(x α 1 1 xαn n ) = n α i ln x i n α i x i = c (c > ). F (x, λ) = n ( n ) α i ln x i λ α i x i c, : F xi = F λ = = α i x i = λα i, n α i x i c =, = x i = n c, i = 1,, n. α i D : x i, n α i x i = c, f, f D,, ln(x α 1 1 xαn n ) n c α i ln n α, i

41 x α1 1 xα n n ( ) α1 + +α α1 x α n x n n. α α n 1. (, b, c > ): (1) f(x, y) = xy, x + y = 1; (2) f(x, y) = x/ + y/b, x 2 + y 2 = 1; (3) f(x, y) = x 2 + y 2, x/ + y/b = 1; (4) f(x, y) = cos π(x + y), x 2 + y 2 = 1; (5) f(x, y, z) = x 2y + 2z, x 2 + y 2 + z 2 = 1; (6) f(x, y, z) = 3x 2 + 3y 2 + z 2, x + y + z = 1; (7) f(x, y, z) = x y b z c, x, y, z, x + y + z = l. 3. x2 2 + y2 b 2 = x2 2 + y2 b 2 + z2 c 2 = x + 2y + z + 9 =, 2x y 2z 18 =. 6. x 2 + 4y 2 = 4 x + y = n 8. 1, 2,, n, u = i x i. n x 2 i = r 2 9. p > 1, 1 p + 1 q = 1, Hölder n ( n u i v i u p i ) 1 p ( n v q i ) 1 q, u i, v i, i = 1, n. 1. ( ) A = ( ij )n n n, b = (b 1, b 2,, b n ) R n, f(x) = xax + bx = n ij x i x j + i,j=1 (1) f R n ; (2) f n x 2 i = 1. n b i x i, x R n

42 R n n 1 v 1,, v n 1, w, w v i = (v1, i, vn) i (1 i n 1)., L : R n R: x 1 x n v1 1 v 1 n L(x) = v 1 n 1 vn n 1 L, w = (w 1,, w n ). L(x) = x w = n x i w i. (12.28) v1 1 vi 1 1 vi+1 1 v 1 n w i = ( 1) i , v 1 n 1 v n 1 i 1 v n 1 i+1 vn n 1 i = 1,, n. (12.29) L L(v i ) = (1 i n 1). (12.28) w v i. w v 1,, v n 1, : w = v 1 v n 1. {v i } n 1 v1 v n 1 =., {v i } n 1, L(x), w = L = ;, {v i } n 1, R n {u, v i 1 i n 1}. L(u), u w, w. v i, λ, µ R v i, ṽ i, v 1 (λv i +µṽ i ) v n 1 = λv 1 v i v n 1 +µv 1 ṽ i v n 1. i < j, v i v j, v 1 v j v i v n 1 = v 1 v i v j v n 1.

43 38, σ {1,, n 1}, v σ(1) v σ(n 1) = ( 1) σ v 1 v n 1, σ ( 1) σ = 1, σ ( 1) σ = 1. [ v 1 v n 1 = det ( v i vj) ] 1/2. (n 1) (n 1), {v i } n 1 w =, det ( v i v j) (n 1) (n 1) = ; {vi } n 1, w 2 = L(w) v i w = w w ( ) w w w 4 = det v 1. v n 1 v 1. v n 1 = det = w 2 det ( v i v j) (n 1) (n 1), w 2 = det ( v i v j) (n 1) (n 1). ( vi v j) (n 1) (n 1) {e i } n R n, e 1 e i 1 e i+1 e n = ( 1) i 1 e i. D R n 1, f : D R. (f ) ϕ : D R n, ϕ(x) = ( x, f(x) ), x = (x 1,, x n 1 ) D. v i = ϕ xi = (,,, 1,,,, f xi ) ( i 1). {e i } n, vi e i + f xi e n. N : N = v 1 v n 1 = (e 1 + f x1 e n ) (e n 1 + f xn 1 e n ) n 1 = e 1 e n 1 + e 1 f xi e n e n 1 = ( 1) n 1 e n n ( 1) n f xi e i = ( 1) n( f x1,, f xn 1, 1 ). 2 f,.. A = ( ij )n n n, Q : Rn R, Q(x) = x, Ax = n 1 S n 1 = { x R n. n ij x i x j. i,j=1 n x 2 i = 1}, Q S n 1

44 Q(x) S n 1 A.. λ = min S n 1 {Q(x)}, x S n 1, λ = Q(x ). Q(x) λ x 2, x R n., y R n, t R, ϕ(t) = Q(x + ty) λ x + ty 2. ϕ(t) t, t =, ϕ () =. y, Ax + x, Ay λ ( x, y + y, x ) =. A y, Ax λ x =. y = Ax λ x, Ax λ x =. λ A. : V R n, AV V, V A. µ = inf{q(x) x V, x = 1}, µ A. V, V = {y R n y, x =, x V } A ; µ, V (µ) = {x R n Ax = µx}.,, A λ λ 1 λ r, r n. λ i = min{q(x) x V (λ ) V (λ 1 ) V (λ i 1 ), x = 1}, i A n, (1) A ; (2) A.. (1) (2) =.

45 4 (2) = : A λ,, λ r, R n x R n, x, x R n = V (λ ) V (λ 1 ) V (λ r ). x = x + x x r, x i V (λ i ), Q(x) = r λ i x i 2 >. i= A. :,, A A n, ( ) k, V R n, x V (x ), Q(x) >, dim V k.. ( ) dim V > k, A V, dim V = k. P : V V, dim V > dim V, ker P {}. x V (x ) x V , Q(x),. : A = ( ij ) n n det ( ij ) 1 i k 1 j k. = A, ( ) ij 1 i k 1 j k ( ). >, 1 k n.,, = n. n = 1. n 1, n,, ( ) ij 1 i n 1, A R n 1 = {x R n x n = } 1 j n , A n 1. det A >,, A ,.. f i : U R (1 i m) U R n. F : U R m, F (x) = ( f 1 (x),, f m (x) ), x = (x 1, x 2,, x n ) U. V F (U) Φ : V R, Φ(F ) = Φ(f 1, f 2,, f m ), Φ(y), y V,

46 {f i } U.. {f i },, Ψ, {f i }, f j, f j = Ψ(f 1, f 2,, f j 1, f j+1,, f m ), f 1, f 2, f 3, n n f 1 (x) = x k, f 2 (x) = x 2 k, f 3 (x) = x i x j. k=1 k=1 1 i<j n. f 1 (x) 2 = f 2 (x) + 2f 3 (x), {f 1, f 2, f 3 } ( Φ(y) = y1 2 y 2 2y 3 ). x U, {f i }, {f i }.?,. {f i }, Φ(F ) =, Jcobi JΦ JF (x) =, x U. JΦ = Φ, RnkJF (x) < m, x U, Rnk., m n, {f i }. RnkJF (x) = m, x U, {f i } ;, m = n det JF (x) x U, {f i }.. RnkJF (x) = m { f i (x)} m f i (x) = n j=1 ij x j (1 i n) R n, det A = {f i } ; det A {f i }, A = ( ij )n n. : x = ϕ(u), {f i (x)} {f i (ϕ(u))}.,, JF RnkJF (x) l < m, x U.

47 42, {f i } x U., JF (x ) = l, ( ) fi det (x ). x j 1 i l 1 j l G : U R n, G(x 1, x 2,, x n ) = (f 1 (x), f 2 (x),, f l (x), x l+1,, x n ), det JG(x ),,, G x, ϕ = G 1, x = ϕ(u)., F F ϕ(u) = (u 1, u 2,, u l, F l+1 (u),, F n (u)). RnkJ(F ϕ) = RnkJF = l, F i u j, l + 1 i, j n., u = G(x ), F i (l + 1 i n) u 1,, u l., F ϕ u = G(x ), F x. RnkJF = mx RnkJF (x), x U RnkJF F U. JF x U, x JF RnkJF., {f i } F. JF x U l, l x, l.

48 .,,,., R 2. :, Riemnn [, b], [c, d] R, I = [, b] [c, d] R 2, d(i) σ(i) d(i) = (b ) 2 + (d c) 2, σ(i) = (b )(d c). π 1 : = x < x 1 < < x m = b, π 2 : c = y < y 1 < < y n = d, x = x i ( i m) y = y j ( j n) I mn I ij = [x i 1, x i ] [y j 1, y j ], 1 i m, 1 j n. I, π = π 1 π 2. π π = mx d(i ij ). i,j y d π 2 c π 1 b x

49 ( Riemnn ). f : I R I, A, ε >, δ >, π < δ, f(ξ ij )σ(i ij ) A < ε, ξ ij I ij, i,j f f I Riemnn, A f I, A = f = f(x, y) dxdy. I I (1) f(ξ ij )σ(i ij ) f π Riemnn, i,j S(f, π, ξ). f, f = lim S(f, π, ξ). π I (2), f I Riemnn f. f I. f. M ij = sup p I ij f(p), m ij = inf p I ij f(p), I S(π) = S(π, f) = i,j M ij σ(i ij ), s(π) = s(π, f) = i,j m ij σ(i ij ), S(π) s(π) f π Drboux Drboux., ω ij = M ij m ij = sup p I ij f(p) inf p I ij f(p) f I ij. f S(π) s(π) = i,j ω ij σ(i ij ). [, b] π 1 π 1, [c, d] π 2 π 2, [, b] [c, d] π = π 1 π 2 π = π 1 π 2., π π, s(π) s(π ) S(π ) S(π),,.

50 13.1 Riemnn I π 1, π 2, s(π 1 ) S(π 2 ).. π 1 = π 1 π 2, π 2 = π 1 π 2, π = π 1 π 2 = (π 1 π 1) (π 2 π 2), π π 1, π 2, s(π 1 ) s(π) S(π) S(π 2 ),.,. S(f) = inf π S(π), s(f) = sup s(π). π S(f), s(f) f I f(x) = k, I Riemnn kσ(i),.,. k, f + k f, S(f + k) = S(f) + kσ(i), s(f + k) = s(f) + kσ(i) (Drboux). f I, lim S(π) = S(f), lim π s(π) = s(f). π.. f,, f M. ε >, π, S(π ) < S(f) + ε 2. δ >, I ij = [x i 1, x i ] [y j 1, y j ] π, δ, ( ) 13.3 I δ ij = (x i 1 + δ, x i δ) (y j 1 + δ, y j δ),

51 46 J δ = I i,j I δ ij, J δ ( ), σ(j δ ), δ,. δ, σ(j δ ) < ε 2M + 1. π < δ, π,, J δ, π ( ), ( )., π : S(f) S(π) Mσ(J δ ) + S(π ) M ε 2M S(f) + ε 2 < S(f) + ε. lim S(π) = S(f). π.,? Drboux, ( ). f I, : (1) f I Riemnn. (2) f I. (3) lim ω ij σ(i ij ) =. π i,j (4) ε >, I π, S(π) s(π) = i,j ω ij σ(i ij ) < ε., (Riemnn). f I, f ε, η >, I π, σ(i ij ) < ε. {I ij π ω ij η}, f I, f Riemnn.,.

52 13.1 Riemnn ( ). A R 2. ε >, {I i }, A I i, σ(i i ) < ε, i 1 i 1 A., (1) ; (2) ; (3) ; (4) ; (5) φ [, b], grph(φ) = {(x, φ(x)) R 2 x [, b]}..,. (4) δ >, δ δ,,,. δ,,. (5) ε >, φ, [, b] π : = x < x 1 < < x n = b n ω i x i < ε, ω i = M i m i φ [x i 1, x i ], M i m i φ. {(x, φ(x)) x [x i 1, x i ]} [x i 1, x i ] [m i, M i ] = I i, 1 i n. n grph(φ) I i. n n n σ(i i ) = (x i x i 1 )(M i m i ) = ω i x i < ε, grph(φ). y b x 13.4

53 48.,,. f : A R, x A. f x ω(f, x) = lim r + sup{ f(x 1) f(x 2 ) : x 1, x 2 B r (x) A}., f x ω(f, x) =. δ >, D δ = {x A ω(f, x) δ}, f ( ) D f = n=1 D 1 n (Lebesgue). f I. f f D f.. Lebesgue, Drboux, (1) f I, J I, f J ; (2) I {J i }, f J i, f I. ( ),. A R 2, χ A : R 2 R : 1, x A, χ A (x) =, x / A ( ). A, I A. A χ A I, A, σ(a) χ A I. A I. A, A, A A., A, Ā.. I Ā, Ā I., χ A I A. Lebesgue A A. Ā A, A Ā.

54 13.1 Riemnn 49,.,. A, f : A R A, f R 2 f A : f(x), x A, f A (x) =, x R 2 \ A A f, I A. f A I, f A, f A I, f = f A. A I I. A, f : A R A. f f A.. I Ā, Ā I., f A f A I. Lebesgue, f A. I \ Ā f A, f A Ā. A, f A f A A \ A. A \ A f A = f, f A f A \ A, f A , f, g I, fg I. 2., f, g [, b] [c, d], f(x)g(y) [, b] [c, d]. 3. Riemnn. 4. f(x, y) [, b] [c, d], x [, b], f(x, y) y, y [c, d], f(x, y) x. f. 5. f I, f I. 6. f [, b], ϕ : I [, b] I, f ϕ I.

55 5 7. f I, ϕ = (ϕ 1, ϕ 2 ) : J I J, f ϕ J. 8. I, J, I J. χ I J, I. 9. A R. A R R γ : (, b) R 2, γ(t) = (x(t), y(t)). x(t) y(t), γ ( (, b) ) R D(x) [, b] Dirichlet,? 12. I = [, b] [c, d] Dirichlet D(x, y) : x, y D(x, y) = 1, D(x, y) =. D(x, y), Lebesgue n R n, I = [ 1, b 1 ] [ 2, b 2 ] [ n, b n ] n, d(i) v(i) d(i) = (b 1 1 ) (b n n ) 2, v(i) = (b 1 1 )(b 2 2 ) (b n n ). [ i, b i ] (i = 1, 2,, n) π i : i = x i < x i 1 < < x i m i = b i, x i = x i j (i = 1, 2,, n; j =, 1,, m i) I m 1 m 2 m n n I i1 i n = [x 1 i 1 1, x 1 i 1 ] [x n i n 1, x n i n ], 1 i 1 m 1,, 1 i n m n. π = π 1 π n, π. π = mx i 1 i n d(i i1 i n ),

56 (n Riemnn ). f : I R n I. A, ε >, δ >, π < δ, f(ξ i1 i n )v(i i1 i n ) A < ε, ξ i1 i n I i1 i n, i 1 i n f I Riemnn, A f I, A = f = f(x) dx = f(x 1,, x n ) dx 1 dx n. I I I,, R n R 2, A R n, f, g A. fg, f, λf + µg, λ, µ R, f g, A f A A (λf + µg) = λ g., A A f.,. f + µ A, I A, I \ A, v(i \ A) = v(i) v(a). v(a B) = v(a) + v(b) v(a B). A A f. A, B, A B, A B,. I, χ I\A = 1 χ A, I \ A, v(i \ A) = (1 χ A ) = v(i) v(a)., χ A B = χ A χ B A B. I χ A B = χ A + χ B χ A B g. A B, v(a B) = v(a) + v(b) v(a B) {I i }, I. {I i }., I i (i k) I. ( ) v(i) v I i v(i i ), i k i k, I.

57 A ε >, {I i }, A I i, i v(i i ) < ε.. A, A, A A. A A. ε >, {I i }, A I i, v(i i ) < ε. i 1 i i 1 A,, k A I i, v(i i ) < ε. i k i k,, Ā, Ā., A Ā, A. ( ) v(a) v I i v(i i ) < ε, i k i k ε v(a) = ( ). A, f, g A. g A, µ R, fg = µ g, inf f µ sup f. A A. g, fg (inf A A f)g(x) f(x)g(x) (sup f)g(x), x A. A (inf f) g A A A A fg (sup f) g, A A g =, fg A, µ. A g >, A [ ] 1 inf f g fg sup f, A A A A µ. [ µ = A ] 1 g fg, A

58 A, f A. g A, ξ A, fg = f(ξ) g, A A R 2, R n. 2., n R n ; R n. 3. f n I, J I. f J. 4. A, A. 5. A, f, g A, mx{f, g} min{f, g} A. 6. A, B A B, B \ A, v(b \ A) = v(b) v(a). 7. A, Ā v(ā) = v(a). 8. A 1,, A k, A 1 A k, v ( A 1 A k ) = k 1 j 1 < <j i k I ( 1) i 1 v ( A j1 A ji ).. 9. f I, f, f. 1. f > I, f >. 11., f, x + y 1 1 lim r + πr 2 I dxdy 1 + cos 2 x + cos 2 y < 2. x 2 +y 2 r 2 f(x, y) dxdy.

59 f(x, y) I = [, b] [c, d]. x [, b], f(x, y) [c, d] y, [c, d] ϕ(x) ψ(x), [, b] f(x, y) I, ϕ(x) ψ(x) [, b], I f = b ϕ(x) dx = b ψ(x) dx.. π 1, π 2 [, b] [c, d] : π 1 : = x < x 1 < < x m = b, π 2 : c = y < y 1 < < y n = d, I π = π 1 π 2. f I, ε >, δ >, π < δ I f ε < ij f(ξ ij )σ(i ij ) < f + ε, ξ ij = (ξ i, η j ) I ij., π 1 < δ/ 2, π 2 < δ/ 2,. f ε inf f(ξ i, η j ) x i y j I η j [y j 1, y j ] ij sup f(ξ i, η j ) x i y j f + ε, ij η j [y j 1, y j ] I j=1 I n inf f(ξ i, η j ) y j f(ξ i, y) [c, d] Drboux, η j [y j 1, y j ] n inf f(ξ i, η j ) y j ϕ(ξ i ). η j [y j 1, y j ] j=1 n sup j=1 η j [y j 1, y j ] f(ξ i, η j ) y j ψ(ξ i ). m m f ε ϕ(ξ i ) x i ψ(ξ i ) x i f + ε. I I ϕ(x) ψ(x) [, b], f I.

60 f(x, y) I. x [, b], y f(x, y) [c, d], I f = b dx d c f(x, y) dy., y [c, d], x f(x, y) [, b], I f = d c dy b f(x, y) dx f(x, y) I, I f = b dx d c f(x, y) dy =,. d c dy b f(x, y) dx,,., I = [, 1] [, 1], y dxdy. (1 + x 2 + y 2 ) 3 2., y dxdy 1 = dx (1 + x 2 + y 2 ) 3 2 I I 1 1 y dy (1 + x 2 + y 2 ) 3 2 ( 1 = 1 ) dx = ln x 2 x I = [, 1] [, 1], f,. f, f = 1 x y, x + y 1, f(x, y) =, x + y > 1. I = = dx dx 1 1 x I f(x, y) dy (1 x y) dy 1 2 (1 x)2 dx = 1 6.

61 I = [, 1] 3 = [, 1] [, 1] [, 1], dxdydz (1 + x + y + z) 3.., dxdydz 1 (1 + x + y + z) 3 = dx I = = 1 1 I dx dy 1 [ 2 dz (1 + x + y + z) 3 1 (1 + x + y) 2 1 ] (2 + x + y) 2 [( x x = 1 (5 ln 2 3 ln 3). 2 ) ( x x dy )] dx, A R 2, f : A R. A x I = {x R y (x, y) A}. x I, A x = {y R (x, y) A} ( ), f = dx A I f(x, y) dy. A x, A y J = {y R x (x, y) A}. y J, A y = {x R (x, y) A} ( ), f = dy f(x, y) dx. A J A y. A, f, f. A [, b] [c, d], f A [, b] [c, d]. x I, f A (x, y) y [c, d] f(x, y) y A x. x [, b] I c f A (x, y) =., x [, b], f A (x, y) y [c, d], b d f = f A = dx f A (x, y) dy A [,b] [c,d] d = dx y. I c c f A (x, y) dy = dx f(x, y) dy. I A x

62 y A x { A y 2 (x) A { I x x y 1 (x) b f A, f(x, y) y A x,,.. y 1 (x) y 2 (x) [, b], A = {(x, y) R 2 y 1 (x) y y 2 (x), x b},, A. A x,, y 1, y 2 A. f : A R, x [, b], y, f = y2(x) y 1 (x) b dx f(x, y) dy y2(x) A y 1 (x) f(x, y) dy..,., A {(x, y) R 2 x 1 (y) x x 2 (y), c y d}, f = d dy x2 (y) A c x 1 (y) f(x, y) dx. n, ( ),. d A x 1 (y) x 2 (y) y c x 13.6 y

63 I = x 2 y 2 dxdy, A A y =, x = 1 y = x.. x, I = 1 dx x x 2 y 2 dy = x5 dx = I = A y 2 dxdy, y y = x A 1 x x = A 2x y 1 = x = y 2.. 2x y 1 = x = y 2 ( 1 4, 1 ) (1, 1). 2 y, 1 y I = dy y 2 [1 dx = 1 2 y y2 (y + 1) y 4] dy = y 1 x = 1 2 (y + 1) x = y 2 y y = x 2 y = x A x A 1 x I = 1 dy y y sin x x.. sin x y = x x y = x 2, I = 1 dx x x 2 sin x x dy = 1 dx. sin x x (x x2 ) dx = 1 sin 1.

64 y. (x x ) 2 + (y y ) 2 r 2 (x, y ), r. x I = [x r, x + r], x I, 2 r 2 (x x ) 2, v = = 2 x+r x r π 2 π 2 2 r 2 (x x ) 2 dx r 2 cos 2 θ dθ = πr 2. x r r (x, y ) 13.9 x x + r A A = {(x, y) R 2 x, y, x + y } ( > ) xy dxdy.., xy dxdy = dx A x xy dy = 1 2 x( x)2 dx = ( > ) A = {(x, y, z) R 3 x, y, z, x + y + z } xyz dxdydz. A A., xyz dxdydz = = dx y,z, y+z x xyz dydz 1 24 x( x)4 dx = n n () ( > ), z y A 13.1 x n () = {(x 1,, x n ) R n x 1,, x n, x x n }.. n () {x R n x 1, x 2 x 1,, x n x 1 x n 1 }, v( n ()) = dx 1 x1 x1 x n 1 dx 2 dx n.

65 6 y n = x x n,, y 2 = x 1 + x 2, y 1 = x 1, 13.3 v( n ()) = dy 1 y 1 dy 2 y n 1 dy n = dy 1 dy 2 ( y n 1 ) dy n 1 y 1 y n 2 1 = = ( y 1 ) n 1 dy 1 (n 1)! = n n!. 1. : x 2 (1) dxdy, I = [, 1] [, 1]; I 2 y2 (2) sin(x + y) dxdy, I = [, π/2 ] [, π/2 ] ; I (3) sin 2 x dxdy, A y =, x = π/2, y = x ; (4) (5) A A A x 2 dxdy, A x = 2, y = x xy = 1 ; y2 xy 2 dxdy, A y 2 = 2px, x = p/2 (p > ). 2. x =, x = 1, y =, y = 2, z = z = xy x = 1, x = 1, y = 1, y = 2, z = z = x 2 + y x + y + z = ( > ). 5. z = xy, z = x + y, x + y = 1, x =, y =. 6. A = {(x, y, z, w) x, y, z, w, x + y + z + w }, xyzw dxdydzdw. 7. f, g [, b] [c, d], [,b] [c,d] A [ b ][ d f(x)g(y) dxdy = f(x) dx c ] g(y) dy.

66 f(x, y) [, b] [, b], b dx x 9. f [, b], b dx f(x, y) dy = x 1. f [, b], b f(x) dx f(y) dy = x 11. f [, b], b f(x 1 ) dx 1 x1 b b dy b y f(x, y) dx. f(y)(b y) dy. f(y) dy = 1 [ b 2. f(x) dx] 2 xn 1 f(x 2 ) dx 2 f(x n ) dx n = 1 [ b n! f(x) dx] n. 12. n : (1) 1 x1 x 1 dx 1 x 2 dx 2 xn 1 x n dx n ; (2) [,1] n (x x n ) 2 dx 1 dx n. 13. [, 1] [, 1] f(x, y) : 1, x, f(x, y) = 3y 2, x. f, 1 dx 1 f(x, y) dy. 14. [, 1] [, 1] f(x, y) :, x, y, f(x, y) = 1 p, x = r p, y, 1 x, y = s q. q, f,. 15. f [, b], A f = {(x, y) R 2 x b, y f(x)}., A f f, A f f [, b].

67 62 v 3 v 2 v 1 v 2 v ,,., :.?, : { n } P (v 1, v 2,, v n ) = x i v i R n x i 1, i = 1, 2,, n. (13.1) v i (1 i n) R n. n = 2 v 1, v 2, P (v 1, v 2 ).,.? (1). v R n, ϕ : R n R n, ϕ(x) = x + v., A R n, ϕ(a),,. (2). λ i R (1 i n) ϕ : R n R n, ϕ(x 1, x 2,, x n ) = (λ 1 x 1, λ 2 x 2,, λ n x n ), (x 1, x 2,, x n ) R n. I = [ 1, b 1 ] [ n, b n ] ϕ ( ), v(ϕ(i)) = λ 1 λ n v(i) = det(ϕ) v(i). I,, ( ). A R n, ε >, {I i } {J j }, I i A, v(i i ) > v(a) ε; J j A, i i j v(j j ) < v(a) + ε, j

68 {I i }.. A I,, χ A = v(a). I, ε >, I π = {I ij }, χ A (ξ ij )v(i ij ) v(a) < ε, ξ ij I ij. ij, inf χ A (ξ ij )v(i ij ) = v(i ij ), ξ ij I ij ij I ij A 13.12, π v(a) ε < I ij A v(i ij ) v(a).. sup χ A (ξ ij )v(i ij ) = ξ ij I ij ij v(a) I ij A I ij A v(i ij ) < v(a) + ε. v(i ij ),., A 2ε., >, n n (), n () = {x R n x 1,, x n, x x n }..,. ϕ(x) = x v( n ()) = v( n (1)) n., v( n (1)) = v n, v n., v n+1 = 1 v( n (1 x 1 )) dx 1 = v n 1 (1 x 1 ) n dx 1 = 1 n + 1 v n. v 1 = 1 v n = 1 n!, v( n()) = 1 n! n,.

69 x R n, r n B r (x ), B r (x ) = {x R n (x x ) (x x ) r 2 }., x. r n ω n (r). ω n (r) = ω n (1)r n,, ω n = ω n (1)., ω n+1 = 1 1 ω n ( 1 x 2 1 ) dx1 1 = ω n (1 x 2 1) n/2 dx 1 1 = 2ω n π 2, J n : J 2k = ω 1 = 2 cos n+1 t dt = 2ω n J n+1. (2k 1)!! π (2k)!! 2, J 2k+1 = (2k)!! (2k + 1)!!, k. ω 1 = 2, ω 2 = π, ω n+2 = ω n 2π n + 2, n 1. ω 2k = (2π)k (2k)!! = πk k!, ω 2k 1 = 2k π k 1, k 1. (13.2) (2k 1)!! : 2 n n n/2 ω n 2 n. (13.3), [ r, r] n nr, (2r) n ω n ( nr) n ;, [ r, r] n r, ω n r n (2r) n i (1 i n), E( 1,, n ) = { (x 1,, x n ) R n x 2 } x2 n n. x i = i t i (1 i n) ( 1 2 n )ω n, ω n n. (3). ϕ : R n R n, R n {e i } ϕ ϕ(e 1, e 2,, e n ) = (e 1, e 2,, e n )O, O, OO = O O = I n I n..

70 ( ). A R n, ε >, n {B i } {B j }, B i A, v(b i ) > v(a) ε; B j A, v(b j ) < v(a) + ε, i i j j {B i }.. v(a) > , I = [, b] n A I. I m n, m, A {Ii 1} v(ii 1 ) > 1 2 v(a). i I 1 i B 1 i,, v(bi 1 ) = ω n 2 n v(ii 1 ) > ω n v(a). 2n+1 < q < 1, i i q = 1 ω n 2 n+1, ( < v A \ i B 1 i ) < qv(a) A \ i B1 i, A \ i B1 i {Bi 2 }, ( < v A \ Bi 1 \ ) ( Bi 2 < qv A \ ) B i 1 < q 2 v(a). i i i. ε >, q k (k ), k, n {B i }, ( < v A \ ) B i < q k v(a) < n n/2 ε/2. i, Ã = A \ i B i, I m n, m, Ã {I j }, I j B j 2, j v(i j ) < v(ã) + n n/2 ε/2 < n n/2 ε. j v(b j 2 ) = ω nn n/2 2 n v(i j ) < ω nn n/2 2 n n n/2 ε ε. j

71 66 {B i, B j 2 } A n,. v(a) =, n n, ( ).,,.,, P n, E(P ) = {x = (x 1, x 2,, x n ) R n xp x 1}..,, O, P = Odig ( ) λ 1, λ 2,, λ n O, λ i >, 1 i n. y = xo, E(P ) {y R n λ 1 y λ n yn 2 = ydig ( ) λ 1, λ 2,, λ n y 1}, v ( E(P ) ) = ω n (λ 1 ) 1 2 (λn ) 1 2 = ωn (det P ) 1 2., P,, A P (A), v(p (A)) = (det P )v(a),,. (4). ϕ : R n R n, R n n, ϕ. A R n, ϕ(a)., ϕ, ϕ(a) R n,. det ϕ, ϕϕ,,, ϕϕ ϕϕ = P 2,

72 P, det P = det ϕ. O = P 1 ϕ, O,, ϕ(a) = P (O(A)), v(ϕ(a)) = v(p (O(A))) = (det P )v(o(a)) = (det P )v(a), v(ϕ(a)) = det ϕ v(a). (13.4) {v i } n R n, { n P (v 1,, v n ) = x i v i R } n xi 1, i = 1, 2,, n.. R n {e i } n, ϕ : Rn R n, ϕ(e i ) = v i. { n P (v 1,, v n ) = x i v i R } n x i 1, i = 1, 2,, n. = { ϕ ( n = ϕ(p (e 1,, e n )). P (e 1,, e n ) = [, 1] n,, (13.4) ) x i e i R } n x i 1, i = 1, 2,, n. v(p (v 1,, v n )) = det ϕ v([, 1] n ) = det ϕ., n = 2, v 1 = ( 1, b 1 ), v 2 = ( 2, b 2 ), v(p (v 1, v 2 )) = 1 b 2 2 b 1, {v i } n R n, { n (v 1,, v n ) = x i v i R } n x i (1 i n), x x n 1., ϕ : R n R n, ϕ(e i ) = v i, (v 1,, v n ) = ϕ( n (1)), n (1). (13.4) v ( (v 1,, v n ) ) = det ϕ v( n (1)) = 1 det ϕ. n!

73 r >, E r = {(x, y, z) R 3 x 2 + y 2 + z 2 + xy + yz + zx r 2 }.. E r E r = {(x, y, z) R 3 (x + y) 2 + (y + z) 2 + (z + x) 2 2r 2 }, ϕ : R 3 R 3 : ϕ(x, y, z) = (x + y, y + z, z + x), ϕ(e r ) 2r, v(e r ) = det ϕ π( 2r) 3 = 3 4 2πr ,. ϕ : R n R n Lipschitz, ρ, ϕ(x) ϕ(y) ρ x y, x, y R n ϕ Lipschitz, A R n. A, ϕ(a). A, ϕ(a), v(ϕ(a)) ρ n v(a).. A. ε >, B i = B ri (x i ), A B i, v(b i ) = ω n ri n < ε. i 1 i 1 i 1 ϕ Lipschitz ϕ(b i ) B ρri (ϕ(x i )), v( B ρri (ϕ(x i ))) = ω n (ρr i ) n < ρ n ε, i 1 i 1 ϕ(a)., A, ϕ(a) , ε >, A n {Br j j }, ω n rj n < v(a) + ε. j

74 ϕ(a) {B j ρr j }, v(ϕ(a)) j ω n (ρr j ) n = ρ n v(a) + ρ n ε, ε v(ϕ(a)) ρ n v(a).., Lipschitz. D R n, ϕ : D R n C 1., ϕ Lipschitz,. Jϕ, ϕ. A Ā D, ϕ(a)., ϕ(a) = ϕ(a) \ int[ϕ(a)] ϕ(ā) \ ϕ(inta) ϕ( A). A A, ϕ( A), ϕ(a), ϕ(a). ϕ, x D. Jϕ, ε >, δ >, x x < δ Jϕ(x) Jϕ(x ) < ε. L : R n R n, L(x) = ϕ(x ) + Jϕ(x )(x x ), L. R(x) = ϕ(x) L(x), x x < δ JR(x) = Jϕ(x) Jϕ(x ) < ε., R(x) R(y) ε x y, x, y B δ (x )., x, y B δ (x ) L 1 ϕ(x) L 1 ϕ(y) = (x y) (Jϕ) 1 (x ) [ R(x) R(y) ] x y + (Jϕ) 1 (x ) R(x) R(y) ( 1 + (Jϕ) 1 (x ) ε ) x y. A B δ (x ), v ( L 1 ϕ(a) ) ( 1 + (Jϕ) 1 (x ) ε ) n v(a). (13.4) v(ϕ(a)) = det Jϕ(x ) v ( L 1 ϕ(a) ) (13.4). det Jϕ(x ) v(a) + O(ε)v(A).

75 ϕ : D R n C 1, Jϕ. A, Ā D, ϕ(a), v(ϕ(a)) = det Jϕ.. ϕ 1 : ϕ(d) D C 1. A. ϕ 1 v(a) = v ( ϕ 1 (ϕ(a)) ) det(jϕ) 1 (x ) v(ϕ(a)) + O(ε)v(ϕ(A)) det(jϕ) 1 (x ) v(ϕ(a)) + O(ε)v(A), A v(ϕ(a)) det Jϕ(x ) v(a) + O(ε)v(A). v(ϕ(a)) = det Jϕ(x ) v(a) + O(ε)v(A). (13.5) A,, (13.5), ε +, ( ). ϕ : D R n C 1, Jϕ. A, Ā D, f : ϕ(a) R, f = f ϕ det Jϕ. (13.6) ϕ(a)., ϕ(a), (13.6). A,, A. A π = {A ij }, A f ϕ det Jϕ = ij A ij f ϕ det Jϕ = ij f ϕ(ξ ij ) det Jϕ. A ij f ϕ det Jϕ = f ϕ(ξ ij )v(ϕ(a ij )) A ij = f + [f(ϕ(ξ ij )) f]. ϕ(a) ij ϕ(a ij) f ϕ(a) ( ), π, (13.6).. f ϕ(a) Riemnn,.

76 A x =, y = x + y = 1, I = A e x y x+y dxdy.., A {(, )}, A., x y = u, x + y = v, x = (u + v)/2, y = (v u)/2, (x, y) (u, v) Jcobi (x, y) (u, v) = 1/2 1/2 1/2 1/2 = 1/2., D = {(u, v) R 2 u + v, v u, v 1}. I = = 1 2 e u 1 v 1/2 dudv = 2 D 1 1 v dv e u v du v v(e e 1 ) dv = 1 4 (e e 1 ). y 1 v 1 D v + u = v = 1 x + y = 1 A 1 x v u = u 13.14

77 72 y xy = xy = b y 2 = qx A y 2 = px x < p < q, < < b. y 2 = px, y 2 = qx xy =, xy = b A. I = xy dxdy. A.,, y 2 /x = u, xy = v, (u, v) (x, y) Jcobi (u, v) (x, y) = y 2 /x 2 y 2y/x x = 3y2 /x = 3u, (x, y) (u, v) Jcobi (3u) 1., [p, q] [, b], I = q p du b v (3u) 1 dv = 1 6 (b2 2 ) ln q p., R 2 (x, y) (r, θ), x = r cos θ, y = r sin θ, r, θ 2π., Jcobi (x, y) (r, θ) = cos θ r sin θ sin θ r cos θ = r. (r, θ) [, R] [, 2π] (x, y) x 2 + y 2 R 2.,, r =.,

78 (, + ) (, 2π), , D = {(x, y) R 2 x 2 + y 2 R 2 } I = e (x2 +y 2) dxdy. D.., I = R dr 2π e r2 r dθ = 2π R e r2 r dr = π(1 e R2 ) x 2 + y 2 + z 2 2 x 2 + y 2 = x,..,, z = 2 x 2 y 2 D = {(x, y) R 2 y, x 2 + y 2 x}, V x = r cos θ, y = r sin θ V = 4 2 x 2 y 2 dxdy = 4 D π 2 dθ = π 2 cos θ 2 r 2 r dr (1 sin 3 θ) dθ = 4 3 ( π 2 2 3) 3.,. (,, ) (,, ) x2 2 + y2 = 1 (, b > ). b2. x = r cos θ, y = br sin θ, r [, 1], θ [, 2π], Jcobi (x, y) (r, θ) = cos θ b sin θ r sin θ br cos θ = br, 1 dr 2π br dθ = πb.

79 74 R 3. : x = r cos θ, y = r sin θ, z = z, Jcobi r D x 2 + y 2 + z 2 = 4 x 2 + y 2 = 3z. I = z dxdydz.., D D A = {(r, θ, z) r 2 + z 2 4, r 2 3z}, A (r, θ) { r 3, θ 2π}. (r, θ), z [ r 2 /3, 4 r 2 ], I = A rz drdθdz = 2π dθ 3 r dr 4 r 2, R 3 : z dz = π. 3 r2 x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ, r, θ [, π], ϕ [, 2π]. Jcobi sin θ cos ϕ r cos θ cos ϕ r sin θ cos ϕ (x, y, z) (r, θ, ϕ) = sin θ sin ϕ r cos θ sin ϕ r sin θ cos ϕ = r 2 sin θ. cos θ r sin θ D x 2 + y 2 + z 2 = R 2 x 2 + y 2 = z, I = (x 2 + y 2 + z 2 ) dxdydz. D., D A = { (r, θ, ϕ) r R, ϕ 2π, θ π/4 }, I = A r 2 r 2 sin θ drdθdϕ = R r 4 dr π 4 sin θ dθ 2π dϕ = π 5 (2 2)R 5.

80 x2 2 + y2 b 2 + z2 1 (, b, c > ). c2. : x = r sin θ cos ϕ, y = br sin θ sin ϕ, z = cr cos θ, Jcobi bcr 2 sin θ, {(r, θ, ϕ) r [, 1], θ [, π], ϕ [, 2π]}, V = 1 dr π bcr 2 sin θ dθ 2π dϕ = 4 3 πbc.., R n (n 3) x 1 = r cos θ 1, x 2 = r sin θ 1 cos θ 2, x 3 = r sin θ 1 sin θ 2 cos θ 3, x n 1 = r sin θ 1 sin θ 2 sin θ n 2 cos θ n 1, x n = r sin θ 1 sin θ 2 sin θ n 2 sin θ n 1, r R, θ 1,, θ n 2 π, θ n 1 2π, R n ( ). Jcobi :. Jcobi (x 1, x 2,, x n ) (r, θ 1,, θ n 1 ) = rn 1 sin n 2 θ 1 sin n 3 θ 2 sin θ n 2. n, 1 ω n = r n 1 dr sin n 2 θ 1 sin n 3 θ 2 sin θ n 2 dθ 1 dθ 2 dθ n 1, A n A n = {(θ 1,, θ n 1 ) θ 1,, θ n 2 π, θ n 1 2π}, A n sin n 2 θ 1 sin n 3 θ 2 sin θ n 2 dθ 1 dθ 2 dθ n 1 = nω n, (13.7). nω n n 1.

81 B : x 2 + y 2 + z 2 + w 2 1 x 2.., B x 2 dxdydzdw = 1 4 (13.7), 13.4 B x 2 dxdydzdw = 1 4 B (x 2 + y 2 + z 2 + w 2 ) dxdydzdw, 1 r 5 4ω 4 dr = 1 6 ω 4 = 1 12 π : R n, n. 3. A. v(a) >, n {B i }, v(a) = i 1 v(b i ) b 2 2 b 1,. ( 1 x + b 1 y + c 1 ) 2 + ( 2 x + b 2 y + c 2 ) 2 = 1 5. < p < q, < < b, x + y = p, x + y = q y = x, y = bx. 6. n() x 1 x 2 x n dx 1 dx 2 dx n, n () n. 7. ( > ): (1) { x + y }; (2) { x + y + z }, (3) { x x n }. 8. < p < q, < < b, y 2 = px, y 2 = qx x 2 = y, x 2 = by. 9. : (1) (x y) dxdy, A y 2 = 2x, x + y = 4, x + y = 12 ; A (2) (x 2 + y 2 ) dxdy, A x 2 y 2 = 1, x 2 y 2 = 2, xy = 1, xy = 2 ; A (3) (x y 2 ) dxdy, A y = 2, y 2 y x =, y 2 + 2y x =. A

82 (, b > ): (1) (x 2 + y 2 ) dxdy, A = {(x ) 2 + y 2 2 }; A (2) (x 2 + y 2 ) dxdydz, A x 2 + y 2 = 2z z = 2 ; (3) A A A x2 / 2 + y 2 /b 2 dxdy, A = { x 2 / 2 + y 2 /b 2 1 } ; (4) (x + y) dxdy, A = {x 2 + y 2 x + y}; 11. (, b, c > ): (1) ( x 2 / 2 + y 2 /b 2 + z 2 /c 2) 2 = x 2 / 2 + y 2 /b 2 ; (2) (x 2 + y 2 + z 2 ) 2 = z; (3) x 2 / 2 + y 2 /b 2 + z/c = 1, z = ; (4) x 2 + z 2 = 2, x + y =. 12. (r, R,, b, c > ): (1) x2 + y 2 + z 2 dxdydz, A = {x 2 + y 2 + z 2 R 2 }; A (2) (x 2 + y 2 ) dxdydz, A = {r 2 x 2 + y 2 + z 2 R 2, z }; (3) A A 1 ( x 2 / 2 + y 2 /b 2 + z 2 /c 2) dxdydz, A = { x 2 / 2 +y 2 /b 2 +z 2 /c 2 1 }. 13. f(u), 14. f(u), k = 2 + b 2. x + y 1 x 2 +y 2 1 f(x + y) dxdy = f(x + by) dxdy = 2 f(u) du. 1 1 u2 f(ku) du, 15. f(u),, b, c, f(x + by + cz) dxdydz = π 1 1 x 2 +y 2 +z 2 1 k = 2 + b 2 + c 2. (1 u 2 )f(ku) du, 16. A n {v i }, det A = v 1 v n. R n Jcobi.