2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n

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1 ɛ-n 1 ɛ-n lim n a n = α n a n α 2 lim a n = 1 n a k n n k= ɛ-n a n α a n n α lim n a n = α ɛ N(ɛ) n > N(ɛ) a n α < ɛ (1.1.1) ɛ n > N(ɛ) a n α < ɛ (1.1.2) N(ɛ) ( ɛ > N(ɛ) n > N(ɛ) = a n α ) < ɛ (1.1.3) 1 1 2

2 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n α ɛ N(ɛ) N(ɛ) N(ɛ) ɛ N ɛ N ɛ lim n a n = a n a n n lim n a n = + M N(M) n > N(M) a n > M (1.1.4) lim a n = + lim a n = {a n } n n n n N n N

3 3 N N N = 1 4 N = 1 1 N = 1 1 N n a n = 1/n n n n ɛ > n a n α ɛ ɛ ɛ ɛ = 1 6 ɛ = 1 14 ɛ = 1 2 N ɛ a n α N ɛ 3 a n α a n α n a n = 1/n ɛ =.1 n > 1 n > 1 a n α <.1 ɛ = 1 6 n > 2 n > 2 a n α < 1 6 ɛ = 1 12 n > 1 2 ɛ = 1 1 n > 1 3 ɛ > lim n a n = α ɛ = 1 3 N lim n a n = α N ɛ ɛ-n N ɛ n a n α n a n α ɛ a n α N n ɛ a n α ɛ-n ɛ N(ɛ) ɛ N n = 1, 2, 3,... a n = 1 n, b n = 1 log(2 + log(2 + log n)), c 1 n = log(2 + log(2 + log n)) (1.1.5) n n a n b n c n

4 4 a n b n c n b n c n n n a n 1/n b n log n c n 1 8 n n n N ɛ a n α ɛ ɛ n n a n α ɛ ɛ-n n N(ɛ) n = 1, 2, 3,... a n = 3, b n = 1 n, c n = 1, d n = 1 n n n 1, 1 2, 1 3, 1 4, 1 5, 1 6,... e n = (1.1.6) (1.1.7) (1.1.5) n f n = n + 3 n, g n = sin n n, h n = n + 1 n, p n = 2n + 1 n + 1, q 1 n = log(n + 1) (1.1.8) ɛ-n ɛ-n lim a n = α, lim b n = β lim (a n + b n ) = α + β. n n n lim a n = α, lim b n = β lim a nb n = αβ. n n n lim a n = α, lim b a n n = β β lim = α n n n b n β. b n m b m = {b n } a n = n a n n lim a n = α lim a n = β n n α = β

5 5 ɛ-n a n b n = 1 n n k=1 a k lim n a n = α lim n b n = α ɛ-n lim a a 1 + a a n n = α = lim = α n n n a 1 a n ρ 1, ρ 2, ρ 3,... ( n ) / ( n ) b n := ρ j a j ρ j j=1 lim a n = α lim b n = α ρ 1, ρ 2, ρ 3,... n n ρ 1 = ρ 2 = ρ 3 =... = 1 j=1 1.2 ɛ-δ 4 n a n x x a f(x) f(x) a, b f(x) x a b lim x a f(x) = b ɛ δ(ɛ) < x a < δ(ɛ) x f(x) b < ɛ (1.2.1) ( ɛ > δ(ɛ) > < x a < δ(ɛ) = f(x) b ) < ɛ (1.2.2) x a > x = a f(x) a f(a) b f(a) = b x a 4

6 6 δ(ε 2 ) b ε 2 ε 2 ε 1 ε 1 x δ(ε 1 ) a ɛ-n < x a < δ(ɛ) f(x) b < ɛ < x a δ(ɛ) f(x) b ɛ < x a ɛ-n ɛ, δ ɛ, δ x a f(x) b ɛ-n ɛ δ ɛ-n α f(x) b < ɛ δ(ɛ) δ(ɛ) 1) lim x x, a > ( 2) lim x 2 2x + 3 x ) ( ), 3) lim x 2 2x + 3. (1.2.3) x 1 1 x 2 1 4) lim, 5) lim x 1 + x x 1 x 1, 6) lim sin 1 x x, (1.2.4) x 3 a 3 7) lim x a x a 1 + x 1 x 8) lim x x 9) lim x x (1.2.5) f(x) lim f(x) x ɛ-δ.1 x = 1 1, 1 2, 1 3, 1 4,... f(x) := x { } { } lim f(x) = α lim g(x) = β lim f(x) + g(x) = α + β lim f(x)g(x) = αβ x a x a x a x a ɛ-δ

7 () a 1, a 2, a 3,... {a n } {a n } {a n } 1, 2, 3, 4, 5, 6,... 1, 3, 5, 7, 9,... 1, 4, 9, 16, 25,... 1, 2, 5, 1, 1, 132, , () {a n } L n a n < L K n a n > K K, L {a n } n a n L n K () {a n } {b n } {b n } a 1, a 2, a 3,... K L accumulation point a 11 a23 K a 1 a 4 a 2 a3 a 5 a 8 a 15 a 12 a 9 a 1 L

8 8 a n 2 n (1.3.1) a 1 = 1.4, a 2 = 1.41, a 3 = 1.414,... 2 II Bolzano-Weiertrass lim a n = α a n n α ( ) ɛ > N(ɛ) n > N(ɛ) a n α < ɛ (1.4.1) α e ( e = lim n (1.4.2) n n) e x = 1 + x + x2 2! + x3 3! + = lim N N n= x n n! (1.4.3) x e x e x 6 lim N N n= x n n n! lim N N n= x n n n! (1.4.4) e x

9 () a 1 a 2 a 3... a n... a n (monotone) increasing (monotone) decreasing (monotone) non-decreasing (monotone) non-increasing. strictly increasing n n ( 2.2) {a n } lim a n {a n } lim a n n n {a n } lim a n = + {a n } n lim a n = n + ± lim n a n a n 2 n a n 2

10 1 n a n ɛ-δ α {a n } {b k } α {a n } α α {b k } α k b k α (1.4.5) {b k } {a n} {b k } k 1 b k1 > α n 1 k b k b k1 > α b k α {b k } {a n } k n b k = a n (1.4.5) a n = b k a n a n α a n n a n α n m a m a n α {b k } k a n = b k n n a n α (1.4.6) {a n }, {b k } {b n } α ɛ > K(ɛ) > ( ) k > K(ɛ) = b k α < ɛ (1.4.7) k > K(ɛ) α ɛ < b k (1.4.8) a n = b k n α ɛ < a n {a n } n 1 α ɛ < a n1 n > n 1 α ɛ < a n1 a n ɛ > (1.4.7) K(ɛ) K(ɛ) k 1 a n1 = b k1 n 1 n > n 1 α ɛ < a n (1.4.9)

11 11 (1.4.6) ɛ > n 1 > n > n 1 α ɛ < a n < α (1.4.1) lim n a n = α ɛ-δ {a n } α α ( ) a n Cauchy sequence ɛ > N(ɛ) m, n N(ɛ) a m a n < ɛ (1.5.1) ε 1 ε 2 N(ε 1 ) N(ε 2 ) n a n a n a m m, n () a n a n lim sup lim inf 7 4

12 N(ɛ) a n := 1 n b n := 1 n 2 c n := ( 1)n n d n := ( 1)n n {a n }, {b n }, {c n } α c n a n := log n + n k=1 1 k b n := c 1 := 1, n 1 c n+1 := 1 2 n ( 1) k 1 k=1 k (c n + α c n ) c n a n, b n (1.4.4) e x e x = n= x n n! x x > x sin x = x x3 3! + x5 5! x7 7! + x < r < 1 {a n } a n+2 a n+1 r a n+1 a n n = 1, 2, 3,... n x a () lim x a f(x) f(x) (C) ɛ > δ(ɛ) > < x a < δ(ɛ) < y a < δ(ɛ) x, y f(x) f(y) < ɛ

13 b a f(x) f(x) > a < b f(x)dx [a, b] y = f(x) x- f(x) [a, b] n n a = x < x 1 < x 2 <... < x n 1 < x n = b (x i 1, x i ) i = 1, 2,..., n [a, b] i = x i x i 1 = max (x i x i 1 ) = max i 1 i n 1 i n [x i 1, x i ] ζ i i = 1, 2,..., n ζ 1, ζ 2,..., ζ n ζ, ζ S(f;, ζ) = n f(ζ i ) (x i x i 1 ) = i=1 n f(ζ i ) i (2.1.1) ζ S(f;, ζ), ζ b f(x)dx a b a i=1 f(x)dx lim S(f;, ζ) (2.1.2) f(x) > n = 5 R(f; P, ζ) y=f(x) x ζ 1 x ζ 1 2 x ζ 2 3 x ζ 3 4 x ζ 4 5 x 5 x x f(x) = 1 x 1 f(x)dx (2.1.3) () f(x) [a, b] f [a, b]

14 x, y f(x, y) f(x, y) = xy xy- A = {(x, y) a x b, c y d} f A f(x, y)dxdy z = f(x, y) (x, y, z)- f(x, y) xy A A f(x, y)dxdy 1 A A x- a = x < x 1 < x 2 <... < x n 1 < x n = b y- c = y < y 1 < y 2 <... < y m 1 < y m = dm, n A mn I ij = [x i 1, x i ] [y j 1, y j ] 1 i n, 1 j m = max {(x i x i 1 ), (y j y j 1 )} i,j I ij ζ ij = (ξ ij, η ij ) mn ζ ij ζ, ζ S(, ζ) = n i=1 j=1 A m f(ξ ij, η ij ) (x i x i 1 ) (y j y j 1 ) (2.2.1) ζ A f f(x, y)dxdy = lim S(, ζ) (2.2.2) A f(x, y) f 1 f xy- A A A () f(x, y) A f A

15 xy- A = [a, b] [c, d] xy- B B xy B χ B (x, y) 1 (x, y) B χ B (x, y) = (x, y) B B B B ( ) B (2.3.1) B χ B (x, y) dxdy (2.3.2) B B B χ B B B (B ) B B x = x(t), y = y(t) t 1 x() = x(1), y() = y(1) B x(t), y(t) t C 1 - x = a, x = b y = ϕ(x), y = ψ(x) a x b ϕ(x) ψ(x) ϕ(x) ψ(x) x B y- x- y = c, y = d, x = ϕ(y), x = ψ(y) (2.3.3)

16 16 y y x x ( ) B B f(x, y) f B f(x, y)dxdy f(x, y)χ B (x, y) dxdy (2.3.4) B B f B f B B ( ) B f B f B 2.4 n n A = [a, b] [c, d] f(x, y) A ( ) f(x, y) A x [a, b] A f(x, y)dxdy = F (x) = b a d c F (x)dx = f(x, y)dy (2.4.1) b a [ d c ] f(x, y)dy dx (2.4.2) x, y y [c, d] G(y) = b a f(x, y)dx (2.4.3)

17 17 A f(x, y)dxdy = d c G(y)dy = d c [ b a ] f(x, y)dx dy (2.4.4) z = f(x, y) x- (2.4.2) F (x) (2.4.2) b d a c f(x, y)dy dx (2.4.5) a, b, c, d x, y x, y b a dx d c dy f(x, y) (2.4.6) (Riemann Fubini ) f A b a [ d c ] f(x, y)dy dx = d c [ b a ] f(x, y)dx dy (2.4.7) x- x-y y- y- z y x

18 A F (x) x [a, b] f(x, y) A = [, 1] [, 1] (x 2 + y 2 )dxdy, a) A = [1, 3] [, 2] f(x, y) = xy. b) A = [, 1] [, 1] f(x, y) = A A A f(x, y) dxdy 1 3x + y + 1. c) A x =, y =, x + y = 1 f(x, y) = xy dxdy, (2.4.8) 1 3x + y + 1. d) A y = x y = x 2 f(x, y) = (y x 2 ) f(x, y) A b [ d ] d [ b f(x, y)dy dx = a c f A c a ] f(x, y)dx dy (2.4.9) f (2.4.9) A [, 1] [, 1] S k- ( ) p k k = 1, 2,... 1 m k m, n p k, n p k S T T = k=1 { ( m, n ) } < m < p k, < n < p k p k p k (2.4.1) ( (x, y) T ) f(x, y) = 1 ( (x, y) S\T ) (2.4.11) 1 [ 1 f(x, y) dxdy S ] f(x, y)dy dx = 1 [ 1 ] f(x, y)dx dy = 1 (2.4.12) Riemann Lebesgue Lebesgue Lebesgue Lebesgue Riemann

19 19 B B a x b ϕ(x) < ψ(x) x = a, x = b, y = ϕ(x), y = ψ(x) (2.4.13) B f(x, y) dxdy = b [ ψ(x) a ϕ(x) ] f(x, y)dy dx (2.4.14) Fubini c) x =, y =, 2x + y = 1 x, y x y B (x 1) 2 + y 2 1 y x, y f a, b > a 1/2 bx dx dy f(x, y) = bx x dx dy f(x, y) = x 2 ab 1/4 a a dy dx f(x, y) + dy dx f(x, y), y/b ab y/b dy y y dx f(x, y) + 1/2 1/4 dy 1/2 y B dx f(x, y) x 2 y dxdy f(x) x 1. f (t) f(t) t f(x) = f(a) + x a f (t)dt, f (t) = f (a) + 2. f(x) f(x) = f(a) + f (a)(x a) + 3. f(x) n x a t a f (s)ds (2.4.15) (x s)f (s)ds (2.4.16) [a, x] x 1 f (n+1) (x 1 )(x a) n+1 /(n + 1)! x 1

20 x = x(t) x2 t2 f(x) dx = f(x(t)) x (t) dt (2.5.1) x 1 t 1 t 1, t 2 x(t) x 1, x 2 t x t x (t) (x, y) (u, v) (u, v) (x, y) (u, v) x = x(u, v), y = y(u, v) (2.5.2) x = u + v, y = u v (x, y) A (u, v) B f g(u, v) g(u, v) f(x(u, v), y(u, v)). (2.5.3) f(x, y)dxdy u, v A B A y v x u A B f(x, y) dxdy = g(u, v) dudv (2.5.4) A B [x 1, x 2 ] [t 1, t 2 ] x2 t2 f(x) dx = f(x(t)) dt (2.5.5) x 1 t 1 x2 t2 f(x) dx = f(x(t)) x (t) dt (2.5.6) x 1 t 1 x (t)

21 (2.5.2) J(u, v) x (x, y) J(u, v) (u, v) det u y u (2.5.2) A B x = x(u, v) y = y(u, v) B J(u, v) x v. (2.5.7) y v f(x, y) dxdy = g(u, v) J(u, v) dudv (2.5.8) A B A B A B x (t) (x, y) (r, θ) x = r cos θ, y = r sin θ (2.5.9) J(r, θ) = det [ cos θ sin θ ] r sin θ = r cos 2 θ + r sin 2 θ = r (2.5.1) r cos θ dxdy rdrdθ e (x2 +y 2) dxdy x 2 +y 2 1 x 2 +y 2 1 e (x2 +y 2) dxdy = 1 dr r 2π dθ e r2 = 2π 1 e r2 r dr = 2π ( ) 2 e x2 dx = e x2 dx e y2 dy = ] 1 [ e r2 2 = π(1 e 1 ) (2.5.11) e x2 dx = π (2.5.12) R 2 e (x 2 +y 2) dxdy (2.5.13)

22 22 f(x, y)dxdy xy f(x, y) A h(u, v)dudv uv B h y v x u y+dy v+dv y v x x+dx u u+du uv- [u, u + du] [v, v + dv] xy- (x(u, v), y(u, v)), (x(u + du, v), y(u + du, v)), (x(u, v + dv), y(u, v + dv)), (x(u + du, v + dv), y(u + du, v + dv)) (2.5.14) du, dv x [ ] x(u + du, v) x(u, v) u du x x [ ] y(u + du, v) y(u, v) = u du x(u, v + dv) x(u, v) v dv x y u du y y(u, v + dv) y(u, v) = v dv y u v dv y v [ ] (2.5.15) a b (a, b) (c, d) ad bc c d x u du x v dv x x det y u du y = det u v dudv = J(u, v) du dv (2.5.16) v dv y y u v 2.6

23 23 n 1. n n n n n n n (x 1, x 2,..., x n ), (u 1, u 2,..., u n ) x i u 1 u n f(x 1, x 2,..., x n ) dx 1 dx 2 dx n = g(u 1, u 2,..., u n ) (x 1, x 2,..., x n ) A B (u 1, u 2,..., u n ) du 1du 2 du n (2.6.1) B (u 1, u 2,..., u n ) A g f n n x 1 x 1 x u 1 u 2 1 u n (x 1, x 2,..., x n ) (u 1, u 2,..., u n ) = det x 2 x 2 x u 1 u 2 2 u n x n u 2 x n u 1 x n u n (2.6.2) x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ (2.6.3) (r, θ, φ) r, θ π, φ 2π sin θ cos φ r cos θ cos φ r sin θ sin φ (x, y, z) (r, θ, φ) = det sin θ sin φ r cos θ sin φ r sin θ cos φ = r 2 sin θ (2.6.4) cos θ r sin θ

24 24 n r n-x x x 2 n r 2 V n (r) n- x n r 2 x 2 n (n 1) x n n V n (r) = r r dx n V n 1 ( r 2 x 2 n) (2.6.5) r 1 r n a n V n (r) = r n V n (1) = r n a n (2.6.6) a n a n 1 a 2 = π, a 3 = 4π/3 a n V n (r) 2 n 1 +x x2 n ) dx 1 dx 2... dx n R n e (x π n/2 n drr n 1 c n e r2 c n n 1 n- a n c n = na n π n/2 = n a n dr r n 1 e r2 (2.6.7) a n V n (r) Γ / () f(x) [a, b] f [a, b] f f f(x) f 2.7.1

25 (i) f(x) x = a lim f(x) = f(a) (2.7.1) x a (ii) f(x) [a, b] [a, b] c a c b f c [a, b] lim f(x) = f(c) (2.7.2) x c x = a, b f lim f(x) = f(a) lim x a+ f(x) = f(b) (2.7.3) x b f (2.7.2) (2.7.2) c 2.7.2(ii) ɛ δ c [a, b] ɛ > δ(ɛ, c) > x c < δ(ɛ, c) = f(x) f(c) < ɛ (2.7.4) δ ɛ c c a δ(ɛ, c) f(x) [a, b] δ(ɛ, c) c [a, b] c δ(ɛ) f(x) [a, b] I ( ) f(x) I ɛ > δ(ɛ) > c I x c < δ(ɛ) = f(x) f(c) < ɛ (2.7.5) δ c f(x) = 1 (, 1) (2.7.6) x (, 1)

26 26 Remark. uniform ɛ-δ x x a y y y y y x x c c δ x c c c ( ) a < b [a, b] ɛ > δ(ɛ) > x, y [a, b] x y < δ(ɛ) = f(x) f(y) < ɛ (2.7.7) δ(ɛ) x, y [a, b] [a, b] f(x) = (x, x 1,..., x n ) [x i 1, x i ] f(x) m i (f; ), M i (f; ) n n s(f; ) := m i (f; ) (x i x i 1 ), S(f; ) := M i (f; ) (x i x i 1 ) (2.7.8) i=1 i=1 s(f; ) S(f; ) n = 5 x x 1 x 2 x 3 x 4 x 5 x

27 27 ζ s(f; ) R(f;, ζ) S(f; ) (2.7.9) f [a, b] lim s(f; ) lim S(f; ) (2.7.1) S lim s(f; ) = lim S(f; ) = S (2.7.11) (2.7.9) lim R(f;, ζ) S ζ [a, b] 2 n (n) s(f; ) S(f; ) 1. s(f; ) s(f; ) S(f; ) S(f; ) (2.7.12) s S s S 1 2 s(f; 1 ) S(f; 2 ) (2.7.13) s(f; 1 ) s(f; 12 ) S(f; 12 ) S(f; 2 ) (2.7.14) 2. [a, b] 2 n (n) (2.7.12) s(f; (n) ) n S(f; (n) ) n (2.7.12) 3. s(f; (n) ) S(f; (n) ) n s S s(f; (n) ) S(f; (n) ) s(f; (n) ) s S(f; (n) ) S (2.7.15)

28 28 s S s S 2 n - 4. s = S s(f; (n) ) S(f; (n) ) S(f; (n) ) s(f; (n) ) = 2 n i=1 {M i (f; (n) ) m i (f; (n) )} (x i x i 1 ) (2.7.16) M i (f; (n) ) m i (f; (n) ) [x i 1, x i ] f(x) ɛ ɛ (2.7.7) δ(ɛ) (n) = (b a)/2 n (2.7.7) [x i 1, x i ] x, y f(x) f(y) < ɛ f(x) x, y ɛ ɛ > (n) < δ(ɛ) M i (f; (n) ) m i (f; (n) ) < ɛ (2.7.16) S(f; (n) ) s(f; (n) ) < 2 n i=1 ɛ (x i x i 1 ) = ɛ(b a) (2.7.17) b a ɛ ɛ ɛ s = S (2.7.15) lim {S(f; n (n) ) s(f; (n) )} = (2.7.18) s(f; (n) ) s = S = S(f; (n) ) (2.7.19) 5. (2.7.17) ɛ ɛ (2.7.7) δ(ɛ) 3 M i (f; ) m i (f; ) < ɛ (2.7.17) S(f; ) s(f; ) < ɛ(b a) (2.7.2) 6. S (2.7.17) ɛ > n S(f; (n) ) s(f; (n) ) < ɛ(b a) (2.7.21) 2 (2.7.2) (2.7.19) S(f; 2 ) s(f; 2 ) < ɛ(b a) (2.7.22) s(f; (n) ) S S(f; (n) ) (2.7.23)

29 29 (2.7.13) s(f; (n) ) S(f; 2 ) s(f; 2 ) S(f; (n) ) (2.7.24) s(f; 2 ) S(f; 2 ) (2.7.21) (2.7.23) S(f; (n) ) < s(f; (n) ) + ɛ(b a) S + ɛ(b a) (2.7.25) s(f; (n) ) > S(f; (n) ) ɛ(b a) S ɛ(b a) (2.7.26) S S(f; (n) ) < S + ɛ(b a) S ɛ(b a) < s(f; (n) ) S (2.7.27) (2.7.24) s(f; 2 ) S(f; (n) ) S + ɛ(b a) S(f; 2 ) s(f; (n) ) S ɛ(b a) (2.7.28) (2.7.22) (2.7.22) s(f; 2 ) S(f; 2 ) ɛ(b a) (2.7.28) s(f; 2 ) (2.7.3) 2 s(f; 2 ) S ɛ(b a) ɛ(b a) = S 2ɛ(b a) (2.7.29) S 2ɛ(b a) s(f; 2 ) S + ɛ(b a) (2.7.3) S ɛ(b a) S(f; 2 ) S + 2ɛ(b a) (2.7.31) ɛ S s(f; 2 ) < 2ɛ(b a) (2.7.32) lim s(f; 2) = S (2.7.33) 2 ɛ δ ɛ = 2ɛ(b a) (2.7.31) 2 ɛ δ S S(f; 2 ) < 2ɛ(b a) (2.7.34) lim S(f; 2) = S (2.7.35) 2 lim s(f; 2) = lim S(f; 2) = S (2.7.36) 2 2 S

30 3 f (2.7.17)

31 dx dx (1) x 1 + x2 (2) 1 1 dx = lim x ɛ 1 ɛ 1 x dx, 1 L 1 dx = lim dx (2.8.1) 1 + x2 L 1 + x2 x = ɛ ɛ + [, L] L lim ɛ lim lim a ɛ ɛ + ɛ + x a lim lim lim a x a x a x a x a lim lim lim x a x a+ x a + dxf(x) f(x)dx = L lim K K L + f(x)dx (2.8.2) ( ) A f f(x, y)dxdy A (a) A a f a {A n } n A (b) A {A n } n A lim n f(x, y)dxdy A n (2.8.3) {A n } {A n } A f(x, y)dxdy {A n} f(x, y)dxdy A (a), (b) f A n {A n } A n A n

32 32 α > 1 1 (a) (x 2 + y 2 dxdy, (b) ) α (x 2 + y 2 dxdy (2.8.4) ) α <x 2 +y 2 1 x 2 +y 2 1 (a) A n 1 n 2 x 2 + y 2 1 A n 1 n r 1, θ 2π 1 2π 1 A n (x 2 + y 2 ) α dxdy = dθ drr 1 1 = 2π dr r 1 2α (2.8.5) 1/n r2α 1/n α < 1 n α 1 n α < 1 A n A n A B A 1 (x 2 + y 2 ) α dxdy 1 B (x 2 + y 2 dxdy (2.8.6) ) α A n A n α < 1 (b) 1 x 2 + y 2 n 2 A n 1 A n (x 2 + y 2 ) α dxdy = n 1 drr 1 r 2α = n 1 dr r 1 2α (2.8.7) n α > 1 α 1 (a) α > 1 (a) f(x, y) (a) (b) (i) f(x, y)dxdy A (ii) (a) (b) f(x, y)dxdy A n

33 33 S n := f(x, y)dxdy n > af(x) A n S n S := lim S n n A n A n A n B n A B f(x, y)dxdy f(x, y)dxdy A n B n A n+1 A n A B A n+1 B n f(x, y)dxdy f(x, y)dxdy f(x, y)dxdy A n B n A n+1 lim f(x, y)dxdy lim f(x, y)dxdy n A n n B n f(x, y) dxdy f(x, y)dxdy A f(x, y) dxdy A A n A

34 ( ) n- t n- r(t) = (x 1 (t), x 2 (t), x 3 (t),..., x n (t)) t t [, 1] t 1 t = t = 1 t n- (1, 1) r(t) = (t, t) r(t) = (t 2, t 2 ) r(t) = ( t, t) r(t) = (x 1 (t), x 2 (t), x 3 (t)) r(t) = (x(t), y(t), z(t)) () n- r(t) = (x 1 (t), x 2 (t), x 3 (t),..., x n (t)) (1) x i (t) t (2) r (t) t n- r(t) t 1 (x, y, z) F (x, y, z) = (,, ) a = (a, b, c) Step 1. (a, b, c) F F a 2 + b 2 + c 2 F > F <

35 35 Step 2. a = (a, b, c) F = (F x, F y, F z ) F ( a n = a2 + b 2 + c, b 2 a2 + b 2 + c, c ) F 2 a2 + b 2 + c 2 n F n a b a b (F n)n Step 1 (F n) a 2 + b 2 + c 2 = F x a + F y b + F z c = F a (3.2.1) Step 1 Step 3. r = (,, ) n r 1, r 2,..., r n = a = (a, b, c) r i 1 r i l i l i F i l i Step 2 F i (r i r i 1 ) (a, b, c) n F i (r i r i 1 ) (3.2.2) i=1 Step 4. r(t) = (x(t), y(t), z(t)) t 1 r = (x, y, z) F (r) = (F x (x, y, z), F y (x, y, z), F z (x, y, z)) (x, y, z) Step 3 r =, r 1, r 2,..., r n = a Step 3 r i 1 r i l i l i F (r) r l i F (r) Step 3 = n F (r i ) (r i r i 1 ) (3.2.3) lim i=1 i=1 n F (r i ) (r i r i 1 ) (3.2.4) n i r i r i 1

36 C : r(t) = (x(t), y(t), z(t)) t 1 F (r) C F C r, r 1,..., r n C i = 1, 2,..., n r i 1 r i ζ i ζ 1 ζ n ζ ζ S(, ζ) n F (ζ i ) (r i r i 1 ) (3.2.5) i=1 = max i r i 1 r i ζ C F F (r) dr = lim S(, ζ) (3.2.6) C C (1, 1, 1) y = z = x 2 F z F (x, y, z) = y (3.2.7) F (r) dr C F ( ) C F (x, y, z) x, y, z C x F (r) dr x i (t) F

37 ( ) t F (r) dr = 1 C F (r(t)) r (t) dt (3.3.1) t r (t) r(t) = (x(t), y(t), z(t)) t (x (t), y (t), z (t)) r (t) F (t) η i ζ [, 1] n- t = < t 1 < t 2 <... < t n 1 < t n = 1 [t i 1, t i ] s i t i r i = r(t i ) ζ i = r(s i ) 8 S(, ζ) = t i 1 t i n F (r(s i )) (r(t i ) r(t i 1 ) ) (3.3.2) i=1 r(t i ) r(t i 1 ) r (t i 1 )(t i t i 1 ) (3.3.3) 9 (3.3.2) S(, ζ) n F (r(s i )) r (t i 1 )(t i t i 1 ) (3.3.4) i=1 r (t) t i 1 s i r (t i 1 ) r (s i ) S(, ζ) n F (r(s i )) r (s i )(t i t i 1 ) (3.3.5) 1 (3.3.5) i=1 F (r(t)) r (t) dt 1 F (r(t)) r (t) dt 8 (3.2.5) t i, s i (3.3.2) t F (r(t i )) 9

38 r (t) dt (3.3.6) a = (x, y, z) a = x 2 + y 2 + z 2 F ( r(t) ) = r (t) r (t) (3.3.7) 1 f(x, y, z) 1 f(x, y, z) r (t) dt (3.3.8) F (r(t)) = r (t) r f(r(t)) (3.3.9) (t) (3.3.8) (3.3.8) (3.3.1) 1 F (r) dr = F (r(t)) r (t) dt (3.3.1) C F C ρ C F (r) 1 ρ(r(t)) r (t) dt (3.3.11) F (r) 1 F (r(t)) r (t) dt (3.3.12) f(r) C f(r) ds C f(r) dr 1 f(r(t)) r (t) dt (3.3.13) (3.3.1) 1 1 F (r) dr = F (r(t)) r r (t) (t) dt = F (r(t)) C r (t) r (t) dt = (F t)ds (3.3.14) C

39 t(r) r (t) r (t) (F t)ds F F t ds C F (r(t)) r (t)dt = C C C f(r(t)) r (t) dt = F (r)dr F C f(r) dr f 3.4 S ρ(r) S v(r) 1 f(r) dσ(r) = lim f (3.4.1) S dσ(r) r

40 4 S 1, S 2, S 3,... S i η i S i S i η i S i τ i (η i ) S(, η) = i f(η i ) τ i (η i ) (3.4.2) f(r)dσ(r) = lim S( ; η) = lim f(η i ) τ i (η i ) (3.4.3) S η S i

41 41 ( ) F (r) ds(r) F (r) n(r) dσ(r) (3.4.4) S S n(r) r 1 F n F ds = ndσ dσ ds Step 1. S u Su Step 2. u u 1 n u n Step 1 (u n) S Step 3. u S i S i S i 1 n i S i Step 2 (u i n i )S i u i S i (u i n i ) S i (3.4.5) i u i n i S i Step 4. S i Step 3 S i η i η i S i S i τ i (η i ) S i τ i (η i ) S i τ i 1 n i u i = u(η i ) (u i n i )τ i (η i ) S(, η) = i (u i n i )τ i (η i ) = i ( u(ηi ) n(η i ) ) τ i (η i ) (3.4.6) η S u(r) ds(r) = lim ( u(ηi ) n(η i ) ) τ i (η i ) (3.4.7) f(r) u(r) n(r) (3.4.3) u(r) ds(r) = u(r) n(r) dσ (3.4.8) (3.4.4) S i S

42 dσ n a (a 1 a 2, a 3 ) b (b 1, b 2, b 3 ) a b a b = (a 2 b 3 a 3 b 2, a 3 b 1 a 1 b 3, a 1 b 2 a 2 b 1 ) (3.5.1) a b sin θ θ a, b a b a, b a b r = r(u, v) (u, v) r(u) z = f(x, y) x = u, y = v, z = f(u, v) S (u, v) U f(r)dσ(r) U uv- S U f(r(u, v)) dudv (3.5.2) r S (u, v) U f r r(u, v) dσ dudv f(x, y)dxdy (u, v) A g(u, v)dudv J(u, v) g(u, v) J(u, v) dudv dxdy dudv (u, v)- dudv xy- uv- uv- (u, v) (u 2, v 2 ) = (u + u, v + v) u, v (u, v) (x(u, v), y(u, v), z(u, v)) (u 2, v 2 ) (x(u 2, v 2 ), y(u 2, v 2 ), z(u 2, v 2 )) ( x (x(u 2, v), y(u 2, v), z(u 2, v)) (x(u, v), y(u, v), z(u, v)) u (x(u, v 2 ), y(u, v 2 ), z(u, v 2 )) (x(u, v), y(u, v), z(u, v)) y z ) u, u, u u u = ( x u, y u, z ) u (3.5.3) u ( x y z ) ( x v, v, v v v v = v, y v, z ) v (3.5.4) v r = (x, y, z) r ( x u := u, y u, z ), u r ( x v := v, y v, z ) v (3.5.5) r u u r v (3.5.6) v

43 43 r u r v u v r = u r u v (3.5.7) v dσ dudv dσ r u r v f(r)dσ(r) = f ( r(u, v) ) r u r dudv (3.5.8) v S U u, v (3.4.4) dσ n (3.5.6) r u r v (3.5.9) 1 r n(r) = ± u r v r u r v (3.5.1) ± (u, v) S F (r) n dσ = ± F (r(u, v)) U r u r v r u r r u r dudv = ± v v U ( r F (r(u, v)) u r ) dudv (3.5.11) v n ± n F r u r v r x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ θ, φ r θ = r(cos θ cos φ, cos θ sin φ, sin θ), r = r( sin θ sin φ, sin θ cos φ, ) (3.5.12) φ r θ r φ = r2 sin θ (sin θ cos φ, sin θ sin φ, cos θ) (3.5.13) r θ r φ = r2 sin θ ˆr (3.5.14) ˆr = r/ r r n U f(r) r 2 sin θ dθ dφ (3.5.15)

44 44 r 2 sin θ dr dθ dφ F (r) ˆr r 2 sin θ dθ dφ (3.5.16) U θ φ

45 gradient, divergence, rotation A A = (A x, A y, A z ) x, y, z d- 1 x i = d j=1 R ijx j R ij d d R ij φ(r) φ φ d- d- (A 1, A 2,..., A d ) A i = d j=1 R ija j R R B ij 1 i, j d d 2 B ij = d k,l=1 R ikr jl B kl

46 gradient, divergence, rotation r (gradient, divergence, rotation ) (x, y, z) φ(x, y, z) grad φ(x, y, z) ( φ x, φ y, φ ) z φ gradient A (4.1.1) div A(x, y, z) A x x + A y y + A z z A divergence A rot A(x, y, z) = ( Az y A y z, A x z A z x, A y x A ) x y (4.1.2) (4.1.3) A rotation curl gradient, divergence ( x, y, ) z (4.1.4) grad φ = φ, div A = A, rot A = A (4.1.5) 4.2 gradient, potential Gradient Gradient φ t D t φ lim h φ(r + ht) φ(r) h (4.2.1) φ t t t D t φ

47 47 gradient ( gradient) φ(r + ht) φ(r) D t φ = lim = t grad φ (4.2.2) h h t φ = grad φ t t t x, t y, t z φ(r + ht) φ(r) = φ(x + ht x, y + ht y, z + ht z ) φ(x, y, z) φ x ht x + φ y ht y + φ z ht z (4.2.3) grad φ ht h (4.2.2) φ(r + ht) φ(r) D t φ = lim = t grad φ (4.2.4) h h t t grad φ (4.2.2) grad φ (Gradient ) grad φ D t φ t D t φ (4.2.2) (4.2.2) grad φ (4.1.1) F (r) φ(r) F (r) = grad φ(r) = φ(r) (4.2.5) F φ F ( ) φ F C F (r) dr = φ( ) φ( ) (4.2.6) C φ( ) φ( ) C φ C r(t) t 1 F = grad φ F (r) dr = 1 C grad φ ( r(t) ) r (t) dt (4.2.7)

48 48 r = (x, y, z) 1 d φ dx φ(r(t)) = dt x dt + φ y grad φ ( r(t) ) r (t) dt = 1 dy dt + φ z dz dt = grad φ(r) r (4.2.8) d [ ] 1 dt φ(r(t)) dt = φ(r(t)) = φ( ) + φ( ) (4.2.9) rotation 4.3 Divergence Gauss Divergence (x, y, z ) (x +, y +, z + ) S F (r) F (r) ds(r) S F (r) r x- x = x ( 1,, ) x = x + x (1,, ) ( ) F x (x +, y, z ) F x (x, y, z ) 2 F x x 2 = F x x 3 (4.3.1) 12 y- ( ) F y (x, y +, z ) F x (x, y, z ) 2 F y y 3, (4.3.2) z- S ( ) F z (x, y, z + ) F x (x, y, z ) 2 F z z 3 (4.3.3) F (r) ds(r) S ( Fx x + F y y + F ) z 3 + (higher orders) (4.3.4) z (4.1.2) div F 1 div F (r) = lim 3 F (r) ds(r) (4.3.5) S r divergence F 3 F div F divergence F x F x (x +, y, z ) (x, y, z ) F x x, y, z 4 13 S

49 ( ) V S = V V F (r) V div F (r)dxdydz = S F (r) ds(r) (4.3.6) divergence V (divergence ) A(r) r A divergence 1 div A(r) lim A(r) ds(r) (4.3.7) V V V r V V V V V divergence divergence (Green) V V V φ, ψ (r) ( ) φ 2 ψ + φ ψ dxdydz = φ ψ ds (4.3.8) V V ( ) φ 2 ψ ψ 2 φ dxdydz = Green Green V V ( ) φ ψ ψ φ ds (4.3.9) (φ ψ) = φ ψ + ( φ) ( ψ) (4.3.1) V (4.3.8) (4.3.8) φ, ψ (4.3.9) 4.4 Rotation Stokes rotation A r

50 5 r (x, y, z ) (x +, y +, z + ) (x, y, z ) ɛ n = (n x, n y, n z ) C A(r) dr rotation C ( rotation) n ɛ C 1 lim ɛ πɛ 2 A(r) dr = n rot A (4.4.1) C n rot A n C n z- (,, z) y- α, z- β n n n x cos β sin β cos α sin α sin α cos β = sin β cos β 1 = sin α sin β (4.4.2) n y n z 1 sin α cos α C xy- x = ɛ cos t, y = ɛ sin t t 2π r r cos β sin β cos α sin α ɛ cos t cos α cos β cos t sin β sin t r(t) = r + sin β cos β 1 ɛ sin t = r + ɛ cos α sin β cos t + cos β sin t 1 r (t) sin α cos α 1 cos α sin α cos t (4.4.3) cos α cos β sin t sin β cos t r (t) = ɛ cos α sin β sin t + cos β cos t (4.4.4) sin α sin t A(r) C A(r) f(r) = f(r ) + f r (r r ) + O( r r 2 ) { f = f(r ) + ɛ x ( cos α cos β cos t sin β sin t ) + f y ( ) f ( ) } cos α sin β cos t + cos β sin t + sin α cos t z + O(ɛ 2 ) (4.4.5) f = A x, A y, A z O(ɛ 2 ) r (t) t 2π

51 51 C ( A(r) dr = πɛ 2 A x y cos α + A x z sin α sin β + A y x cos α A y sin α cos β z A z x sin α sin β + A z y sin α cos β ) = πɛ 2 ( A x y n z + A x z n y + A y x n z A y z n x A z x n y + A z y n x = πɛ 2 n rot A + O(ɛ 3 ) (4.4.6) rotation r (4.4.1) ) (4.4.1) n n rot A rot A grad φ (Rotation ) rot φ (4.4.1) n (4.4.1) rotation Stokes (Stokes) S C S C F (r) F (r) dr = rot F (r) ds(r) (4.4.7) C S F (r) rotation-free rot F = A B F (r) dr A B C C A B C 1, C 2 C 2 B A C 2 C 1 C 2 A B A F (r) dr = F (r) dr + F (r) dr = F (r) dr F (r) dr (4.4.8) C 1 C 2 C 1 C 2 C 1 C 2 rot F = F (r) dr = rot F ds = (4.4.9) C 1 C 2 S

52 52 S C 1 C 2 F (r) dr = F (r) dr (4.4.1) C 1 C (1) V V ( ) F ds = F dxdydz (4.5.1) V V (2) F (r) = aφ(r) a ( ) φ(r)ds(r) = φ dxdydz (4.5.2) V V (3) F (r) = a A(r) a ( ) ds(r) A(r) = A dxdydz (4.5.3) V V C S C = S S C (4) Stokes ( ) A dr = A ds (4.5.4) S S (5) A(r) = aφ(r) a φ(r)dr = ds φ (4.5.5) S S (6) A(r) = a F (r) a ( ) dr F (r) = ds(r) F (r) (4.5.6) S S

53 A(r) rot E = (rotation-free ) rot E = φ(r) E(r) = grad φ(r) (4.6.1) E E = grad φ rot E = A B C C E(r) dr C φ φ(r) = φ E(r) dr C r (4.6.2) C grad φ = E φ(r) φ div B = (divergence-free ) div B = A(r) B(r) = rot A(r) (4.6.3) B A, A φ A A = grad φ grad φ A(r) A A x1 A(x 1, y 1, z 1 ) = B z (x, y 1, z 1 )dx y1 B x (, y, z 1 )dy x 1 B y (x, y 1, z 1 )dx (4.6.4)

54 54 A B A B = rot A rot A = rot (A grad φ) = rot A rot grad φ = rot A = B (4.6.5) A B = rot A = rot A rot (A A ) = B B = (4.6.6) A A rotation-free A A gradient divergence-free rotation-free ( ) F divergence-free B rotation-free E F (r) = E(r) + B(r), rot E =, div B = (4.6.7) E 1, B 1 E 2, B 2 E 1 E 2 = B 2 B 1 = grad ψ, ψ = (4.6.8) ψ E 1, B 1 (4.6.7) (4.6.8) E 2, B 2 (4.6.7) ( ) F φ A F (r) = grad φ(r) + rot A(r) (4.6.9) E E = grad φ B B = rot A div E = div F, rot E = (4.6.1) rot B = rot F, div B = (4.6.11) div F rot F F (r) E B

55 ψ(r) div E(r) = ψ(r), rot E(r) = (4.6.12) E G(r) div B(r) =, rot B(r) = G(r) (4.6.13) B rot E = φ E = grad φ φ E E(r) = grad φ(r) (4.6.14) div ψ(r) = div E(r) = div grad φ(r) (4.6.15) div grad ( 2 ) div grad φ(x, y, z) = x y z 2 φ(x, y, z) φ(x, y, z) (4.6.16) φ φ(x, y, z) = ψ(x, y, z) (4.6.17) φ E (4.6.17) φ (4.6.17) Poisson ψ(r) φ(r) = 1 4π ψ(q) dv(q) (4.6.18) r q ψ(r) (4.6.17) 14 dv(q) q φ E(r) = grad φ(r) B A B(r) = rot A(r) (4.6.19) rot rot ( rot A(r) ) = rot B(r) = G(r) (4.6.2) A 14 1 q = 4πδ(r q) r q q q

56 56 (4.6.2) rot rot ( rot A(r) ) = A(r) + grad ( div A(r) ) (4.6.21) div A(r) = r (4.6.22) A A(r) = G(r) (4.6.23) (4.6.17) A(r) = 1 4π G(q) dv(q) (4.6.24) r q (4.6.22) E E 1, E 2 i = 1, 2 div E i = ψ, rot E i = (i = 1, 2) (4.6.25) div Ẽ =, rot Ẽ = (4.6.26) Ẽ = E 1 E 2 Ẽ rotation-free φ Ẽ = grad φ (4.6.27) φ = div grad φ = (4.6.28) φ E (4.6.12) φ (4.6.28) E = E + grad φ (4.6.12) φ E φ = φ E = E + grad φ B 1, B 2 B = B 1 B 2 div B =, rot B = (4.6.29) (4.6.26)

57 φ, φ φ =, φ = (4.6.32) E(r) = grad φ(r) + grad φ, φ(r) = 1 4π B(r) = rot A(r) + grad φ, A(r) = 1 4π ψ(q) dv(q) (4.6.3) r q G(q) dv(q) (4.6.31) r q div E = div F rot E = div B = rot B = rot F E, B F = E + B div E = div F, rot E = (4.6.33) E B = F E B div B = div F div E = (4.6.34) E, B F = E + B, div B =, rot E = (4.6.35) E, B Q2 E B = F E Q2 Q Q Poisson Poisson φ(r) = ρ(r) (4.6.36) Poisson V V V φ f φ R 3 φ φ 1, φ 2 ψ(r) = φ 1 (r) φ 2 (r) ψ(r) =, r V (4.6.37) ψ(r) =, r V (4.6.38)

58 58 ψ Green Green ( ψ ψ + ( ψ) 2) dxdydz = ψ ψ ds (4.6.39) V ψ = ( ψ) 2 dxdydz = (4.6.4) V V ψ = V (4.6.41) ψ V

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9 1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),

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