. p.1/11
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- こうじ むこやま
- 4 years ago
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1 . p.1/11
2 [ ] 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
3 [ ] 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) = i x + j y + k z ( ) div F = F. p.1/11
4 . p.2/11
5 F. p.2/11
6 F (x,y,z) h. p.2/11
7 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.2/11
8 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.2/11
9 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.2/11
10 h 3 h 0. p.3/11
11 h 3 h 0 (x,y,z). p.3/11
12 h 3 h 0 (x,y,z) div F(x,y,z). p.3/11
13 h 3 h 0 (x,y,z) div F(x,y,z) [ ]. p.3/11
14 h 3 h 0 (x,y,z) div F(x,y,z) [ ]. p.4/11
15 h 3 h 0 (x,y,z) div F(x,y,z) [ ] ρ(x,y,z) E(x,y,z) E = 1 ε 0 ρ ε 0 ( ). p.4/11
16 [ ] f(x,y,z) f = div (grad f) = ( f). p.5/11
17 [ ] f(x,y,z) f = div (grad f) = ( f) = 2 x y z (Laplacian) 2. p.5/11
18 [ ] f(x,y,z) f = div (grad f) = ( f) = 2 x y z (Laplacian) 2 [ ]. p.5/11
19 [ ] f(x,y,z) f = div (grad f) = ( f) = 2 x y z (Laplacian) 2 [ ] T(x,y,z). p.5/11
20 [ ] f(x,y,z) f = div (grad f) = ( f) = 2 x y z (Laplacian) 2 [ ] T(x,y,z). p.5/11
21 [ ] f(x,y,z) f = div (grad f) = ( f) = 2 x y z (Laplacian) 2 [ ] T(x,y,z) h = κ grad T ( κ ). p.5/11
22 [ ] f(x,y,z) f = div (grad f) = ( f) = 2 x y z (Laplacian) 2 [ ] T(x,y,z) h = κ grad T ( κ ) div( κ grad T) = κ T ( +κ T). p.5/11
23 [ ] f(x,y,z) f = div (grad f) = ( f) = 2 x y z (Laplacian) 2 [ ] T(x,y,z) h = κ grad T ( κ ) div( κ grad T) = κ T ( +κ T) ( ) T t = κ T. p.5/11
24 [ ] ( F F (rotation) F3 rot F = y F 2 F 1 z, z F 3 F 2 x, x F ) 1 = F y. p.6/11
25 [ ] ( F F (rotation) F3 rot F = y F 2 F 1 z, z F 3 F 2 x, x F ) 1 = F y. p.6/11
26 [ ] ( F F (rotation) F3 rot F = y F 2 F 1 z, z F 3 F 2 x, x F ) 1 = F y z y-z (x,y,z) h. p.6/11
27 [ ] ( F F (rotation) F3 rot F = y F 2 F 1 z, z F 3 F 2 x, x F ) 1 = F y z y-z (x,y,z) h z h { F 2 (x+ h,y,z) F 2 2(x h,y,z) F 2 1(x,y+ h,z)+f 2 1(x,y h,z)} +( ) 2. p.6/11
28 [ ] ( F F (rotation) F3 rot F = y F 2 F 1 z, z F 3 F 2 x, x F ) 1 = F y z y-z (x,y,z) h z h { F 2 (x+ h,y,z) F 2 2(x h,y,z) F 2 1(x,y+ h,z)+f 2 1(x,y h,z)} +( ) 2 =h {( F 2 (x,y,z)+ hf 2 2x(x,y,z) ) ( F 2 (x,y,z)+( h)f 2 2x(x,y,z) )} h {( F 1 (x,y,z)+ hf 2 1y(x,y,z) ) ( F 1 (x,y,z)+( h)f 2 1y(x,y,z) )} +( ). p.6/11
29 [ ] ( F F (rotation) F3 rot F = y F 2 F 1 z, z F 3 F 2 x, x F ) 1 = F y z y-z (x,y,z) h z h { F 2 (x+ h,y,z) F 2 2(x h,y,z) F 2 1(x,y+ h,z)+f 2 1(x,y h,z)} +( ) 2 =h {( F 2 (x,y,z)+ hf 2 2x(x,y,z) ) ( F 2 (x,y,z)+( h)f 2 2x(x,y,z) )} h {( F 1 (x,y,z)+ hf 2 1y(x,y,z) ) ( F 1 (x,y,z)+( h)f 2 1y(x,y,z) )} +( ) = h 2 {F 2x (x,y,z) F 1y (x,y,z)} +( ). p.6/11
30 h 2 h 0. p.7/11
31 h 2 h 0 (x,y,z) z- F 2 x F 1 y. p.7/11
32 h 2 h 0 (x,y,z) z- F 2 x F 1 y( F3 rot F = y F 2 F 1 z, z F 3 F 2 x, x F ) 1 y. p.7/11
33 h 2 h 0 (x,y,z) z- F 2 x F 1 y( F3 rot F = y F 2 F 1 z, z F 3 F 2 x, x F ) 1 y [ ]. p.7/11
34 h 2 h 0 (x,y,z) z- F 2 x F 1 y( F3 rot F = y F 2 F 1 z, z F 3 F 2 x, x F ) 1 y [ ]. p.8/11
35 h 2 h 0 (x,y,z) z- F 2 x F 1 y( F3 rot F = y F 2 F 1 z, z F 3 F 2 x, x F ) 1 y [ ] E = ρ ε 0, E = B t, B = 0, c2 B = E t + 1 ε 0 J E : B : J : ε 0 : c :. p.8/11
36 [ ]. p.9/11
37 [ ] F F = gradf f f F. p.10/11
38 [ ] F F = gradf f f F F F = rot f f f F. p.10/11
39 p.60 7 p p.11/11
. p.1/14
. p.1/14 F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y). p.2/14 F(x,y) = (F 1 (x,y),f 2 (x,y)) (x,y) (x,y) h. p.2/14 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
f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) B p.1/14
B p.1/14 f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) B p.1/14 f(x,y) (x,y) x (x,y), y (x,y) f(x,y) x y f x (x,y),f y (x,y) f(x 1,...,x n ) (x 1 x 0,...,x n 0), (x 1,...,x n ) i x i f xi
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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
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
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II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
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26 12 1 23. xyz ϕ f(x, y, z) Φ F (x, y, z) = F (x, y, z) G(x, y, z) rot(grad ϕ) rot(grad f) H(x, y, z) div(rot Φ) div(rot F ) (x, y, z) rot(grad f) = rot f x f y f z = (f z ) y (f y ) z (f x ) z (f z )
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9 E B 9.1 9.1.1 Ampère Ampère Ampère s law B S µ 0 B ds = µ 0 j ds (9.1) S rot B = µ 0 j (9.2) S Ampère Biot-Savart oulomb Gauss Ampère rot B 0 Ampère µ 0 9.1 (a) (b) I B ds = µ 0 I. I 1 I 2 B ds = µ 0
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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
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1 1 1.1 (Isaac Newton, 1642 1727) 1. : 2. ( ) F = ma 3. ; F a 2 t x(t) v(t) = x (t) v (t) = x (t) F 3 3 3 3 3 3 6 1 2 6 12 1 3 1 2 m 2 1 x 1.1: v mg x (t) = v(t) mv (t) = mg 0 x(0) = x 0 v(0) = v 0 x(t)
2.5 (Gauss) (flux) v(r)( ) S n S v n v n (1) v n S = v n S = v S, n S S. n n S v S v Minoru TANAKA (Osaka Univ.) I(2012), Sec p. 1/30
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213 12 1 21 5 524 3-5465-74 nkiyono@mail.ecc.u-tokyo.ac.jp http://lecture.ecc.u-tokyo.ac.jp/~nkiyono/index.html 3 2 1 3.1 ρp, t EP, t BP, t JP, t 35 P t xyz xyz t 4 ε µ D D S S 35 D H D = ε E B = µ H E
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3345 チュートリアル 1 HP テンソル代数 テンソル解析 - - 連続体力学の数理的基礎 - 第 4 講テンソル解析 - テンソル場の微積分 - 登坂宣好 第 4 講概要 2, 3 1 筆者紹介 1971 Engineering cience gradient divergence rotation nabla 3 1 2 3 4 5 6 ol.20, No.4 2015 27 1 [1,2]
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