,,,,., = (),, (1) (4) :,,,, (1),. (2),, =. (3),,. (4),,,,.. (1) (3), (4).,,., () : = , ( ) : = F 1 + F 2 + F 3 + ( ) : = i Fj j=1 2

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1 , R 2, 2 (R 2, B, µ)., R 2,,., 1, 2, 3,., 1, 2, 3,,. () : = (6.1.1).,,, 1

2 ,,,,., = (),, (1) (4) :,,,, (1),. (2),, =. (3),,. (4),,,,.. (1) (3), (4).,,., () : = , ( ) : = F 1 + F 2 + F 3 + ( ) : = i Fj j=1 2

3 ,,,,., f(x, y), :, () : = (6.1.2) f(x, y) = α j χ j (x, y) (6.1.3) j=1., α j,. χ j (x, y) j., f(x, y) f (x, y), f(x, y) f(x, y)., f(x, y), f(x, y) 3

4 (6.1.3), (6.1.2),.,., R = R {± }., f(x, y) R. f(x, y)., ( ) = {(x, y) ; f(x, y) = }, ( ) f(x, y) R 2., f(x, y), (i), (ii) : (i) ( ) B, µ(( )) = 0. (ii) {f (x, y); }, \( ) lim f (x, y) = f(x, y), (x, y) \( ) (6.1.4) (ii) (iii) : (iii) \( ) K ε > 0, 0, 0, f (x, y) f(x, y) < ε, ((x, y) K) (6.1.5), f(x, y), f(x, y). 4

5 6.1.2 (6.1.4),,,, f (x, y) f(x, y) (6.1.1), i δ i = sup{d(p, q); p, q i } (6.1.6), d(p, q) R 2 2 p q., δ = δ() = max δ i (6.1.7) 1 i<,, lim δ() = 0 (6.1.8) 6.1.4,, 0, lim δ() = 0 (6.1.9) 0., 0, δ() 0, 0,., R 2 f(x, y) f(x, y)

6 6.1.1 R 2 f g,., (1) f + g. (2) f g. (3) fg. (4) f/g.,, δ > 0, g δ. (5) αf., α. (6) f. (7) sup(f, g). (8) inf(f, g). (9) f + = sup(f, 0). (10) f = inf(f, 0) R 2 {f n }, 0, f \ 0., f [0, 1] [0, 1] f(x, y), (x, y) 1, (x, y) 0., f(x, y), (x, y) x y 6.2 2, 2. R 2, 2 (R 2, B, µ)., 2. 6

7 R 2, f(x, y) 2,. (1) f(x, y),, f(x, y) () : = (6.2.1) f(x, y) = α j χ j (x, y) (6.2.2) j=1., α j R, (j 1)., f(x, y) 2, f(x, y)dxdy = α j µ( j ) (6.2.3),,. (6.2.3), f(x, y) (6.2.2)., (6.2.3) 2. j=1 (2) f(x, y), f(x, y), f(x, y), \( ) {f (x, y); }, 7

8 , I = lim f (x, y)dxdy (6.2.4), f(x, y) 2, I = f(x, y)dxdy (6.2.5), I f(x, y) {f (x, y); }. {f (x, y); } f (x, y)., f(x, y) 2.,,,,,., f(x, y) 2., 2 (6.2.5) I f(x, y) {f (x, y); }, f(x, y) 2., (6.2.5) I f(x, y) {f (x, y); }, f(x, y) 2. (6.2.4), f(x, y) 2. (6.2.4), f(x, y). 8

9 , f(x, y), f(x, y) , x, y.,, I = f(x, y)dxdy.,., R 2, f(x, y), () : = n, f (x, y), f (x, y) = f(ξ j, η j )χ j (x, y), j=1 ((ξ j, η j ) j, 1 j n) (6.2.6), f(x, y) 2, I = f(x, y)dxdy = lim f (x, y)dxdy = lim f(ξ j, η j )µ( j ) (6.2.7) j=1 9

10 ., R = f(ξ j, η j )µ( j ) (6.2.8) j=1., K = [a, b] [c, d], b d a c f(x, y)dxdy (6.2.9) (6.2.5),, f(x, y), x, y., I.,., R 2. f(x, y). () : = n (6.2.10). i f(x, y) M i, m i, f(x, y) M, m., g (x, y) = M i χ i (x, y), (6.2.11) h (x, y) = m i χ i (x, y) (6.2.12),, lim g (x, y) = f(x, y), (6.2.13) lim h (x, y) = f(x, y) (6.2.14) 10

11 , g (x, y) h (x, y) 2 S, s, S = s = g (x, y)dxdy = h (x, y)dxdy = M i µ( i ), (6.2.15) m i µ( i ). (6.2.16)., µ( i ) i., mµ() s S Mµ() (6.2.17), µ() <,, {s } {S }, S = inf S, s = sup s (6.2.18), s S (6.2.19), ( ),, lim s = s, lim S = S (6.2.20) f(x, y) 11

12 6.2.2 R 2, f(x, y)., f(x, y) s = S. v i = M i m i i f(x, y)., V = S s = v i µ( i ) (6.2.21), s = S,,, lim V = lim v i µ( i ) = 0 (6.2.22) f(x, y) 6.2.2, f(x, y),, 6.2.1, lim V = lim v i µ( i ) = 0 f(x, y) lim V = lim v i µ( i ) = 0, ε > 0,, V = v i µ( i ) < ε (6.2.23) 12

13 6.2.1 R 2 f(x, y), f(x, y) 6.2.3, K 0. //, R 2, ε > 0, δ = δ(), v i = M i m i < ε., V = v i µ( i ) < εµ(). (6.2.24) ε, // R 2 f(x, y) B 0, f(x, y)

14 6.3 2, R 2, f(x, y) g(x, y), (1) (6) : (1) α, β, αf(x, y) + βg(x, y), : {αf(x, y) + βg(x, y)}dxdy = α f(x, y)dxdy + β g(x, y)dxdy. (6.3.1) (2) = 1 + 2, : f(x, y)dxdy = f(x, y)dxdy+ f(x, y)dxdy. 1 2 (6.3.2) (3) f(x, y) 0, f(x, y)dxdy 0 (6.3.3)., f(x, y) (x 0, y 0 ), f(x 0, y 0 ) > 0, : f(x, y)dxdy > 0. (6.3.4) 14

15 (3 ) f(x, y) g(x, y), f(x, y)dxdy g(x, y)dxdy (6.3.5)., f(x, y) g(x, y) (x 0, y 0 ), f(x 0, y 0 ) > g(x 0, y 0 ), f(x, y)dxdy > g(x, y)dxdy. (6.3.6) (4) f(x, y), f(x, y)dxdy f(x, y) dxdy (6.3.7) (5) f(x, y)g(x, y) (6), g(x, y) k > 0 k f(x, y), g(x, y) ( ) R 2, f(x, y), f(x, y) M, m, f(x, y)dxdy = αµ() (6.3.8) α., m α M., f(x, y), α = f(x 0, y 0 ) (x 0, y 0 ) 15

16 6.3.1(3 ), mµ() = mdxdy f(x, y)dxdy Mdxdy = Mµ(). (6.3.9), (6.3.8).// R 2,, f(x, y), φ(x, y),, (ξ, η), f(x, y)φ(x, y)dxdy = f(ξ, η) φ(x, y)dxdy (6.3.10) R 2., f(x, y), f(x, y), R 2, K = [a, b] [c, d]. f(x, y), f(x, y), ((x, y) ), f (x, y) = (6.3.11) 0, ((x, y) K\) 16

17 ., χ (x, y), f (x, y) = f(x, y)χ (x, y)., : f(x, y)dxdy = b d a c f(x, y)χ (x, y)dxdy. (6.3.12) 17

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