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3 C, R, Q, Z, N T : T, x T R n n R n := n R = {x = (x, x 2,, x n T ; x j R (j =, 2,..., n} x = (x,..., x n T R n x x ( n x := j= x 2 j /2 a R n, r > a r B(a; r B(a; r := {x R n ; x a < r} B ( \ B : \ B := {x ; x B}. c R n c := R n \. R n,, : := {x R n ; ε > B(x; ε }, := {x R n ; ε > B(x; ε }, := {x R n ; ε > B(x; ε B(x; ε c }. φ: R m φ C k C k 3, 2, 3, 2
6....................... 6.. (?.............................. 6..2 3................................ 7 ( 2. R n................................ 2........................ 2..2................................. 5..3,,,................. 7..4........................... 2..5, Riemann..................... 2.2 Jordan............................. 27.3 Joran........................ 3.3. (............... 3.3.2..................... 3.3.3 Jodan Lebesgue Jordan.... 34.3.4.3.4.............................. 39.3.5 Jordan...................... 43.3.6..................................... 45.4 Fubini.................................... 45.4............................. 45.4.2................................. 47.4.3 Fubini............................ 54.4.4..................................... 57.5................................... 59.5................................. 59.5.2................................. 6.5.3...................................... 62.5.4 :....................... 69.5.5................ 7.6................................ 7.6. Fubini (................... 7.6.2............... 73.7....................................... 73 3
.7.............................. 74.7.2.............................. 75.7.3..................................... 76.7.4............................... 78.7.5 :.................................. 82.7.6.................. 83.7.7.......................... 87.7.8.7.5............................. 9.8 (............................. 9.8.................................... 9.8.2...................... 94.8.3....................... 96.8.4.............................. 97 99 : 5................... 5.............................. 5..2................................. 6..3................................. 7.2.................................... 7.3................................... 9.3.................................... 2 B 4 B.,...................................... 4 B.2..................................... 4 B.3........................................ 5 B.4......................................... 6 B.5................................... 6 C 7 C........................................ 7 C.2 Heine-Borel................................. 7 C.3................... 8 D 2 D................................ 2 D.2 n.......................... 26 E 29 E.................................... 29 E.2........................ 3 4
F 38 F...................... 38 F.2 Darboux............................... 39 F.3................................ 4 F.4.................................... 42 F.5.................................... 43 F.6 Jordan.................. 45 F.7 (................................... 45 5
2 2. ( R n f f(x dx = f(x, x 2,, x n dx dx 2 dx n 2. ( Gauss-Green-Stokes (. 2 n (n.. (? f(x dx f 2 f(x, y dx dy ( f 6
..2 3 3. 2. 3.. ( f : [a, b] R f b a f(x dx = f x = {(x, y; a x b, y f(x}. 2 R 2, f : R, f f(x, y dx dy = f xy = {(x, y, z; (x, y, z f(x, y}. ( n (n + (measure =, = 2, = 3 measure 7
4 R n 2 R n ( ( 2. 3 Riemann ( ( λf(x + µg(x dx = λ (2 ( f g on f(x dx (3 f(x dx f(x dx + µ g(x dx. g(x dx. f(x dx. (4 = 2, 2 = f(x dx = f(x dx + f(x dx. 2 2 fractal 3 Georg Friedrich Bernhard Riemann (826 866, Breselenz Selasca 9 4 ( Riemann Fourier 8
S Σ sum f : [a, b] R f [a, b] a = x < x < < x i < x i+ < < x n = b = {x i } n i= [a, b] [a, b] = {x i} n i= ξ i [x i, x i ] (i =,, n ξ = {ξ i } n i= f Riemann S(, ξ n S(, ξ := f(ξ i (x i x i i= := max i=,,n (x i x i S(, ξ f [a, b] (Riemann f [a, b] (Riemann b f(x dx f(x dx [a,b] a y = f(x a = x ξ x ξ 2 x 2 x n ξ n b = x n : S(, ξ 5 := := 5 lim lim 9
= 2 Jordan (Jordan measurable set := (= Fubini ( f(x, y dx dy = f(x, y dy dx [a,b] [c,d] [a,b] [c,d] ( = f(x, y dx dy. [c,d] [a,b] ( = ( 3. b a d dx x a f(t dt = f(x F (x dx = [F (x] x=b x=a = F (b F (a. ( b a b f (xg(x dx = [f(xg(x] b a f(xg (x dx ( (fg = f g + fg 2 ( a
( β α f(φ(uφ (u du = b a f(x dx, a = φ(α, b = φ(β f(φ(u det φ (u du = f(x dx, D = φ(d
( f : R ( R n f(x dx = f(x, x 2,..., x n dx dx 2 dx n n = 2 f(x, y dx dy (double integral, n = 3 f(x, y, z dx dy dz (triple integral (double integral. R n.. R n R n R n a < b, a 2 < b 2,, a n < b n 2n a i, b i (i =, 2,, n [a, b ] [a 2, b 2 ] [a n, b n ] := {x = (x, x 2,, x n R n ; a i x i b i (i =, 2,, n} R n R = R a < b 2 a, b [a, b] = {x R; a x b}, R 2 a < b, c < d 4 a, b, c, d [a, b] [c, d] = {(x, y R 2 ; a x b, c y d} R n f : R ( f(x dx 2
.. ( = [a, b] a = x < x < < x l = b {x j } l j= {x j } l j= = (i j := [x j, x j ] (j =, 2,, l (ii := max j=,2,,l (x j x j (iii x j..2 = [, ] l x j = j l (j =,,, l := {x j } l j= = l. R n 2 R n n = 2 (n 3..3 (2 = [a, b ] [a 2, b 2 ] R 2 R [a i, b i ] i (i =, 2 = (, 2 a = x < x < < x l = b, a 2 = y < y < < y m = b 2 = {x j } j=,,,l, 2 = {y k } k=,,,m := ({x j } j=,,,l, {y k } k=,,,m (i jk := [x j, x j ] [y k, y k ] (j =, 2,, l, k =, 2,, m (ii := max{, 2 } Jordan Jordan 3
..4 ( Jordan R n n Jordan = [a, b ] [a 2, b 2 ] [a n, b n ] (n-dimensional Jordan measure µ n ( µ( : (b a (b 2 a 2 (b n a n µ n ( = µ( := (b a (b 2 a 2 (b n a n. Jordan (length 2 Jordan (area 3 Jordan (volume Jordan Jordan Jordan..5 ( R n { j } j=,2,,l µ( = l µ( j. j= (..6 ( R n f : R { j } j=,2,,l f L(f,,, f U(f,, : (. L(f,, := l inf f(x µ( j, U(f,, := x j j= l sup f(x µ( j. x j j=..7 ( inf f(x inf{f(x; x S} inf x S ( inf sup f(x = sup{f(x; x S} x S = [, ], f(x = x, = {j/n} N j= U(f,,, L(f,, ( : U(f,, = (N + /(2N, L(f,, = (N /(2N. N /2 4
2 = [, ] [, ], f(x, y = x 2 y 2, = (, 2, = {j/n} N j=, 2 = {j/n} N j= U(f,,, L(f,, inf sup L(f,, U(f,, (..2..8 ( R, def. { } { } { } { } { } j j j..9 = [, ], = ; j =,, 2, = ; j =,,, 4, = ; j =,,, 6 2 4 6,.,., n n = 2.. (2 R 2 = (, 2, = (, 2 def. 2 2.. P( ( 3 5
(i P(. (ii P(, P( = =. (iii P(, P(, P( =...2 ( R, : s.t.. { } = { } { } (,,..3 = {j/2} 2 j=, = {j/3} 3 j= = {, /3, /2, 2/3, }...4 (2 R 2, : s.t.,. (..5 ( R n f : R, L(f,, L(f,,, U(f,, U(f,,. ( B B, B 2,, B l µ(b = µ(b + µ(b 2 + + µ(b l. B B i inf f(x inf f(x (i =, 2,, l x B x B i inf f(xµ(b = inf f(x (µ(b + µ(b 2 + + µ(b l x B x B = inf x B f(xµ(b + inf x B f(xµ(b 2 + + inf x B f(xµ(b l inf x B f(xµ(b + inf x B 2 f(xµ(b 2 + + inf x B l f(xµ(b l. B L(f,, L(f,,. 6
..6 ( R n f : R, L(f,, U(f,,., L(f,, L(f,,, U(f,, U(f,,. L(f,, U(f,,..3,,,..7 ( R n f : R f (, lower integral L(f,, f (, upper integral U(f, : L(f, := U(f, := sup L(f,, P( inf U(f,, (. P( (, P(..8 ( f(x dx, f(x dx (2..6 2, U(f,, L(f,, U(f, L(f,,. U(f, (.2 U(f, L(f,. L(f, 7
..22..9 ( Riemann R n f : R f (, integrable, summable U(f, = L(f, f L(f, f (integral f(x dx, f(x, x 2,, x n dx dx 2 dx n, f..2 ( b a f(x dx := f(x dx (a < b [a,b] (a = b f(x dx (a > b [b,a] (2 n 2 (n (double integral (3 = [a, b] [c, d] f(x dx f(x, y dx dy, f(x, y dx dy, [a,b] [c,d] a x b c y d f(x, y dx dy (4 Riemann Lebesgue 2 3 (5 f f(x dx Riemann (Georg Friedrich Bernhard Riemann, 826 866, Breselenz Selasca (854, Habilitation ( 2 Henri Léon Lebesgue (875 94, Beauvais Paris (92, (94 Lebesgue 8
(6 f f is integrable (summable on f f f (U(f,.. (Lebesgue Lebesgue ( Lebesgue Fourier Fourier Fourier Lebesgue [2], [] [2] ( [34] 3 [3]..2 ( f : R f(x c f f(x dx = c µ(. { j } j=,2,,l j sup f(x = inf f(x = c x j x j L(f,, = U(f,, = l l cµ( j = c µ( j = c µ(. j= j= L(f, = U(f, = c µ( f f(x dx = c µ(...22 ( (Dirichlet = [, ], f : R { (x Q f(x = (x \ Q L(f,, =, U(f,, = L(f, =, U(f, =. f 3 [3] Lebesgue 9
..4?..23 ( R n f : R f ( ε > ( : s.t. U(f,, L(f,, ε. P( U(f, L(f, = inf U(f,, sup L(f,, P( P( = inf, P( (U(f,, L(f,, f U(f, = L(f, ε >, P( s.t. U(f,, L(f,, < ε. ε > P( s.t. U(f,, L(f,, < ε ( = ( = 4 (.3..24 (, Cauchy (823, Heine (874 R n f : R f R n ( f (a f ( (b f 5 ε >, δ >, x, x x x δ = f(x f(x ε/µ(. 4 ( Jordan f f f 5 C 2
< δ/ n j (j =, 2,, l x, x j x x δ f(x f(x ε/µ( µ( j sup x j f(x inf x j f(x ε µ(. l sup f(xµ( j x j j= l j= inf f(xµ( j x j ε µ( l µ( j. j= U(f,, L(f,, ε. f..5, Riemann.. Riemann Riemann 6..25 ( R n f : R, g : R ( (4 ( αf + βg (αf(x + βg(x dx = α (2 f g f(x dx (3 f f(x dx f(x dx + β g(x dx. f(x dx. g(x dx (4 2, 2 f, 2 f(x dx = f(x dx + f(x dx. 2 6 2
(3 Riemann ( ( ε >, 2, 3, 4 P( s.t. L(f, ε L(f,,, 2 P( s.t. L(g, ε L(g,, 2, 3 P( s.t. U(f,, 3 U(f, + ε, 4 P( s.t. U(g,, 4 U(g, + ε. L(f, ε L(f,, L(f,,, L(g, ε L(g,, 2 L(g,,, U(f,, U(f,, 3 U(f, + ε, U(g,, U(f,, 4 U(g, + ε. L(f, +L(g, 2ε L(f,, +L(g,,, U(f,, +U(g,, U(f, +U(g, +2ε. inf f + inf g inf(f + g, sup(f + g sup f + sup g L(f,, + L(g,, L(f + g,,, U(f + g,, U(f,, + U(g,,. L(f + g,, U(f + g,,. L(f, + L(g, 2ε L(f + g,, U(f + g,, U(f, + U(g, + 2ε. f g f(x dx + g(x dx 2ε L(f + g,, U(f + g,, f(x dx + g(x dx. U(f + g,, L(f + g,, 4ε. f + g L(f + g,, L(f + g, = (f + g(xdx = U(f + g, U(f + g,, ( (f + g(xdx f(x dx + g(x dx 4ε. 22
(2 P( U(f,, U(g,,, L(f,, L(g,, U(f, U(g,, L(f, L(g,. f g f(x dx = U(f, U(g, = g(x dx. (3 f ε > U(f,, L(f,, ε { j } l j= j sup f(x inf f(x sup f(x inf f(x x j x j x j x j µ n ( j j U( f,, L( f,, U(f,, L(f,,. U( f,, L( f,, ε...23 f x f(x f(x f(x f(x dx f(x dx f(x dx. f(x dx f(x dx. (4 n = = [a, b] ε > = {x j } l j= s.t. U(f,, L(f,, ε. = [a, c], 2 = [c, b] x j c U(f,, L(f,, ε. c, c 2, 2 U(f,, = U(f,, + U(f,, 2, L(f,, = L(f,, + L(f,, 2. U(f,, + U(f,, 2 L(f,, + L(f,, 2 ε. 23
[U(f,, L(f,, ] + [U(f, 2, 2 L(f, 2, 2 ] ε U(f,, L(f,, ε, U(f, 2, 2 L(f, 2, 2 ε. f 2 Riemann..26 (Riemann R n f : R { j } l j= {ξ j} l j= ξ j (j =, 2,, l j S(f,,, {ξ j } := l f(ξ j µ( j j= f (, {ξ j } Riemann..27 (Riemann, f, {ξ j } l j= ( ξ j j inf f(x f(ξ j sup f(x (j =, 2,..., l x j x j µ( j L(f,, S(f,,, {ξ j } U(f,, f,, U(f,, Riemann (.3 ( ε > ( {ξ j } l j= U(f,, S(f,,, {ξ j } < ε. j {,..., l} ξ j U(f,, S(f,,, {ξ j } = sup x j f(x f(ξ j j= j= ε µ( l l sup f(xµ( j f(ξ j µ( j x j j= ( l l ε sup f(x f(ξ j µ( j x j µ( µ( j = ε. L(f,, Riemann 24 j=
Riemann ( (ii..28 R n f : R 2 (i f (ii Riemann ( S R ( ε > ( δ > ( :, < δ ( {ξ j } l j=: ξ j j (j =, 2,, l. { j } S = f(x dx S S(f,,, {ξ j } < ε. Darboux 7..29 (Darboux, 875 R n f : R ε >, δ > s.t. ( :, δ U(f,, U(f, ε, L(f,, L(f, ε. lim U(f,, = U(f,, lim ( L(f,, = L(f, n = n ( 8 ( M := sup f, µ := inf f M = µ ( f M > µ ( U(f, ε > ε = {z, z,, z l } (.4 U(f, U(f,, ε U(f, + ε 2. 9 7 Jean Gaston Darboux (842 97, Nimes Paris. Riemann Darboux (875 8 F.2 9 j, k z k z k η x j x j z k, z k [x j, x j ] 25
(2 η := min j l (z j z j < η = {x, x,, x k } [x j, x j ] ε z k z k η > x j x j z k, z k [x j, x j ] (3 ε ε (.5 U(f,, ε U(f,, ε. I j = [x j, x j ] (a ε I j I j ε (b ε z k I j I j ε U(f,, U(f,, ε U(f,, U(f,, ε = I j,l = [x j, z k ], I j,r = [z k, x j ] sup f(x(x j x j x I j sup f(x(z k x j + sup f(x(x j z k x I j,l x I j,r = k: j s.t. z k I j k: j s.t. z k I j k: j s.t. z k I j [ (sup f sup I j I j I j,l f(z k x j + (sup (M µ(z k x j + x j z k (M µ(x j x j. f sup I j,r f(x j z k x j x j z k I j ε z k l U(f,, U(f,, ε l(m µ. δ < δ { } ε δ := min η, 2l(M µ (.6 U(f,, U(f,, ε ε/2. 26 ]
U(f,, U(f, = ( U(f,, U(f,, ( ε + U(f,, ε U(f,, ε + (U(f,, ε U(f, < ε 2 + + ε 2 = ε. lim U(f,, = U(f, ε..28 (i= (ii L(f,, S(f,,, {ξ j } U(f,, Darboux (i f(xdx Riemann f(x dx (ii= (i ε > δ > < δ S U(f,, ε, S L(f,, ε (U(f,,, L(f,, Riemann U(f,, L(f,, 2ε...23 f.2 Jordan R 3 R 2 R n Jordan ( (characteristic function χ (, χ (x := { (x (x R n \ χ Jordan Camille Jordan (838 922. 27
.2. (Jordan R n n Jordan (n-dimensional Jordan measurable χ (x dx ( (2 χ χ (x := { (x (x µ n ( = µ( := χ (x dx n Jordan (n-dimensional Jordan measure.2.2 (Jordan..22 f := [, ] Q f [, ] Jordan.2.3 ( well-defined B B χ χ B χ (x dx = ( B B χ (x dx. χ (x dx.2.4 n Jordan.2. [a, b ] [a n, b n ] (, (2 ( n Jordan (2 χ (x dx = n (b j a j. j= (, (2.2.5 (Jordan R n Jordan χ U(χ, = L(χ,. 28
U(χ, = inf U(χ,,, L(χ, = sup L(χ, j (j =, 2,, l U(χ,, = µ( j = j Jordan, j L(χ,, = j µ( j = j Jordan Jordan ( U(χ, Jordan, L(χ, Jordan.2.6 (Jordan, Dirichlet (839 R n Jordan f : R f ( R n { f(x (x f(x := (x \ f f f f(x dx f(x dx := f(x dx.2.7 R n Jordan Jordan µ( dx ( dx 29
.2.8 ( R n Jordan f : R, g : R (, (2, (3 ( α, β R αf + βg (αf(x + βg(x dx = α f(x dx + β g(x dx (2 f g f(x dx g(x dx. (3 f f(x dx f(x dx...25..25 (4 f(x dx = 2 f(x dx + 2 f(x dx f(x dx 2.3..3 Joran.3. ( f f(x dx (f R n f : R ( Jordan ( f dx Jordan (.3.5 Jordan ( Jordan R 2 R 3 Jordan 3
R n Jordan f(x dx ( f Lebesgue (.3.4 Lebesgue ( f Jordan f : R f(x dx.3. (.3.2 Jordan Jordan Lebesgue.3. (Jordan Lebesgue N R n ( N Jordan ε > {B j } m j= B j R n, m N B j, j= m µ n (B j < ε j= (2 N Lebesgue ε > {B j } j N B j R n, N B j, j= µ n (B j < ε j= Jordan ( Lebesgue 3
( {a} (2 {a n } l n= (3 (, (4 (, ( Jordan.3.2 Jordan Lebesgue ( µ n (B j < ε/2 {B j } j B j 2 /n Jordan Lebesgue ( Lebesgue Jordan (.3.3 (Jordan = Jordan N R n 2 (i N Jordan (ii N Jordan µ n (N =. N (ii χ N χ N (x dx = U(χ N, = L(χ N, = U(χ N, =. U(χ N, L(χ N, χ N L(χ N, U(χ N, = L(χ N, = [(ii= (i ] (ii U(χ N, = ( ε > ( P( U(χ N,, < ε. { j } l j= U(χ N,, = l sup χ N (xµ n ( j = x j j= j N µ n ( j. l N = j (N j j= N j, µ n ( j = U(χ N,, < ε. j N j N 32
j N j {B j } m j= [(i= (ii ] (i ε > m m N B j, µ n (B j < ε j= j= {B j } m j= { j } l j= B j ( n = = [a, b], B j = [α j, β j ] {a, b, α, β,..., α m, β m } a = a < a < < a l = b := {a j } l j= n 2... U(χ N,, = l sup χ N (xµ n ( j = x j j= U(χ N, = j µ n ( j m µ n (B j < ε.3. (Lebesgue Lebesgue Lebesgue Lebesgue R n N N Lebesgue N Lebesgue N Lebesgue Lebesgue Henri Léon Lebesgue (875 94, Beauvais Paris.3.4 (Lebesgue, 92 R n Jordan f : R 2 (i f (ii f Lebesgue Lebesgue Riemann Lebesgue Jordan.3.5 (Jordan R n 3 (i Jordan (ii Lebesgue (iii Jordan j= : = {x R n ; ε > B(x; ε B(x; ε c } 33
χ : R n R (i (ii Jordan Lebesgue (iii = (ii R n (ii = (iii Jordan Lebesgue Jordan.3.6 ( Jordan K R n f : K R f graph f := {(x, f(x; x K} R n+ Jordan ( K = [a, b] ( K f K K ε > δ > ( x K( x K : x x < δ f(x f(x ε µ n (. δ/ n K { j } m j= x K x K j x x n < δ f(x f(x ε/µ n (. ε/µ n ( R I j {(x, f(x; x K j } B j, B j := j I j µ n+ (B j = µ n ( j ε/µ n ( m µ n+ (B j = j= m ε µ n ( j µ n ( ε. j= graph f m B j graph f Jordan j= ( 2 Jordan Jordan.3.3 Jodan Lebesgue Jordan 2 34
Jordan.3.7 R n (, (2, (3 ( Jordan Jordan (2 2 Jordan Jordan (3 Jordan.3.8 n 2, φ: [a, b] R n Lipschitz N := φ([a, b] R n Jordan L R t, s [a, b] φ(t φ(s L t s N [a, b] N I,..., I N ( k {,..., N} ( t, s I k φ(t φ(s L(b a N. φ(i k B(φ(t k ; r, r := M(b a/n 2r n N N k φ(i k φ([a, b] = φ(i k k k= k= ( N µ k k= N µ( k = N (2r n = k= [2M(b a]n N n. n 2 N.3.9 φ C Lipschitz C ( Jordan Jordan n 2 n = Jordan ( φ φ([a, b] Jordan φ: [, ] R 2 φ([, ] = [, ] [, ] ( Peano (89, Hilbert (89 Koch Jordan ( 35
.: Koch Jordan.3., 2 R n Jordan 2, 2, \ 2 Jordan µ n ( 2 = µ n ( + µ n ( 2 µ n ( 2. (.3.5 U := 2, V := 2, W := \ 2 U, V, W 2 2 Jordan 2 Jordan U, V, W Jordan U, V, W Jordan 2 χ 2 = χ + χ 2 χ 2 χ 2 (x dx = χ (x dx + χ 2 (x dx χ 2 (x dx. µ n ( 2 = µ n ( + µ n ( 2 µ n ( 2. 36
.3., 2 R n Jordan f : 2 R (i, (ii (i f, 2 (ii f 2 f(x dx = 2 f(x dx + f(x dx 2 f(x dx. 2 2 Jordan f(x dx = 2 f(x dx + f(x dx. 2 (i (ii f Jordan Jordan Jordan Jordan 2 χ 2 (x = χ (x + χ 2 (x χ 2 (x (x f f(x f(x dx = 2 f(x dx + f(x dx 2 f(x dx 2.3.2 R n Jordan Jordan µ n ( = µ n ( = µ n (. ( = \ ( = \ ( \ =. Jordan Jordan Jordan Jordan ( = \ ( = \ \ =. Jordan Jordan µ n ( = µ n ( = µn (. := [, ] Q Jordan = [, ], = Jordan 37
Lebesgue Lebesgue Lebesgue.3.3 (Jordan Lebesgue ( R n Jordan Lebesgue (2 R n Lebesgue Jordan ( (2 Jordan, Lebesgue ( Q Lebesgue Jordan (.3.4 R n ( (3 ( Lebesgue Lebesgue (2 Lebesgue (3 Lebesgue Q R Lebesgue (, (2 (3 n = N {x j } j N [ B j := x j ε 2, x j+ j + ε ] 2 j+ x j B j N B j µ n (B j = ε 2 j j= µ n (B j = j= j= ε 2 j = ε. n x j B j µ n (B j = ε/2 j B j (3 ( Q (.3.5 (Lebesgue Lebesgue R n Lebesgue Lebesgue N = N j j N N j Lebesgue j= 38
ε > j N N j Lebesgue N j k= B (j k, k= ( µ B (j k ε 2 j {B (j k } k= B(j k (j, k =, 2,... ( 3 N j,k= ( µ B (j k j,k= B (j k = j= k= µ(b (j k j= ε 2 j = ε.. N Lebesgue.3.4.3.4.3.4.3.6 ( Lebesgue R n f : R B f B := {x ; f x } f B Lebesgue. ( ( ε > B Lebesgue R n {U i } i N : B i= U i i= µ n (U i ε 4M, M := sup f(x. x \ B f x R n V x x V x, sup f(y inf f(y < y V x y V x {U i } i= {V x } x \B x ε 2µ n ( {W j } k j= ( R n 4 3 Q 4 C.2 C.2. 39
,..., l W j (.3.7 := U i, 2 := V x 2 [ ] sup f(y inf f(y µ n ( j 2M µ n ( j 2M y j j y j j [ ] ε sup f(y inf f(y µ n ( j µ n ( j y j y j 2µ n ( j 2 j 2 U(f,, L(f,, = ε 4M = ε 2, ε 2µ n ( µ n( = ε 2. [ ] l sup f(y inf f(y µ n ( j ε. y j y j j= f ( f a a f o(f, a ( o(f, a := lim δ sup f(x inf f(x x B(a;δ x B(a;δ f a o(f, a > B m := { x ; o(f, x } m (m N B = m= B m B Lebesgue B m Lebesgue f U(f,, L(f,, < ε m { j } l j= := { j ; j B m } 4
j x j B m. δ > s.t. B(x; δ j. m sup f(y inf f(y y j y j ( µ n ( j j j ( l j= sup f(y y B(x;δ sup f(y inf f(y y j y j sup f(y inf f(y y j y j j inf y B(x;δ f(y o(f, x m. µ n ( j µ n ( j ε. µ n ( j = U(f,, L(f,, < ε m. l i Jordan U,, U k i= l k i U j, i= j= k µ n (U j < ε j= {U j } k j= B m (B m j j j µ n ( j + k µ n (U j < 2ε. j= B m Lebesgue 5 6.3.7 (Lebesgue R n {J i } m i= Rn j J i i {,..., m} f i : R n R f i (x := inf x y (x R n y Ji c x J i f i (x > 5 B m Jordan B Jordan Lebesgue 6 [7] Lebesgue 4
f(x := max{f (x,..., f m (x} (x R n f : R n R f > ( x j {,..., m} s.t. x J i. f j (x > f(x f j (x >. r := min f(x r > < r x n j j x f(x = f i (x i inf x y = f i (x = f(x r > ( j y Ji c j J i.3.5 χ.3.8 ( = R n (a x = {x ; χ x }. x U U χ = on U. x χ (b x x U R n \ U χ = on U. x χ (c x x x U U y U χ (y =. y 2 U (R n \ χ (y 2 =. χ x B χ.3.4 42
.3.4 f f : R f, f D, D D D (D ( Jordan Lebesgue (.3.5 Lebesgue (.3.4 f f D Lebesgue D Lebesgue.3.5 Jordan (.3.9 N R n Jordan N Jordan N Jordan ε > B,..., B m m m N B j, µ n (B j < ε j= m j= B j N j= m B j = j= N Jordan m B j..3.2 Lebesgue = [, ] Q Lebesgue = [, ] Lebesgue.3.2 R n Jordan f : R f(x dx =. f : R n R f(x = { j= f(x (x (x 43
f ( x x V f(x = (x V f x f Jordan Joradn (.3.9 f f(x dx f(x dx = f(x dx sup f(x x dx = sup f(x µ n ( = x.3.22 (, B R n Jordan Jordan µ n (( \ B (B \ = f : R, g : B R B (, (2, (3 ( µ n ( = µ n (B. (2 f g B. (3 f ( g B g(x dx. B f(x dx = C := B, := \ B, B := B \ C Jordan B Jordan = C, C =, B = C B, C B = µ n ( = µ n (C + µ n ( = µ n (C + = µ n (C, µ n (B = µ n (C + µ n (B = µ n (C + = µ n (C µ n ( = µ n (B. f(x dx = f(x dx + f(x dx = C g(x dx = g(x dx + g(x dx = B C B C = B f = g f(x dx = C C C C f(x dx + = f(x dx, C g(x dx + = g(x dx g(x dx. C f(x dx = g(x dx. B 44
.3.23 R n Jordan B R n Jordan f : B R f, B, \ B f(x dx = f(x dx = f(x dx. B \B.3.6 (.7 B = B, B B, (.8 ( B B, B = B ( B, ( B (.9 ( B B, ( B B (.7, (.8, (.9.4 Fubini.4. Fubini Fubini = {(i, j Z 2 ; i, j }. (i, j i 2 j 45
( i 2 j = i 2 2 (2 + j = i = 2 6 (i,j i= j= i= a i b c j d a ij = ( b d a ij i=a j=c 55 = 275. ( (Fubini Fubini 2 = [a, b] [c, d] f : (x, y f(x, y R, f (on f f(x, y dx dy {(x, y, z R 3 ; (x, y, z f(x, y} f(x, y dx dy = = b a d c ( d c ( b a f(x, y dy dx f(x, y dx dy n > 2 f(x, x 2,, x n dx dx 2 dx n = [a,b ] [a 2,b 2 ] [a n,b n] b ( b2 ( bn a a 2 a n f(x, x 2,, x n dx n dx 2 dx (,, repeated integral 46
z y d.2:.4.2 c a d c x b f(x, y dy x [a, b] Fubini (.4.3.4. (, Stolz (886, Fubini R n 2 R n 2 f : 2 (x, y R 2 ( f := 2 (2 x f(x, : 2 y f(x, y R 2 F (x := f(x, y dy 2 (x f(x, y dx dy = 2 F (x dx f(x, y dx dy = 2 n = n 2 = ( f(x, y dy dx. 2 47
.4.2 = [a, b] [c, d] f : R 2 ( f (2 x [a, b] f(x, : [c, d] y f(x, y R [c, d] [a, b] F (x := d c f(x, y dy f(x, y dx dy = f(x, y dx dy = b a b a ( d c (x [a, b] F (x dx f(x, y dy dx..4.3 ( b ( d f(x, y dy dx a b a c dx d c f(x, y dy b d a c f(x, y dy dx f(x, y dx dy [a,b] [c,d].4.4 (a (, (2 ( ( (2 ( [7] 3- (b f (, (2.4.5 (c Fubini Otto Stolz (842 95, ustria Hall ustria Innsburck 886 Fubini 48
.4.5 ( ( = [a, b] [c, d], f : R F (x := [c, d] f(x, y dx dy = d c b a f(x, y dy ( d.4.6 = [, ] [, ], f(x, y = x 2 y c f(x, y dy dx. f(x, y dx dy. f.4.7 = f(x, y dx dy = ( x 2 y dy dx = [, π ] [, π], f(x, y = sin(x + y 2 ] y= [x 2 y2 dx = 2 y= 2 x 2 dx = 6. f(x, y dx dy. f(x, y dx dy = = π/2 π/2 ( π = 2 [sin x] π/2 = 2. sin(x + y dy dx = π/2 ( cos(x + π + cos xdx = 2 [ cos(y + x] y=π π/2 cos x dx y= dx.4.8 ( Fubini D R n Jordan φ φ 2 D φ φ 2 (. := {(x, y; x D, φ (x y φ 2 (x} (, (2 ( R n+ Jordan (2 f : R ( φ2 (x f(x, y dx dy = f(x, y dy dx. D φ (x 49
2.4.9 φ (x φ 2 (x (x [a, b] φ, φ 2 : [a, b] R := {(x, y; x [a, b], φ (x y φ 2 (x} Jordan f : R ( b φ2 (x f(x, y dx dy = f(x, y dy dx. a φ (x.4. ψ (y ψ 2 (y (y [c, d] ψ, ψ 2 : [c, d] R := {(x, y; y [c, d], ψ (y x ψ 2 (y} Jordan f : R ( d ψ2 (y f(x, y dx dy = f(x, y dx dy. c ψ (y y y = ϕ 2 (x y = ϕ (x O a b x.3:.4. D, φ, φ 2 (..4.2 ( K R n Jordan f : K R K f V := {(x, y; x K, y f(x} Jordan ( f(x µ n+ (V = dx dy = dy dx = V K 5 K f(x dx.
.4.3 3 (,, (,, (, I := x 2 y dx dy ( = {(x, y; x, y x} I = ( x x 2 y dx dy = ] y=x [x 2 y2 x 4 dx = 2 y= 2 dx = [ x 5 ] =. y y = x O x.4:.4.3.4.4 ( = {(x, y; x 2 + y 2 R 2 } ( D = [ R, R], φ (x = R 2 x 2, φ 2 (x = R 2 x 2 φ φ 2 = {(x, y; x D, φ (x y φ 2 (x} µ( = dx dy = D ( R 2 x 2 R 2 x 2 dy R dx = 2 R2 x 2 dx. R x = R sin θ (θ [ π/2, π/2] dx = R cos θ dθ, R2 x 2 = R 2 R 2 sin 2 θ = R 2 cos 2 θ = R cos θ = R cos θ π/2 π/2 µ( = 2 R cos θ R cos θ dθ = 2R 2 cos 2 θ dθ = πr 2. π/2 π/2.4.5 ( = {(x, y, z; x 2 + y 2 + z 2 R 2 } ( D := {(x, y; x 2 + y 2 R 2 }, φ (x, y := R 2 x 2 y 2, φ 2 (x, y := R 2 x 2 y 2 (x, y D φ (x, y φ 2 (x, y = {(x, y, z; (x, y D, φ (x, y z φ 2 (x, y} 5
( φ2 (x,y µ( = dx dy dz = dz dx dy = D = 2 D D φ (x,y R2 x 2 y 2 dx dy. ψ (x := R 2 x 2, ψ 2 (x := R 2 x 2 ψ (x ψ 2 (x (x [ R, R] D = {(x, y; x [ R, R], ψ (x y ψ 2 (x} D (φ 2 (x, y φ (x, y dx dy ( R ψ2 (x ( R R 2 x 2 µ( = 2 R2 x 2 y 2 dy dx = 2 R2 x R ψ (x R 2 y 2 dy dx. R 2 x 2 a a a2 x 2 dx = a = πa2 2 a = R 2 x 2 R π(r 2 x 2 R µ( = 2 dx = 2π (R 2 x 2 dx = 2π (R 2 R R3 = 4 R 2 3 3 πr3 7.4.6 = {(x, y, z R 3 ; x 2 + y 2 + z 2 x} x dx dy dz ( (/2,,, /2 D = { (x, y; (x /2 2 + y 2 }, = {(x, y, z; (x, y D, x x 2 y 2 z } x x 2 y 2. D D = [, ], D = {(x, y; x D, x x 2 y } x x 2. 7 ( rchimedes, BC 287 BC 22, Syracuse Syracuse 52
x dx dy dz = = = = D ( x x 2 y 2 x dz x x 2 y 2 dx dy = ( x x 2 2x x x 2 y 2 dy dx x x ( 2 x x 2 4x (x x2 y 2 dy dx 4x π(x x2 dx = π 4 D (x 2 x 3 dx = π 2. 2x x x 2 y 2 dx dy Fubini b x a ( d c f(x, y dy dx = d c ( b a f(x, y dx dy ( (.4.7 I := e y2 dy dx ( = {(x, y; x, x y } = {(x, y; y, x y} ( y I = e y2 dx dy = e y2 dx dy = [ e y2 [x] x=y x= dy = ye y2 dy = 2 ] e y2 = e 2. = {(x, y; a x b, φ (x y φ 2 (x} = {(x, y; c y d, ψ (y x ψ 2 (y}, φ (x φ 2 (x (x [a, b], ψ (y ψ 2 (y (y [c, d] 2 ( b φ2 (x f(x, y dy dx = a φ (x f(x, y dx dy = d c ( ψ2 (y f(x, y dx dy (,.8. d f f(x, t dx = (x, t dx dt t 53 ψ (y
y y y = ϕ 2 (x d x = ψ (y x = ψ 2 (y y = ϕ (x c O a b x O x.5: ( 8 [, X f, g X f g (, convolution f g X (f g(x := x f(x yg(y dy (, (2 9 (x [, ( f, g X f g = g f (. (2 f, g, h X (f g h = f (g h (..4.3 Fubini Fubini (.4. = [a, b], = [c, d] : a = x < x < < x l = b, : c = y < y < < y m = d i := [x i, x i ], ij := i [y j, y j ] ( i l; j m = max{, } 8 ( 9 54
M ij := sup f(x, y, m ij := inf f(x, y (x,y ij (x,y ij ξ i i m ij f(ξ i, y M ij (y [y j, y j ] [y j, y j ] m ij (y j y j j =, 2,..., m yj y j f(ξ i, ydy M ij (y j y j. m m ij (y j y j j= m j= yj y j f(ξ i, ydy m M ij (y j y j, j= m m ij (y j y j d j= c µ( i f(ξ i, ydy m M ij (y j y j. j= l m m ij (y j y j µ( i i= j= l d i= c f(ξ i, ydy µ( i l m M ij (y j y j µ( i, i= j= L(f,, l F (ξ i µ( i U(f,,, F (x := i= d c f(x, y dy. f F Riemann i F [a, b] b a ( d b a F (x dx = Fubini.4.8 c f(x, y dx dy f(x, y dy dx = f(x, y dx dy. f(x, y dx dy F (ξ i µ( i 55
( K := {(x, y R n R; x D, y = φ (x}, K 2 := {(x, y R n R; x D, y = φ 2 (x}, K 3 := {(x, y R n R; x D, φ (x y φ 2 (x} = K K 2 K 3 K K 2.3.6 Jordan K 3 Jordan D Jordan D Jordan ε > { i } l i= s.t. i, l D i, i= l µ( i < ε i= à i := i [m, M], m := min φ (x, x D M := max φ 2 (x x D K 3 l à i, i= l µ(ãi < (M mε. K 3 Lebesgue i= (2 n = (n 2 = [a, b] [c, d] f { f(x, y ((x, y f(x, y := ((x, y \ x D f(x, : [c, d] y f(x, y R 2 φ(x, ψ(x [c, d].4. ( d f(x, y dx dy = f(x, y dx dy = f(x, y dy dx = D [a,b] ( ψ(x f(x, y dy dx. φ(x c 56
.4.4 ( Jordan.4.8 ( Fubini, 2 R n, R n 2 Jordan f : 2 R 2 ( f := 2 (2 x f(x, : 2 y f(x, y R ( f(x, y dx dy = 2 f(x, y dy 2 dx. j =, 2 j j j := 2 f : R { f(x, y ((x, y 2 f(x, y := ((x, y 2 ( f x f(x, : 2 y f(x, y R (i x f(x, 2 (ii x f(x, : 2 R (2 2 x f(x, 2 Fubini ( f(x, y dx dy = f(x, y dx dy = f(x, y dy dx 2 ( 2 = f(x, y dy dx ( 2 = f(x, y dy dx. 2.4.9, B R n, R m Jordan B R n R m Jordan µ n+m ( B = µ n (µ m (B. 57
( (.4.2 D R n+ Jordan η R D η := {x R n ; (x, η D} ( y = η y = R n Jordan f : D R ( d f(x, y dx dy = f(x, y dx dy D c D y c := inf y, d := sup (x,y D D y f(x, y dx = µ n+ (D = d c ( y D y = y (x,y D D y dx dy = d c µ n (D y dy. R n D [c, d] f : R { f(x, y ((x, y D f(x, y = ((x, y D y [c, d] f(x, y = { f(x, y (x D y (x D y D f(x, y dx dy = ( = [a,b] d c f(x, y dx dy = D y f(x, y dx dy. d c ( f(x, y dx dy.4.2 ( R n+ Jordan D, E y R µ n (D y = µ n (E y µ n+ (D = b µ n (D y dy = b a a 58 µ n (E y dy = µ n+ (E.
2.5 (.5. b a f(x dx C φ: [α, β] [a, b] [a, b] f (. b f(x dx = β a α f(φ(uφ (u du ( φ(α = a, φ(β = b. f F β α f(φ(uφ (u du = f(φ(uφ (u = F (φ(uφ (u = d du F (φ(u β α d du F (φ(u du = [F (φ(u]β α = [F (x]b a = b a f(x dx. x = φ(u dx = φ (u du ( dx = dx du du F (udu = f(φ(uφ (udu = f(xdx. u x α β a b ( 2 Bonaventura Francesco Cavalieri (598 647, Milan Bologna Galileo 59
.5.2.5. (, n = 2: Euler (769, n = 3: Lagrange (773 D, R n Jordan U D R n φ: U R n C (i, (ii, (iii (i φ(d = (ii det φ (u (u D (iii φ f : R (.2 f(x dx = f(φ(u det φ (u du. D 2 ( N.5.2 ( ( D, R n Jordan N µ(n = D U D R n φ: U R n C (i, (ii, (iii (i φ(d = (ii det φ (u (u D \ N (iii φ D\N f : R (.3 f(x dx = f(φ(u det φ (u du. D.5.3 ( (.3 φ ( det φ (u (. φ (u (. a = min{a, b}, b = max{a, b}, α = min{α, β}, β = max{α, β} 2 r = det φ = D θ = 2π θ =, 2π 6
v (u,v D u ϕ y (x,y x.6: n = 2 : uv D xy 6
f(φ(u φ (u du = f(x dx [α,β ] [a,b ].5.4 ( (u,..., u n (x,..., x n (x,..., x n (u,..., u n n = 2 ( (x, y (u, v = det x u y u x v y v = x u y v x v y u. ( dx dx n = (x,..., x n (u,..., u n du du n dx = dx du du.5.3 22 xy P(x, y O OP r, OP (x θ { x = r cos θ (r, θ [, 2π. y = r sin θ (θ θ ( π, π] ( ( r x φ: θ y φ (r, θ = ( x r y r x θ y θ = ( cos θ sin θ r sin θ r cos θ 22 62
(x, y (r, θ = det φ = cos θ r cos θ ( r sin θ sin θ = r. dx dy = r dr dθ (r, θ (, [, 2π φ det φ r = θ = 2π φ det φ =.5..5.2 = {(x, y; x 2 + y 2 R 2 } D := {(r, θ; r R, θ 2π} 2π ( R f(x, y dx dy = f(r cos θ, r sin θ r dr dθ = f(r cos θ, r sin θr dr dθ N := {(r, θ D; r = θ = 2π}.5.2 (rθ xy.5.5 ( I = 2π ( I = x 2 +y 2 x 2 dx dy (r cos θ 2 dr dθ = 2π cos 2 θdθ = 4 4 π = π 4..5.6 r = r(θ [α, β] (β α 2π xy µ( = r r(θ, α θ β dx dy = β α ( r(θ r dr dθ = 2 β α r(θ 2 dθ ( 2 r2 θ Cardioid r = + cos θ ( θ 2π µ( = 2 2π ( + cos θ 2 dθ = 3 2 π. 63
.5.5.5 2 -.5 -.7: Cardioid ParametricPlot[{(+Cos[t]Cos[t],(+Cos[t]Sin[t]}, {t,,2pi}].5.7 I = ( x 2 y 2 dx dy, = {(x, y; x 2 + y 2 x, y } Fubini ( x 2 + y 2 x x 2 x + y 2 x 2 + y 2 2 4 (/2, /2 x = 2 + r cos θ, y = r sin θ dx dy = r dr dθ D = {(r, θ; r /2, θ π} I = = π ( ( 2 2 + r cos θ (r sin θ 2 r dr dθ /2 ( 3 dθ 4 r r2 cos θ r 3 dr = = 5π 64. D { x = r cos θ (r, θ [, 2π] y = r sin θ 64
{ D = (r, θ; r cos θ, θ π } 2 ( r = cos θ x θ y = x tan θ φ(r, θ = (x, y φ(d = I = ( x 2 y 2 dx dy = D ( r 2 r drdθ = π/2 dθ cos θ (r r 3 dr = 5π 64. R 3 P(x, y, z OP r [,, OP z θ [, π], P xy P, OP ( r sin θ x ϕ [, 2π 23 x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ (r, θ π, ϕ 2π r, θ, ϕ P (3 (, spherical coordinate.8: θ, ϕ 23 OP e 3 θ, ϕ cos θ =, OP 65 ( x = y ( cos ϕ sin ϕ sin ϕ cos ϕ ( ρ (ρ := x 2 + y 2.
φ: [, [, π] [, 2π] (r, θ, ϕ (x, y, z R 3 x r x θ x ϕ sin θ cos ϕ r cos θ cos ϕ r sin θ sin ϕ φ (r, θ, ϕ = y r y θ y ϕ = sin θ sin ϕ r cos θ sin ϕ r sin θ cos ϕ. z r z θ z ϕ cos θ r sin θ 3 det φ r cos θ cos ϕ r sin θ sin ϕ sin θ cos ϕ r sin θ sin ϕ = cos θ + ( ( r sin θ r cos θ sin ϕ r sin θ cos ϕ sin θ sin ϕ r sin θ cos ϕ cos ϕ sin ϕ cos ϕ sin ϕ = cos θ r cos θ r sin θ + ( ( r sin θ sin θ r sin θ sin ϕ cos ϕ sin ϕ cos ϕ = r 2 cos 2 θ sin θ + r 2 sin 3 θ = r 2 sin θ θ [, π] sin θ r 2 sin θ dx dy dz = det φ drdθ dϕ = r 2 sin θ dr dθ dϕ..5.8 ( = {(x, y, x; x 2 +y 2 +z 2 R 2 } D = {(r, θ, ϕ; r R, θ π, ϕ 2π} = φ(d µ( = dx dy dz = det φ dr dθ dϕ = r 2 sin θ dr dθ dϕ D D R ( 2π ( π ( R ( 2π ( π = r 2 sin θ dθ dϕ = r 2 dr dϕ sin θ dθ = R3 4πR3 2π 2 = 3 3..5.9.5..5.2 D := [, R] [, π] [, 2π], N = {(r, θ, ϕ D; r = or θ = or θ = π or ϕ = 2π} {(x, y, z; x 2 + y 2 + z 2 R 2, z } /8 {(x, y, z; x 2 + y 2 + z 2 R 2, x, y, z } ( r, θ, ϕ a, b, c (x, y, z R 3 3 = {(x, y, z; (x x 2 + (y y 2 + (z z 2 } a 2 b 2 c 2 ( x x y y z z = r sin θ cos ϕ, = r sin θ sin ϕ, = r cos θ a b c x x y y z z = u, = v, = w a b c {(u, v, w; u 2 + v 2 + w 2 } 66
24.5. (,, (4,, (2, 3 I = ( x = 4u + 2v, y = u + 3v ( ( (x, y (u, v = det x r x θ 4 2 = det = 4 3 2 = 3 y r y θ x dx dy dx dy = du dv = du dv (,, (,, (, ( v I = (4u + 2v du dv = (4u + 2vdv du = ( 3u 2 + u 2 + du =..5. xy y = 2x, y = x 2, x + y = 2 I = (x + ydx dy ( (,, (4/3, 2/3, (2/3, 4/3 φ: ( x y = ( u v, := ( 4 3 2 3 det φ det = 4 uv 3 (,, (,, (, 3 = {(u, v; u, v, u + v } [( 4 I = 3 u + 2 ( 2 3 v + 3 u + 4 ] 3 v 4 3 du dv = 8 (u + vdu dv 3 = 8 ( u (u + vdv du = 8 3 9..5. ( 3 (x, y, B(x 2, y 2, C(x 3, y 3 ( µ( := dx dy, X := x dx dy, Y := y dx dy 24 R n R n R n ( 2 3 4 3 ( u v ( x y 67
( φ: R 2 R 2 ( ( ( x x 2 x x 3 x u φ(u, v := + y y 2 y y 3 y v := {(u, v R 2 ; u, v, u + v }, φ( = ( φ x 2 x x 3 x (u, v =, det φ (u, v = D y 2 y y 3 y D := (x 2 x (y 3 y (x 3 x (y 2 y D du dv = 2, u du dv = v du dv = 6 µ( = dx dy = X = x dx dy = = D (x = D Y X D du dv = D du dv = D 2, [x + (x 2 x u + (x 3 x v] D du dv u du dv + (x 3 x v du dv = x + x 2 + x 3 D. 6 du dv + (x 2 x ( x 2 + (x 2 x 6 + (x 3 x 6 Y = y + y 2 + y 3 D. 6 (x, y x := x dx dy, y := µ( µ( y dx dy (B.4 x = D /2 x + x 2 + x 3 6 D = x + x 2 + x 3 3, y = y + y 2 + y 3. 3 68
.5.4 : ( f : R n R m ( (6 ( f M(m, n; R s.t. f(x = x (x R n. (2 m = n, M(n; R s.t. f(x = x (x R n (i (vii (i f (ii f (iii f (iv rank f = n (rank = n (v ker f = {} (vi det (vii (f f f (x = x (x R n f C (3 m = n = 2, M(2; R s.t. f(x = x (x R 2 25 (i f(r 2 = R 2 ( rank = 2 (ii f(r 2 ( O rank = (iii f(r 2 = {} ( = O rank = (4 m = n = 2, GL(2; R s.t. f(x = x (x R 2 (a (d ( : GL(2; R = { M(2; R; det } (a f (b f ( (c f ( (d f ( ( x 25 y ( a b = c d ( x y 69
(5 m = n = 2, M(2; R s.t. f(x = x (x R 2 (( ( (( ( a b f =, f = = c d ( a c b d. ( : (6 R 2 {p + ta + sb; t [, ], s [, ]} det(a, b. M(2; R f : R 2 x x R 2 26 det (det < ta + ( tb B (4 ( : 2 a, b (t R f (ta + ( tb = tf (a + ( tf (b (3 (6 3.5.2 ( f : R n x x R n det f n Jordan det n = 2 (6 := {ta + sb; t [, ], s [, ]} µ 2 ( = a b sin θ = a b cos 2 θ = a 2 b 2 (a, b 2 = (a 2 + a 2 2(b 2 + b 2 2 (a b + a 2 b 2 2 = (a b 2 a 2 b 2 = a b 2 a 2 b = det(a, b. f : x x {ta + sb; t [, ], s [, ]} µ 2 (f( = det(a, b = det [(a, b] = det det(a, b = det det(a, b = det µ 2 (. 2 µ 2 ( µ 2 ( def. = dx dy µ 2 ( = a b sin θ C ( R 2 (p; a, b := {p + ta + sb; t, s, t + s } (2 R 3 T (p; a, b, c := {p + ta + sb + rc; t, s, r, t + s + r } (3 R n (n N, n 4 26 Jordan 7
.5.5 ( D. l D D : D j. j u j ( l = φ(d φ j = j= j= l φ( j. j= f(x dx l j= φ( j f(x dx = l f(x dx j= φ( j l f(φ(u j µ(φ( j. j= u u j φ(u φ(u j + φ (u j (u u j µ(φ( j det φ (u j µ( j. g(x = x + b µ(g( j = det µ( j l f(x dx f(φ(u j det φ (u j µ( j f(φ(u det φ (u du. D j= ( Riemann.6.6. Fubini ( R 3 I := (, (2 f(x, y, z dx dy dz ( (3 Fubini xy D := {(x, y R 2 ; z R s.t. (x, y, z } φ : D R, φ 2 : D R φ φ 2 (on D = {(x, y, z; (x, y D, φ (x, y z φ 2 (x, y} I = D ( φ2 (x,y f(x, y, z dz dx dy. φ (x,y 7
(2 ( x R a := inf x, b := sup x. (x,y,z (x,y,z x := {(y, z R 2 ; (x, y, z } ( x = x yz I = b a ( x f(x, y, z dy dz dx. = {(x, y, z; x 2 + y 2 + z 2 R 2 } ( ( R 2 x 2 y 2 I = f(x, y, z dz dx dy, D := {(x, y; x D R 2 + y 2 R 2 }. 2 x 2 y 2 (2 I = R R ( x f(x, y, z dy dz dx, x := {(y, z; y 2 + z 2 R 2 x 2 }..6. = {(x, y, z; x 2 + y 2, x z 2x + } ( ( D := {(x, y; x 2 + y 2 }, φ (x, y := x, φ 2 (x, y := 2x + φ φ 2 (on D, = {(x, y, z; (x, y D, φ (x, y z φ 2 (x, y} µ 3 ( = = dx dy dz = r, θ 2π D ( 2x+ x dz dx dy = (r cos θ + r dr dθ = 2π D (x + dx dy r dr = π..6.2 = {(x, y, z; x 2 + y 2 z 2, z a} ( (2 z [, a] µ 3 ( = z := {(x, y; (x, y, z } = {(x, y; x 2 + y 2 z 2 } dx dy dz = a ( dx dy dz = z a µ 2 ( z dz = a πz 2 dz = πa3 3. 72
.6.2 R 3, (.6.3 ( R 3 z = x 2 + y 2 z = x z = x 2 + y 2 z = x = {(x, y, z; x 2 + y 2 z x}. D := {(x, y; x 2 + y 2 x}, φ (x, y := x 2 + y 2, φ 2 (x, y := x φ (x, y φ 2 (x, y ((x, y D 27 µ( = dx dy dz = = {(x, y, z; (x, y D, φ (x, y z φ 2 (x, y} D (φ 2 (x, y φ (x, y dx dy = D (x x 2 y 2 dx dy. D (/2,, /2 x = r cos θ + /2, y = r sin θ r /2, θ < 2π dx dy = rdr dθ, x x 2 y 2 = /4 r 2 µ( = = 2π ( 4 r2 r dr dθ = 2π 4 ( 2 2 ( 4 = π 2 4 2 32. r /2 θ 2π ( /2 ( 4 r2 r dr z = x 2 + y 2 z = x 2 y 2 (z = f(r, r = x 2 + y 2 y = (xz z = f(x.7 2 Jordan 27 D 73
(, 2, Laplace, Fourier, Lebesgue Riemann.7..7. α > x α dx? [, [, R] R dx = lim xα R dx := lim xα R R [ ] R α x α+ (α [log x] R (α = x α dx (α > = α ( < α..7.2 β > x β dx? f(x = x β lim f(x = + x f (, ] < ε < f [ε, ] dx := lim xβ ε ε x dx β [ ] dx = lim β x β+ (β xβ ε ε [log x] ε (β = = x α x β + (β β ( < β <. dx dx 74
dx xα.7.3 + x 2 dx. [, [, R] dx := lim + x2 R R + x 2 dx [ dx = lim + x2 Tan x ] R = π R 2. ( Tan tan arctan = tan.7.4 ( I = e x2 dx. R e x2 dx = lim e x2 dx R f(x := e x2 (x [, ] g(x := (x [, x 2 28 f(x g(x (x [, g(x dx < I ( I.7.2 K n (n {K n } n N f(x dx := lim f(x dx n K n K n 28 x 2 e x2 /e x 2 e x2 /e. e x2 x 2. 75
.7.5 x 2 +y 2 dx dy x2 + y 2 = {(x, y; x2 + y 2 } f(x, y = / x 2 + y 2 (, lim f(x, y = +. (x,y (, (, K n := { (x, y; } n 2 x2 + y 2 (n N Jordan f f(x, y dx dy = K n /n r θ 2π r dr dθ = r 2 /n 2π ( dr dθ = 2π. n ( f(x, y dx dy = lim f(x, y dx dy = lim 2π = 2π. n K n n n dx dy.7.6 x 2 +y 2 (x 2 + y 2 = {(x, y; 2 x2 + y 2 } f(x, y = /(x 2 + y 2 2 ( f(x, y K n := { (x, y; x 2 + y 2 n 2} (n N Jordan n f(x, y dx dy = r dr dθ = K n (r 2 2 r dr 3 r n θ 2π 2π [ dθ = 2π ] n 2 r 2 = π ( n. 2 f(x, y dx dy = lim f(x, y dx dy = lim π ( n = π. n K n 2 n.7.3 ( 76
K K 2 K n K n+ 29 : K n := {x; n N s.t. x K n }. n= K n = n N (a m N K m K n L n= :, B n= K n K n, (b n N K n L n= = B ( B B B ( x x B (a, (b.7.6 K n = (.7.5 n= K n = \ {(, } n= 3 Jordan (Jordan N f \ N K n = \ N n= 29 3 K n \ {(, } n K n. (x, y \ {(, } < x 2 + y 2 < /n < x 2 + y 2 n N n (x, y K n. (x, y n K n. n N K n n K n. 77
( Jordan ( Jordan \ N K K 2, K n = n= {K n } n N 3 (.4 lim f(x dx n K n {K n } ( (a {K n } (b {K n } (.4 (a ( (b 32. ( f f (.4 {K n } 2. ( ( f {K n }.7.4 {K n } n ( l 3.7.6 K n 32 78
.7.7 ( R l {K n } n N (i (iii (i K K 2. K n =. n= (ii n N K n R l Jordan (iii K K K n n N (i (ii K n Jordan ( (iii.7.3 ( (.7.3 (iii (i K n = 33 n= (iii K n Kn+ n= K n K K n= K n. {Kn} K n,..., n r s.t. K r j= K n j. n,..., n r n j {,..., r} K nj K n K r k= K n k = Kn K n. (iii [.7.8 := (,, K n := n, ] {K n } n N n.7.9 := R l, K n := {x R l ; x n} {K n } n N.7. := R l \ {}, K n := {x R l ; /n x n} {K n } n N 34 33 x K := {x} (iii x K n n N 34 I (http://nalab.mind.meiji.ac.jp/~mk/lecture/kaisekigairon-/ 79
Jordan.7. ( Jordan R l Jordan R B(; R R l ( Jordan ( B(; R = {x R l ; x < R} Jordan Jordan Jordan.7.2 ( R l Jordan, f : R N N := (f ( f = ( \ {x ; {y n } n N s.t. lim n y n = x (a-(c (a N Jordan lim f(y n = } n (b f \ N Jordan K ( f(x dx (c \ N K \ N {K n } n N lim f(x dx K n f f(x dx := lim f(x dx n K n n f (improper integral f f f f 35 ( x 2 +y 2 dx dy x2 + y2 (b ( Lebesgue 35 Riemann Lebesgue Lebesgue Riemann (Lebesgue Riemann Riemann Lebesgue 8
f \ N Jordan f f(x dx 36 ( \ N f f.7.3 ( R l, f : R ( f f Jordan K {K n } n N, {K n} n N lim n f(x dx lim K n n K n f(x dx ( f ( n N a n := f(x dx, a n := f(x dx K n K n {K n } {K n} {a n } n N {a n} n N : a a 2, a a 2 (iii N N a N a n. n N s.t. K N K n. sup n N a n sup a n n N (. sup n N a n sup a n n N ( sup n N a n = sup a n. n N 36 8
.7.4 ( f(x, y = x2 + y 2 ((x, y (, N f (, ((x, y (, f(x, y = x2 + y 2 ((x, y = (, f f f ( ( f f N.7.5 : {a n } n= a n := lim n n a k. a n a n n= n= k= a n n= n a k (n N k= a n =. a n a n n= n= a n a n (Dirichlet, n= n= n= 837 φ: N N a φ(n = n= n= a n (. a n λ R {, } n= φ: N N a φ(n = λ n= 82
(Riemann, 854 {a n } a a 2, lim n a n = ( n a n = a a 2 + a 3 a 4 + n= (Leibniz, 682.7.6.7.3 {K n } n N lim f(x dx K n (.7.5 sin x x dx n n := [(n π, nπ] n n f ( f >, n n f ( f < sin x a n := n x a 2k >, a 2k <,.8.6.4.2-3 -2-2 3 -.2 dx a 2k =, k= nπ a n a 2k =, k= (n π, a a 2 a n a n+, a n = π 2 n= ( ( π/2 37 K n := n k= k = [, nπ] {K n } = [, sin x n K n x dx = sin x n k x dx = a k π 2. k= k= 37 83
R sin x ( lim R x dx = π 2 π/2 n a n λ R {, } φ: N N a φ(k = λ (Riemann K n := n k= φ(k {K n} n N K n k= k= sin x n x dx = sin x n φ(k x dx = a φ(k λ k= (n. ( = k= 2k, 2 = k= 2k f f, f 2 f sin x x dx = sin x a 2n =, 2 x dx = a 2n =. n= K n := [, nπ] f f ( {K n } n= K n f(x dx.7.6 ( (, f, N (a, (b, (c \ N φ (i \ N f φ. ( φ (ii φ f x \ N f + (x := max{f(x, }, f (x := max{ f(x, } f +, f f + f \ N Jordan f(x = f + (x f (x, f + (x f(x φ(x, f (x f(x φ(x. 84
f + (x dx φ(x dx φ(x dx <, K n K n f (x dx φ(x dx φ(x dx <. K n K n n f + (x dx, f (x dx f + f K n K n f = f + f.7.7 φ := f.7.8 = {(x, y; x, y }, f(x, y = e x cos(xy I = f(x, y dx dy ( f cos(xy f(x, y = e x cos(xy e x. φ(x, y := e x f(x, y φ(x, y. K n := [, n] [, ] {K n } n N ( n φ(x, y dx dy = e x dx dy = [ e x] n = e n (n. K n φ φ(x, y dx dy = lim φ(x, y dx dy = <. n K n φ I n := f(x, y dx dy = K n n n f(x, y dx dy ( n e x cos xy dx dy e x cos xy dx = [ e x cos xy ] n x=n ( e x ( y sin xydx x= = e n cos ny + y e x sin xy dx = [ e x sin xy ] n n = e n sin ny + + y 85 n n e x sin xy dx, ( e x (y cos xydx e x cos xy dx
n I n = ( e x cos xy dx = e n cos ny y e n sin ny + y n n e x cos xy dx n = + e n (y sin ny cos ny y 2 e x cos xy dx. e x cos xy dx = y sin ny cos ny + e n. + y2 + y 2 ( y sin ny cos ny + e n dy = π4 + y2 + y + y sin ny cos ny 2 e n dy. + y 2 y sin ny cos ny e n dy + y 2 e n e n y sin ny cos ny + y 2 dy + y + y 2 dy 2e n (n f(x, y dx dy = lim n I n = π 4. ( n e x cos xy dx = + y y 2 [, ].7.9 ( OK I = dx dy x 4 + 2x 2 y 2 + y 4 + x 2 + x 2 +y 2 {K n } K n x 4 + 2x 2 y 2 + y 4 + x 2 + x 4 + 2x 2 y 2 + y = 4 (x 2 + y 2 2 x 2 +y 2 dx dy (x 2 + y 2 2 I 86
.7. ( = ( 38 ( f f (29 9 2 [3] E. V. Hobson, The Theory of Functions of a Real Variable, third edition, Cambridge (927, pp. 58 533 (http: //ia342.us.archive.org/2/items/theoryfunctrealhobsrich/theoryfunctrealhobsrich. pdf [3], (, (952, pp.398 4..7.7., f, N ( f f f N N Jordan 2. f ( 3 f (a \ N {K n } n N ( f(x dx K n 38 [5], [3] pp.26 27 87
(b f φ φ φ(x dx K n 3. \ N {K n } n N f(x dx K n.7.2 ( dx dy I = x2 + y 2 x 2 +y 2 = {(x, y; x 2 + y 2 }, f(x, y = f K n = {(x, y; /n 2 x 2 + y 2 }, N = {(, } x2 + y2 {K n } n N \ N K n x = r cos θ, y = r sin θ (r, θ [, 2π] D n = {(r, θ; /n r, θ [, 2π]} f(x, y dx dy = K n I = 2π. D n r r drdθ = 2π ( dθ dr = 2π 2π /n n (n..7.2 ( y dx dy x 2 + y2 x 2 +y 2 y dx dy x 2 + y2 K n x 2 +y 2 ( K n = {(x, y; /n 2 x 2 + y 2 } y r sin θ dx dy = r dr dθ = 4( /n < 4 x 2 + y2 r 2 /n r θ [,2π]. ( x dx 88 R
.7.22 ( α dx dy I = (x 2 + y 2 α x 2 +y 2 = {(x, y; x 2 + y 2 }, f(x, y =, N = ( f (x 2 + y 2 α K n = {(x, y; x 2 + y 2 n 2 } {K n } n N K n x = r cos θ, y = r sin θ (r, θ [, 2π] D n = {(r, θ; r n, θ [, 2π]} 2π n f(x, y dx dy = r drdθ = dθ r 2α dr K n D n r2α [ ] r 2( α n = n2( α ( 2α = 2π 2( α 2( α [log r] n = log n ( 2α = { π (α > α (n. (α α > π/(α, < α.7.23 ( 39 e x2 dx = {(x, y R 2 ; x, y }, f(x, y = e x2 y 2 f(x, y dx dy {K n }, {C n } : K n = [, n] [, n], C n = {(x, y; x, y, x 2 + y 2 n 2 }. 39 m, σ 2 f(x = [ ] exp (x m2 2πσ 2 2σ R 2 89
f C n C n f(x, y dx dy = π 4 ( e n2 π 4. f f x 2 +y 2 +z 2 ( π n 2 4 = lim f(x, y dx dy = lim e dx x2. n K n n dx dy dz x 2 + y 2 + z 2 e x2 dx = dx dy dz α x 2 +y 2 +z 2 (x 2 + y 2 + z 2 α π 2..7.8.7.5 a n n = ((n π, nπ { sin x (sin x/x a n { > (n < (n > (n < (n a n (.5 nπ sin x dx = n (n π sin x dx = ( n 2 = 2 n 2 nπ x (n π (x n 9
sin x (> sin x sin x dx < dx < nπ n n x sin x dx. (n π n n sin x nπ sin x dx < sin x n n x dx < (n π sin x dx. n (.5 2 nπ < sin x n x dx < 2 (n π i.e. 2 nπ < a n < 2 (n π. a n b n := 2 nπ b n > a n > b n (n 2. n = a > 2 π = b a > b > a 2 > b 2 > a 3 > a n n a 2k >, a 2k > a 2k <, a 2k > 2 (2k π a 2k > a 2k k= k= 2 (2k π = 2 (2k π a 2k =. 2 (2k π a 2 2k < (2k π 2 a 2k = a 2k =. (2k π k= k=.8 (.8. 3 ( 2 3 k= k= lim f(x, y, lim (x,y (a,b lim x a y b 9 f(x, y, lim lim f(x, y. y b x a
(.8. 4,2 f(x = ( a n (x x n n= f n (x := n a k (x x k, k= f(x = lim n f n (x d f(t, x dx = dt ( B lim f n (x dx = n df n lim n dx (x = d ( dx f (t, x dx t lim f n(x dx n lim f n(x n 2 f x y (x, y = 2 f (x, y y x ( ( f(x, y dy dx = f(x, y dx dy B (, (, (, (.8. ( ( ( df n f n (x dx = f n (x dx, dx (x = d f n (x dx n= n= 4 92 n= n=
.8.2 (lim ( n N.8.3 (lim R f n (x := + (x n 2 (x R lim f n (x dx = lim π = π, n n lim f n(x dx = n dx =. (2 n N f n : R ( (x n 2 x ( x /n f n (x = n 2 (2/n x (/n x 2/n (x 2/n x R lim f n (x = n 2 lim f n(x dx =. n n N f n 2 f n (x dx = 2 2 n n =. 2 lim n f n (x dx =. 2 lim f n(x dx lim n n 2 f n (x dx. lim 93
.8.2.8.4 ( R m, f : R N {f n } n N n N f n : R N ( n {f n } n N f x lim f n (x = f(x n (2 n {f n } n N f lim sup f n (x f(x = n x (Weierstrass, 84 (.8.2,.8.3 n {f n } n N f(x R.8.5 (.8.6 (, Weierstrass (86 R m, f : R N {f n } n N n N f n : R N f n {f n } n N f f x f x f(x f(x f(x f N (x + f N (x f N (x + f N (x f(x 2 sup f(y f N (y + f N (x f N (x y n {f n } n N f ε > N N f N x δ > s.t. sup f(y f N (y < ε y 3. x B(x ; δ f N (x f N (x < ε 3. 94
x B(x ; δ f(x f(x 2 sup f(z f N (z + f N (x f N (x 2 ε z 3 + ε 3 = ε..8.7 ( n N (x n f n (x := nx ( n x n (x n f n : R R f n R x R lim f n(x = f(x := n f (x < (x = (x > (.8.8 (Weierstrass M {f n } n N R N {M n } n N s.t. sup f n (x M n and x x M n <. n= f n n= f n (x M n (n N, M n < n= f n (x ( S(x n= n S(x f n (x = k= n f k (x f k (x = f k (x k= k= k=n+ n S(x f k (x k= k=n+ f k (x k=n+ M k. 95
M n n= ε > N N ( n N : n N k=n+ M k < ε. x n S(x f k (x < ε. k= n k= f k(x S(x.8.3 (.8.9 ( R n Jordan {f n } n N n f f lim f n (x dx = f(x dx n ( f(x dx f n (x dx = (f(x f n (x dx f(x f n (x dx sup f(x f n (x dx = µ( sup f(x f n (x. x x.8. ( R I = [a, b] C {f n } n N ( {f n } n N n f I (2 {f n} n N n g I f I C f = g x [a, b] f n (x = f n (a + x a f n(t dt 96
n f(x = f(a + x a g(t dt. g f (x = g(x. ( a n (z c n = na n (z c n n= n= (.8.4.8. (, R n Jordan I R K = I, f : I (x, t f(x, t R (, (2 ( F (t = f(x, t dx I (2 f t K ( f C F C (I F (t = f (x, t dx t d f(x, t dx = dt f (x, t dx. t I [a, b] ( K f K ε > δ > (x, s K (y, t K (x, t (y, s < δ = f(x, t f(y, s < ε. t s < δ = f(x, t f(x, s < ε (x. 97
(2 f t F (t F (s F I K ( f(x, t f(x, s dx G(t := f (x, t dx t I s s ( f G(t dt = (x, t dx dt = a a t = [f(x, t] t=s t=a dx = (f(x, s f(x, a dx = F (s F (a. F (s = G(s. F C ([a, b]. ( s a dx εµ(. f (x, t dt dx t ( 4 d dx x a f(x, t dt = x a f (x, t dt + f(x, x x ( : F (x, y := y a f(x, t dt F x, F y 4 I 98
( 2 2 [9] ( ( ( [3], [4] ( 42 ( 43 [28] [7] 42 43 99
( [4] (? nalysis by its history [33] 44 [5], [9] ( 44
[],, (23. [2],, (963. [3],, (28. [4],, (96. [5],,, (22. [6], I, II, (2, 2. [7],, 2, (26. [8],, (99. [9],, (2, 2 22. [], I, II,, (994, 995. [] L., 7, (97 97. [2], 3, (99. [3], I, (98. [4], II, (985. [5],,,,, (989. IV II ( (. [6] J.,,,, (25. [7],,, (972. (27 [8], (, (983.
[9],, (987. [2],,. [2],,, (962, 963. [22],, (989, 99, 99, 99. [23],,, III, (986. [24], 3,, (996. [25], (,, (976. [26],, (955. [27], I, (23. [28], II, (23. [29],, (27. [3],, (25. [3],, (99. [32],,,, (975.. L. Cauchy, Résumé des Leçons sur le Calcul Infinitésimal (823. [33] E., G.,,, (997. [34],,, 3, (977. H. Lebesgue, Intégrale, Longueur, ire (92 [35],,,, (24. 2
barycenter, 6 Cardioid, 63 center of gravity, 6 center of mass, 6 improper integral, 8 repeated integral, 46 spherical coordinate, 64 arccos, cos, 6 arcsin, sin, 6 arctan, tan, 6, 66, 94, 94, 2, 6, 7, 94, 74, 89, 4, 5, 7, 8 (Jordan, 29, 6, 33, 34, 23, 9, 64, 35, 2, 6 (n, 27 (, 64 (, 62, 64 (, 8, 8, 96, 96, 7, 2, 5, 2 3, 64, 5, 6, 8, 46 (, 53 (, 97, 4, 7 Jordan, 28 Jordan, 28 Jordan (, 4 Jordan, 3, 8 (Jordan, 29, 8 (Jordan, 29, 97,, 4 3
, 54, 5 Darboux, 25, 59,, 28, 2 Heine-Borel, 8 cosh, 7 sinh, 7 tanh, 7 (, 3, 97 Fubini, 47 Fubini (, 49, (, 3 (, 3, 2, 2, 33, 42 (, 6, 2, 4, 4, 6, 7, 2 Riemann, 8 Riemann, 24, 46 Lebesgue, 8 Lebesgue, 3 Weierstrass M, 95 4
:... e (i, (ii, (iii ( (i e = lim + n. n n (ii e = n= (iii lim h a h h n! =! +! + 2! + +! = 3! = a e 2.. (e (ii e 8 n = 4 : e = 2.7828 8284 5945 23536 2874 7352 66249 77572 e y = e x exp x e x = exp x. R x e x (, log : x = log y y = e x. a a = e log a log x x 3 e x+y = e x e y, e =, (e x y = e xy log(xy = log x+log y, log =, log x y = y log x a a y = a x a x = ( e log a x = e (log ax = e x log a = exp (x log a e e x = exp x e 4 Napier Euler 2 e 3 x ln x 4 e 5
..2 ( ( T N t N t/t ( N = N 2 ( 2 = exp log 2 ( t N = N exp T log ( = N exp t log 2 2 T exp..3 ( y = ky x = y = y y = a x a = e k y = a x = (e k x e kx..2 [ π/2, π/2] x sin x [, ] sin arcsin x = sin y y = sin x π 2 x π 2. [, π] x cos x [, ] cos arccos x = cos y y = cos x x π. ( π/2, π/2 x tan x (, tan arctan x = tan y y = tan x π 2 < x < π 2. f(x = sin x ( ( (2 (f(x/a (3 f( 3/2 (4 y = f(x f(x = tan x ( f (x = + x 2 (2 f (x (3 y = f(x ( 6
..3 cosh, sinh, tanh cosh x = ex + e x, sinh x = ex e x, tanh x = sinh x 2 2 cosh x...4 sinh x = log ( x + x 2 + ( y = sinh x x = sinh y = ey e y. Y = e y x = (Y /Y /2 2 Y 2 2xY =. 2 x ± x 2 + Y > Y = x + x 2 +. ( sinh x = y = log Y = log x + x 2 +...5 cosh x = log ( x + x 2 ( cosh: R R f : [, x cosh x [, cosh y = cosh x x = cosh y = ey + e y. y 2 x Y = e y Y, x = (Y + /Y /2. Y 2 2xY + =. 2 2 Y = x + x 2, Y 2 = x x 2 x > Y > Y 2 < ( Y Y 2 = Y > Y 2 < Y 2 Y = Y = x + x 2. ( cosh x = y = log Y = log x + x 2. tanh tanh ( tanh x = 2 log + x x (2 ( tanh x = x 2.2 7
f(x f (x c c x n n nx n x n n,x nx n x α α,x > αx α sin x cos x cos x sin x tan x cos 2 x e x log x sin x e x x x 2 cos x tan x sinh x cosh x tanh x x 2 + x 2 cosh x sinh x (cosh x 2.2. ( f, g f +g, f g, f g, f/g g (f + g (x = f (x + g (x, (f g (x = f (x g (x, (f g (x = f (xg(x + f(xg (x, ( f (x = g(xf (x g (xf(x. g g(x 2.2.2 ( f, g g f g f (g f (x = g (f(xf (x. z = g(y,y = f(x z = g(f(x = (g f(x dz dx = (g f (x, dz dy = g (y, 8 dy dx = f (x
dz dx = dz dy dy dx.2.3 ( f f ( f (y =, y = f(x. f (x y = f(x x = f (y dy dx = f (x, dx dy = ( f (y dx dy = dy dx.2.4 f(x = log x + x 2 + k f (x =.2.5 a y = sin ( x a = ( x + x2 + k + x + x 2 + k = 2 (x2 + k /2 (x 2 + k x + x 2 + k x + x2 + k x + x 2 + k = x2 + k. x a = sin y, π/2 y π/2. dx ( x 2 dy = a cos y = a sin 2 y = a = a2 x a 2. ( ( x sin dy = a dx = dx dy = a2 x 2. π/2 y π/2 cos y cos y = sin 2 y, a > a = a 2.3 C 9
f(x f(x dx x α (α x e x x α+ α + log x sin x cos x cos x sin x sec 2 x = / cos 2 x tan x cosec 2 x = / sin 2 x cot x x 2 + a (a > 2 a tan a (a > a2 x2 x sin a (k R x2 + k log x + x 2 + k ( a2 x 2 (a > x a 2 x 2 + a 2 sin x a ( x2 + k (k R x x2 + k + k log x + x 2 + k.3. Mathematica / x 2 + k 5 a > dx ( x = sinh dx ( x (x R, = cosh (x > a x2 + a 2 a x2 a 2 a e x 6 x = a sinh u dx ( x = sin (x ( a, a a2 x 2 a (f(x + g(x dx = f(x dx + g(x dx, kf(x dx = k f(x dx 5 k > rcsinh x k < log ( x + k + x 2 k 6 / x 2 a 2 x < a cosh dx x2 a = sign x 2 cosh x a. log x + x2 a 2
f(u du = f(φ(xφ (x dx f(φ(xφ (x f(x F (x f(ax + b dx = F (ax + b. a f (x f(x dx = log f(x tan x dx = log cos x, cot x dx = log sin x f (xg(x dx = f(xg(x f(xg (x dx ( e.3. xe 2x 2x dx = x dx = x e2x 2 2 (x e2x xe2x dx = e 2x dx = 2 2 2 xe 2x 2 e2x 4. n x n e 2x dx n.3.2 log x dx = (x log x dx = x log x x (log x dx = x log x x x dx = x log x x..3.3 e x cos x dx = (e x cos x dx = e x cos x e x (cos x dx = e x cos x+ e x sin x dx = e x cos x+e x sin x e x cos x dx. 2 e x cos x dx = ex (cos x + sin x. 2
.3.4 k x2 + k dx = (x x 2 + k dx = x ( x 2 + k x x2 + k dx = x x 2 + k x 2 x2 + k dx = x x 2 + k k x 2 + k x2 + k dx = x x2 x 2 + k + k dx + k dx x2 + k. 2 2 x2 + k dx = ( x dx x 2 2 + k + k x2 + k = ( x x 2 2 + k + k log x + x 2 + k..3. ( / ( ( R(x, y x, y R(cos θ, sin θdθ, R x, x 2 dx.3.5 I = dx x 2. x 2 = (x (x + = ( 2 x x + I = ( 2 x dx = (log x log x + x + 2 = 2 log x x +..3.6 I = x 4 x 3 3x + 2 dx. x 4 = x(x 3 3x + 2 + 3x 2 2x 2
x 4 x 3 3x + 2 = x(x3 3x + 2 + 3x 2 2x x 3 3x + 2 I = x2 2 + J, J := 3x 2 2x x 3 3x + 2 dx. x 3 3x + 2 = (x 2 (x + 2 3x 2 2x x 3 3x + 2 = x + 2 + = x + 3x2 2x x 3 3x + 2. B (x + C 2 x 3x 2 2x = (x 2 + B(x + 2 + C(x (x + 2. x = B = 6. x = 2 = 3 9. x2 3 = + C C = 3 = 3 6 9 = 9. 3x 2 2x x 3 3x + 2 = 6 9(x + 2 + 3(x 2 + 9(x. J = 6 9 log x + 2 3 x + log x 9 I = x2 2 + 6 9 log x + 2 3(x + log x. 9 3
B ( B., ( R n (Jordan n Jordan µ n ( = dx n = 2 (area n = 3 (volume.4.8.4.2 (2 3 B.2 f(x dx f R 3 (x, y, z ( ρ(x, y, z M = ρ(x, y, z dx dy dz Riemann l ρ(ξ j µ( j j= ρ(ξ j µ( j j B.3. ( g cm 3 4
B.3 R n Jordan w : R f : R f w (weighted mean, weighted average f(xw(x dx w(x dx w f(x dx = dx f(x dx µ( f (mean, average n = 2 ( i f iw i i w i B.3. ( f : R a a r B r f f(x dx µ(b r B r r f(a f(x dx f(a = f(x dx f(a dx µ(b r B r µ(b r B r µ(b r B r = [f(x f(a] dx µ(b r B r f(x dx f(a µ(b r f(x f(a dx B r µ(b r B r sup f(x f(a dx x B r µ(b r B r = sup x B r f(x f(a. f a r + lim f(x dx = f(a. r + µ(b r B r 5
f : [a, b] R lim h h c+h c f(x dx = f(c (c [a, b] B.4 R 3 2, (x, y, z ρ = ρ(x, y, z (barycenter, center of gravity, center of mass g = (x, y, z x = xρ(x, y, z dx dy dz, M y = yρ(x, y, z dx dy dz, M z = zρ(x, y, z dx dy dz M M := ρ(x, y, z dx dy dz (. g = M xρ(x dx, M = ρ(x dx. :.5. (p.67 B.5 R 3, (x, y, z ρ = ρ(x, y, z l (x, y, z p(x, y, z l I I = ρ(x, y, zp(x, y, z 2 dx dy dz l R I = MR 2 (M ( R := I/M 2 6
C C. ( R n K K a ( Heine-Borel a K R n R n {U λ } λ Λ K λ Λ U λ λ,..., λ r Λ K r j= U λ j ( (B R n ( : I (http://nalab.mind.meiji. ac.jp/~mk/lecture/kaisekigairon-/ R n (B 2 ( ( [33] C.2 Heine-Borel Heine-Borel K K K 2 Heine-Borel ( Heine-Borel 7