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1 r phone: , Office hours: 4/8 I.. ɛ-n. ɛ-δ II

2 r III... IV.. grad, div, rot. grad, div, rot 3., B grad, div, rot I, II ɛ-δ web page (

3 r : 6 ma{, } % III

4 r 4 ɛ-n a n n () a n <. n () a n < 5 n (3) ɛ > a n < ɛ n (4) lim n a n b n, c n n ɛ-n b n + n n +, c n log(n + ) 4 3 8: B5

5 r 5 * * sin 3 lim ɛ-δ 4* a n, b n n 3 6 sin a n n k k n b n 5* c n n c n + n ( ) k k n e e c n e n e /e e O.K. e e e c n 6***..7 a n lim n a n b n b n n lim n b n k! n k k n a k k

6 r : B5 a n. ǫ ǫ ǫ n () n. n n a n <. n > a n >. a n <. a n > <. n n > n <. n < (.) 4 < 4 n 4 < n n > a n >. () (). 5 n > a n < 5 (3). 5 ɛ n, ɛ > n < ɛ n < ɛ < ɛ n ɛ < n n > ɛ a n < ɛ (4) ɛ > n > ɛ a n < ɛ

7 r 7 lim a n N(ɛ) lim n ɛ a n n (4) ɛ > n > ɛ a n < ɛ lim n a n b n ǫ ǫ ǫ n c n ǫ ǫ ( ) ep ǫ n b n n b n < ɛ b n + n + b n < ɛ n + < ɛ n + > ɛ n > ɛ ɛ > n > ɛ b n < ɛ lim b n N(ɛ) lim n ɛ b n n ɛ > N(ɛ) ɛ n > N(ɛ) b n < ɛ lim n b n b n c n n c n < ɛ c n < ɛ log(n + ) < ɛ log(n + ) > ɛ ( ) n > ep ɛ ( ) ɛ > n > ep ɛ c n < ɛ

8 r 8 ) lim c n N(ɛ) ep( n ɛ lim c n n ( ) ɛ > N(ɛ) ep ɛ n > N(ɛ) c n < ɛ lim n c n c n b n b n < ɛ b n < ɛ b n b n < ɛ n b n ɛ n b n < ɛ n a n < ɛ n ɛ > N(ɛ) n > N(ɛ) a n < ɛ N(ɛ) b n c n b n N(ɛ) ɛ ( ) c n N(ɛ) ep ɛ

9 r 9 7 f(, y)ddy (a) f(, y) + y y + y 3 (b) f(, y) y y y (c) f(, y) + y y y 5 8: B5 3 > 6 sin > < 6 sin ( ) 6 sin sin 6 ( ) ɛ > δ 6ɛ < δ sin 6 < ɛ sin lim sin lim

10 r f() sin f() f () cos f() f() g() sin ( 3 /6) g(), g () cos +, g () sin + g () g () g () g () g () g() g() g() sin 3 3! + 5 5! 7 7! + < < π/4 > sin E B H C O D O, OB O OC CB sin B sin < (cos ) (sin ) cos 4 8 DHE CD + CE DHE OD OH cos cos CD + CE sin sin cos < < a a a /8 ( a )

11 r sin. lim. sin, cos 3. 4 a n b n a n a n n k n k d log n n a n a n, b n a n a m a m }{{ 4 } }{{ 8 } m + + m + m }{{} 4 m m a n b n b n n b n + n k b n n k + d + n < b n m m+ k k ( m ) m m k m b n m b n b n + k k m+ k m k m m n n k + n k(k ) + ( k ) + k n k

12 r b n m > n < b m b n m kn+ ɛ > N /ɛ m k n d n m n m > n > N b m b n < ɛ b n b n 5 m > n n c m c n kn+ ( ) k k! n kn+ ( k! + ) (n + )! n + + (n + )(n + 3) + (n + )(n + 3)(n + 4) + n (n + ), (n + 3), (n + 4) 3 ( + 3 (n + )! ) 3 + (n + )! 3 < (n + )! > ɛ > N /ɛ m > n > N c m c n < (n + )! < ɛ {c n } {c n } c n n n c n+ c n (n + )! + ( ) (n + )! ( ) (n + )! n + n + < d n : c n d n e n : c n n c n, d n, e n c n n k k! d n e n d n, e n c n d n e n d n e n c n c n (n)! n d n e n c n

13 r 3 8 y, y, u + y, v y ( + y)e y ddy (u, v) 9 a > y, y, a ( + y) cos(y) ddy a) cos(y) ddy y cos(y) ddy b) u y, w y (u, w) (, y) w) (u, (u, w) (, y) p.3, 4, 5 8 8: B5

14 r 4 7 y y y (a) (b) (c) (a) y ( + y)ddy 3 d 3 dy( + y) 3 d [y + y ] y3 t 3 3 y 3 d {(3 ) + { } [ dt (3 t)t + t 3t ] t3 3 + t3 3 [ 3t ] 6 t (3 ) } (b) (y)ddy d dy(y) [ y ] y d y [ d 3 4 ] 8 8 (c) ( + y)ddy d d dy( + y) d [ y + y } [ { ] y y ] d { ( ) + 4 }

15 r 5 () y- () ( + y ) () ddy α α α ( ddy α α + y ) α B y- () B ( + y ) () B ddy α α α ( ddy α α + y ) α 6 4 8: B5 8 + y y u + v u v

16 r 6 y v 4 B 4 u, y u, v (, y) (u, v) 4 dv (u, v) (, y) (u, v) (, y) 4 v du u e v 4 (, y) (u, v) v (4 v) dv e 4 4 e v (4 v) dv e4 3 9 w y a a B a a u a) cos(y)ddy y cos(y)ddy y sin(y) sin(y) a a dy d a y dy a y sin(y) d [ sin(y) ] a y d sin( ) y d sin(y) a dy [ sin(y) ] a a dy{sin(ay) sin(y )} y

17 r 7 a d sin( ) + a dy{sin(ay) sin(y )} a dy sin(ay) [ cos(ay) ] a cos(a ) a a b) u y y y a, y u, w u ± u + 4w, y u ± u + 4w, y u, w u w y > y y a u + u + 4w, y u + u + 4w u + u + 4w a u + u + 4w w w u a w + au a u a w w + au a u a ( + y) cos(w) B [ cos(a ] a au) a (u, v) (, y) + y (, y) (u, v) + y + y dudw cos w dudw B cos(a ) a a) a du a au dw cos w a du sin(a au)

18 r 8 () ().8. () y > y α > + y + y ( + y ) α ( + y ) α () () () α < () < α < α α y y n n n y B n n y C n n y B n C n n n ( + y ) α ddy B n ( + y ) α ddy + C n ( + y ) α ddy B n C n

19 r 9 B n y + y α B n ( ) α ddy B n ( ) α ddy B n ( + y ) α ddy B n ( ) α ddy ddy B n α ddy B n α d dy α d α d α 4 d α α > α < 3/ B n α < 3/ C n y α C n y α ddy C n ( y ) α ddy C n y α ddy C n ( + y ) α ddy y dy d y y α dy y / α 4 C n y α ddy dy y / α / α > α < 3/ C n α < 3/ n α < 3/ () α < 3/ y t, t () () α α > y n n y n B n n y C n n y B n C n n n ( + y ) α ddy B n ( + y ) α ddy + C n ( + y ) α ddy B n C n B n + y α ddy B n α B n ( + y ) α ddy ddy B n α ddy B n α

20 r ddy B n α n d dy α n d α 4 n d α n α < α > 3/ B n α > 3/ C n y α C n y α ddy C n ( + y ) α ddy C n y α ddy C n y α ddy n y dy d n y y α 4 dy y / α n / α < α > 3/ C n α > 3/ n α > 3/ () α > 3/

21 r 7/8 7/ 7/4.5 C z + y z ( + y) F F (, y, z) (y, z, ) F (r) dr C (,, ) C 3 S z +y +y 4 (, y, z) S +z F (r) ds(r) G(r) ds(r) a) F (, y, z) (,, 3), b) G(, y, z) (, y, z) S S 7 5: B5

22 r 7 4 grad div () () grad div 7 9 5: B5 C z + y ( + y) ( ) + (y ) y + cos θ, y + sin θ, z 4 + (cos θ + sin θ) θ [, π] r (θ) ( sin θ, cos θ, (cos θ sin θ)) r (θ) F (r(θ)) sin θ( + sin θ) + cos θ{4 + (cos θ + sin θ)} (cos θ sin θ)( + cos θ) cos θ sin θ sin θ + 8 sin θ cos θ cos θ sin θ + cos θ + 4 sin θ π dθ { cos θ sin θ + cos θ + 4 sin θ} π 3 S, y r cos θ, y r sin θ, z r ( r, θ π)

23 r 3 cos θ r r sin θ, r r θ r sin θ r cos θ, r r r θ r cos θ r sin θ r r cos θ F (r), G(r) r sin θ S S dg(r) ds(r) df (r) ds(r) dr π dr 3 π dθ 3r π r dr 3r 6π π dθ { r 3 cos θ r 3 sin θ + r 3 } dr π dθ (, y) df (r) ds(r) F S z S S y- S y- 4π 3 4π 3 π

24 r 4 4 () grad f e, e y, y grad f e + y e y (a), y r, θ (b) e, e y e r, e θ (a) r r + θ θ y r r y + θ θ y r, θ, y ] [ r θ r y θ y (r, θ) (, y) (, y) (r, θ) [ ] [ (, y) cos θ (r, θ) sin θ (r, θ) (, y) ] [ r sin θ r cos θ cos θ sin θ r sin θ r cos θ ] sin θ cos θ r θ r y cos θ sin θ + r θ r [ ] [ cos θ y sin θ sin θ r cos θ r ] [ ] r θ (b) [ ] [ ] [ ] e r cos θ sin θ e sin θ cos θ e θ grad f e + y e y [ e r e θ ] [ r e y [e e y ] [ y ] [ ] [ ] r r e r + r θ e θ θ [ e e y ] ] [ cos θ sin θ e r e θ sin θ cos θ [ cos θ sin θ ] [ cos θ sin θ ] [ sin θ cos θ sin θ r cos θ r e r e θ ] ] [ ] r θ div F grad F F e + F y e y div F F + F y y

25 r 5 (b ) F e r, e θ (a ), y r, θ grad e F F r e r + F θ e θ F r, F θ F F r e r +F θ e θ F r (cos θ e +sin θ e y )+F θ ( sin θ e +cos θ e y ) (F r cos θ F θ sin θ)e +(F r sin θ+f θ cos θ)e y [ F F y ] [ cos θ sin θ ] [ sin θ cos θ F r F θ ] div F F + F y y (F r cos θ F θ sin θ) + y (F r sin θ + F θ cos θ) (a ), y r, θ div F cos θ r (F r cos θ F θ sin θ) sin θ r + sin θ r (F r sin θ + F θ cos θ) + cos θ F r r + F r r + F θ r θ r r r (rf r) + r f div (grad f) F grad f θ (F r cos θ F θ sin θ) θ (F r sin θ + F θ cos θ) F θ θ F r e r + r θ e θ F r r, div F θ r θ f div F F r r + F r r + F θ r θ f r + r r + f r θ

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2014 S hara/lectures/lectures-j.html r 1 S phone: , 14 S1-1+13 http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html r 1 S1-1+13 14.4.11. 19 phone: 9-8-4441, e-mail: hara@math.kyushu-u.ac.jp Office hours: 1 4/11 web download. I. 1. ϵ-δ 1. 3.1, 3..

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