. p.1/14
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- ともひろ ちづ
- 5 years ago
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1 . p.1/14
2 F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y). p.2/14
3 F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y) (x,y) h. p.2/14
4 F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y) (x,y) h h { F 2 (x+ h,y) F 2 2(x h,y) F 2 1(x,y+ h)+f 2 1(x,y h)} +( ) 2. p.2/14
5 F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y) (x,y) h h { F 2 (x+ h,y) F 2 2(x h,y) F 2 1(x,y+ h)+f 2 1(x,y h)} +( ) 2 =h {( F 2 (x,y)+ hf 2 2x(x,y) ) ( F 2 (x,y)+( h)f 2 2x(x,y) )} h {( F 1 (x,y)+ hf 2 1y(x,y) ) ( F 1 (x,y)+( h)f 2 1y(x,y) )} +( ). p.2/14
6 F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y) (x,y) h h { F 2 (x+ h,y) F 2 2(x h,y) F 2 1(x,y+ h)+f 2 1(x,y h)} +( ) 2 =h {( F 2 (x,y)+ hf 2 2x(x,y) ) ( F 2 (x,y)+( h)f 2 2x(x,y) )} h {( F 1 (x,y)+ hf 2 1y(x,y) ) ( F 1 (x,y)+( h)f 2 1y(x,y) )} +( ) = h 2 {F 2x (x,y) F 1y (x,y)} +( ). p.2/14
7 D. p.3/14
8 D h 0. p.3/14
9 D h 0 ( F2 F dr = C D x F ) 1 ds y C D. p.3/14
10 [ ] C y = x y = x 2 (0 x 1) I = (xy + x 2 )dx + x 2 dy C 1. I 2. I. p.4/14
11 [ ] C y = x y = x 2 (0 x 1) I = (xy + x 2 )dx + x 2 dy C 1. I 2. I [ ]. p.4/14
12 [ ] C y = x y = x 2 (0 x 1) I = (xy + x 2 )dx + x 2 dy C 1. I 2. I [ ] 1. C 1 : r 1 (t) = (t,t 2 ), C 2 : r 2 (t) = (t,t) (0 t 1). p.4/14
13 [ ] C y = x y = x 2 (0 x 1) I = (xy + x 2 )dx + x 2 dy C 1. I 2. I [ ] 1. C 1 : r 1 (t) = (t,t 2 ), C 2 : r 2 (t) = (t,t) (0 t 1) I = (xy + x 2 )dx + x 2 dy + (xy + x 2 )dx + x 2 dy C 1 C 2. p.4/14
14 [ ] C y = x y = x 2 (0 x 1) I = (xy + x 2 )dx + x 2 dy C 1. I 2. I [ ] 1. C 1 : r 1 (t) = (t,t 2 ), C 2 : r 2 (t) = (t,t) (0 t 1) I = (xy + x 2 )dx + x 2 dy + (xy + x 2 )dx + x 2 dy C 1 C 2 = 1 0 (t 3 + t 2 + 2t 3 )dt (t 2 + t 2 + t 2 )dt = p.4/14
15 2.C { D I = x (x2 ) } y (xy x2 ) D dxdy. p.5/14
16 2.C { D I = D x (x2 ) } y (xy x2 ) = xdxdy D dxdy. p.5/14
17 2.C { D I = D x (x2 ) } y (xy x2 ) = xdxdy D 1 { x } = xdy dx 0 x 2 1 = x(x x 2 )dx = dxdy. p.5/14
18 [ ] D a 2 x 2 + y 2 b 2 C I = 2xydx + (x 3 y)dy C 1. I 2. I. p.6/14
19 [ ] D a 2 x 2 + y 2 b 2 C I = 2xydx + (x 3 y)dy C 1. I 2. I [ ]. p.6/14
20 [ ] D a 2 x 2 + y 2 b 2 C I = 2xydx + (x 3 y)dy C 1. I 2. I [ ] 1. C 1 :(b cost,bsin t), C 2 :(a cos t,asin t) (0 t 2π). p.6/14
21 [ ] D a 2 x 2 + y 2 b 2 C I = 2xydx + (x 3 y)dy C 1. I 2. I [ ] 1. C 1 :(b cost,bsin t), C 2 :(a cos t,asin t) (0 t 2π) I = 2xydx + (x 3 y)dy + 2xydx + (x 3 y)dy C 1 C 2. p.6/14
22 [ ] D a 2 x 2 + y 2 b 2 C I = 2xydx + (x 3 y)dy C 1. I 2. I [ ] 1. C 1 :(b cost,bsin t), C 2 :(a cos t,asin t) (0 t 2π) I = 2xydx + (x 3 y)dy + 2xydx + (x 3 y)dy C 1 C 2 2π { = 2b 2 costsin t( b sin t) + (b 3 cos 3 t b sin t)b cost } dt 0 0 { + 2a 2 cos t sin t( a sin t) + (a 3 cos 3 t a sin t)a cos t } dt 2π. p.6/14
23 [ ] D a 2 x 2 + y 2 b 2 C I = 2xydx + (x 3 y)dy C 1. I 2. I [ ] 1. C 1 :(b cost,bsin t), C 2 :(a cos t,asin t) (0 t 2π) I = 2xydx + (x 3 y)dy + 2xydx + (x 3 y)dy C 1 C 2 2π { = 2b 2 costsin t( b sin t) + (b 3 cos 3 t b sin t)b cost } dt 0 0 { + 2a 2 cos t sin t( a sin t) + (a 3 cos 3 t a sin t)a cos t } dt 2π = 3 4 (b4 a 4 )π. p.6/14
24 [ ] D a 2 x 2 + y 2 b 2 C I = 2xydx + (x 3 y)dy C 1. I 2. I [ ] 1. C 1 :(b cost,bsin t), C 2 :(a cos t,asin t) (0 t 2π) I = 2xydx + (x 3 y)dy + 2xydx + (x 3 y)dy C 1 C 2 2π { = 2b 2 costsin t( b sin t) + (b 3 cos 3 t b sin t)b cost } dt 0 0 { + 2a 2 cos t sin t( a sin t) + (a 3 cos 3 t a sin t)a cos t } dt 2π = 3 4 (b4 a 4 )π 2.. p.6/14
25 . p.7/14
26 S C S C. p.7/14
27 S C S C S F. p.7/14
28 S C S C S F F dr = rotf ds C S. p.7/14
29 [ ]. p.8/14
30 [ ] ( ) E t = 0 B j rot B = 1 ε 0 c 2j. p.8/14
31 [ ] ( ) E t = 0 B j rot B = 1 ε 0 c 2j C S B dr = 1 j ds ε 0 c 2 C S. p.8/14
32 [ ] ( ) E t = 0 B j rot B = 1 ε 0 c 2j C S B dr = 1 j ds C ε 0 c 2 S C C. p.8/14
33 [ ] ( ) E t = 0 B j rot B = 1 ε 0 c 2j C S B dr = 1 j ds C ε 0 c 2 S C C B dr = 1 ε 0 c2(c ) C. p.8/14
34 [ ] ( ) E t = 0 B j rot B = 1 ε 0 c 2j C S B dr = 1 j ds C ε 0 c 2 S C C B dr = 1 ε 0 c2(c ) C. p.8/14
35 [ ]. p.9/14
36 [ ] I B r I e r. p.9/14
37 [ ] I B r I [ ] e r. p.9/14
38 [ ] I B r I [ ] r e r. p.9/14
39 [ ] I B r I [ ] r B ds = 2πrB (B = B ) e r. p.9/14
40 [ ] I B r I [ ] r B ds = 2πrB (B = B ) B = I 2πε 0 c 2 e r. p.9/14
41 [ ] I B r I [ ] r B ds = 2πrB (B = B ) B = I 2πε 0 c 2 i.e. B = 1 I e r 2πε 0 c 2 r e r. p.9/14
42 . p.10/14
43 F. p.10/14
44 F (x,y,z) h. p.10/14
45 F (x,y,z) h h 2{ F(x + h,y,z) i + F(x h,y,z) ( i) 2 2 +F(x,y + h,z) j + F(x,y h,z) ( j) 2 2 +F(x,y,z + h) k + F(x,y,z h ) ( k) + ( )} 2 2. p.10/14
46 F (x,y,z) h h 2{ F(x + h,y,z) i + F(x h,y,z) ( i) 2 2 +F(x,y + h,z) j + F(x,y h,z) ( j) 2 2 +F(x,y,z + h) k + F(x,y,z h ) ( k) + ( )} 2 2 =h 2{( F 1 (x,y,z)+ hf 2 1x(x,y,z) ) ( F 1 (x,y,z)+( h)f 2 1x(x,y,z) ) + ( F 2 (x,y,z)+ hf 2 2y(x,y,z) ) ( F 2 (x,y,z)+( h)f 2 2y(x,y,z) ) + ( F 3 (x,y,z)+ hf 2 3z(x,y,z) ) ( F 3 (x,y,z)+( h)f 2 3z(x,y,z) )} +( ). p.10/14
47 F (x,y,z) h h 2{ F(x + h,y,z) i + F(x h,y,z) ( i) 2 2 +F(x,y + h,z) j + F(x,y h,z) ( j) 2 2 +F(x,y,z + h) k + F(x,y,z h ) ( k) + ( )} 2 2 =h 2{( F 1 (x,y,z)+ hf 2 1x(x,y,z) ) ( F 1 (x,y,z)+( h)f 2 1x(x,y,z) ) + ( F 2 (x,y,z)+ hf 2 2y(x,y,z) ) ( F 2 (x,y,z)+( h)f 2 2y(x,y,z) ) + ( F 3 (x,y,z)+ hf 2 3z(x,y,z) ) ( F 3 (x,y,z)+( h)f 2 3z(x,y,z) )} +( ) = h 3 div F+( ). p.10/14
48 D F D h. p.11/14
49 D F D h. p.11/14
50 D F D h h 0 div FdV = F ds D S D ds D S. p.11/14
51 [ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ). p.12/14
52 [ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ) E(x,y,z) = grad ϕ = Q 1 4πε r 2e r (e r (x,y,z) ). p.12/14
53 [ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ) E(x,y,z) = grad ϕ = Q 1 4πε r 2e r (e r (x,y,z) ) S 0 (Q S ) E ds = S Q (Q S ) ε 0. p.12/14
54 [ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ) E(x,y,z) = grad ϕ = Q 1 4πε r 2e r (e r (x,y,z) ) S 0 (Q S ) E ds = S Q (Q S ) ε 0 S D. p.12/14
55 [ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ) E(x,y,z) = grad ϕ = Q 1 4πε r 2e r (e r (x,y,z) ) S 0 (Q S ) E ds = S Q (Q S ) ε 0 S D E ds = dive dv S D. p.12/14
56 [ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ) E(x,y,z) = grad ϕ = Q 1 4πε r 2e r (e r (x,y,z) ) S 0 (Q S ) E ds = S Q (Q S ) ε 0 S D E ds = dive dv = div(grad ϕ) dv S D D. p.12/14
57 [ ] O Q ϕ(x,y,z) = 1 Q 4πε 0 r ( r = x 2 + y 2 + z 2 ) E(x,y,z) = grad ϕ = Q 1 4πε r 2e r (e r (x,y,z) ) S 0 (Q S ) E ds = S Q (Q S ) ε 0 S D E ds = dive dv = div(grad ϕ) dv = 0 S D D. p.12/14
58 S r S S S D. p.13/14
59 S r S S S D 0 = dive dv D. p.13/14
60 S r S S S D 0 = dive dv = D S S E ds. p.13/14
61 S r S S S D 0 = dive dv = D S S E ds = E ds E ds S S. p.13/14
62 S r S S S D 0 = dive dv = E ds = E ds E ds D S S S S E ds= Q 1 S 4πε 0 S r 2e r ds. p.13/14
63 S r S S S D 0 = dive dv = E ds = E ds E ds D S S S S E ds= Q 1 S 4πε 0 S r 2e r ds = Q 1 4πε 0 S r 2e r nds. p.13/14
64 S r S S S D 0 = dive dv = E ds = E ds E ds D S S S S E ds= Q 1 S 4πε 0 S r 2e r ds = Q 1 4πε 0 S r 2e r nds = Q 4πε 0 S 1 r 2dS. p.13/14
65 S r S S S D 0 = dive dv = E ds = E ds E ds D S S S S E ds= Q 1 S 4πε 0 S r 2e r ds = Q 1 4πε 0 S r 2e r nds = Q 1 4πε 0 S r 2dS = Q ε 0. p.13/14
66 S r S S S D 0 = dive dv = E ds = E ds E ds D S S S S E ds= Q 1 S 4πε 0 S r 2e r ds = Q 1 4πε 0 S r 2e r nds = Q 1 4πε 0 S r 2dS = Q ε 0 E ds = Q S ε 0. p.13/14
67 p.92 1 p p.14/14
. p.1/11
. p.1/11 [ ] F(x,y,z) = (F 1 (x,y,z),f 2 (x,y,z),f 3 (x,y,z)) = F 1 i+f 2 j+f 3 k div F = F 1 x + F 2 y + F 3 z F (divergence). p.1/11 [ ] F(x,y,z) = (F 1 (x,y,z),f 2 (x,y,z),f 3 (x,y,z)) = F 1 i+f 2 j+f
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2.4 ( ) ( B ) A B F (1) W = B A F dr. A F q dr f(x,y,z) A B Γ( ) Minoru TANAKA (Osaka Univ.) I(2011), Sec p. 1/30
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F(r)=QE(r) Q ρ( r ') 3 E= ke 3 ( r r ') d r ' V r r' () Er () = Fr Q E E x y Ez ρ = x y z ε E E z y Ex E E z y Ex =, =, = y z z x x y A A x y Az diva = x y z A A z y A A x A z y Ax rot A =,, y z z x x
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145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
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1 1 f (t) lim t a f (t) = 0 f (t) t a 1.1 (1) lim(t 1) 2 = 0 t 1 (t 1) 2 t 1 (2) lim(t 1) 3 = 0 t 1 (t 1) 3 t 1 2 f (t), g(t) t a lim t a f (t) g(t) g(t) f (t) = o(g(t)) (t a) = 0 f (t) (t 1) 3 1.2 lim
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
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II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
Chap11.dvi
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応用数学A
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米田 戸倉川月 7 限 193~21 西 5-19 応用数学 A 積分定理 Gaussの定理 divbd = B nds Stokesの定理 E bds = E dr Green の定理 g x f y dxdy = fdx + gdy = f e i + ge j dr Gauss の発散定理 S n FdS = Fd 1777-1855 ドイツ Johann arl Friedrich Gauss
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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..
5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................
5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)
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1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3)
< 1 > (1) f 0 (a) =6a ; g 0 (a) =6a 2 (2) y = f(x) x = 1 f( 1) = 3 ( 1) 2 =3 ; f 0 ( 1) = 6 ( 1) = 6 ; ( 1; 3) 6 x =1 f(1) = 3 ; f 0 (1) = 6 ; (1; 3) 6 y = g(x) x = 1 g( 1) = 2 ( 1) 3 = 2 ; g 0 ( 1) =
x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)
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grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))
sec13.dvi
13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
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A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9
f (x) x y f(x+dx) f(x) Df 関数 接線 x Dx x 1 x x y f f x (1) x x 0 f (x + x) f (x) f (2) f (x + x) f (x) + f = f (x) + f x (3) x f
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9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x
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