Q&A Question and Answer Q&A
ii Q&A 1999 Q&A Q&A Q&A 8 Q&A 2007 2
1. 1.1... 1 1.1.1... 1 1.1.2... 6 1.2... 7 1.2.1... 7 1.2.2... 11 1.3... 18 1.3.1... 18 1.3.2... 20... 25 2. 2.1... 26 2.2... 33... 38 3. 3.1... 39
iv 3.2... 48 3.3... 56... 65 4. 4.1... 66 4.2... 69 4.2.1... 69 4.2.2... 72 4.3... 74 4.4... 79 4.4.1... 79 4.4.2... 81 4.5... 86 4.5.1... 86 4.5.2... 87... 90 5. 5.1... 92 5.2... 101... 111... 112 A.1... 112
v A.2... 117 A.3... 119 A.4... 121 A.5... 124 A.6... 127... 131... 135... 136... 173
1 vector scalar 1.1.1 1.1 xyz ( 1 ) x, y, z 1 x, y, z base vectori, j, k i, j, k 1 (1.1) i = j = k =1 (1.1) ( 2 ) 1.1 A
2 1. z x A x A z A k j A y y i O 1.1 A i, j, k (1.2) A = A x i + A y j + A z k (1.2) i, j, k A x, A y, A z A x, y, z ( 3 ) 1.2 O(0, 0, 0) P(x, y, z) r P position vector (1.3) r = xi + yj + zk (1.3) z Pxyz x O r y 1.2 r ( 4 ) A, B, C, a, b, c A, B, C, a, b, c A (1.4) A = A = A 2 x + A 2 y + A 2 z (1.4)
1.1 3 ( 5 ) 1.3 A, B A B (1.5) A = B A = B, A / B (1.5) A B 1.3 A = A x i + A y j + A z k, B = B x i + B y j + B z k A B (1.6) A = B A x = B x,a y = B y, A z = B z (1.6) ( 6 ) 0 zero vector 0 A = A x i + A y j + A z k 0 A (1.7) A = 0 A x =0,A y =0,A z =0 (1.7) 0 0 ( 7 ) 1 unit vector A = A x i + A y j + A z k A = A = A 2 x + A 2 y + A 2 z =1 (1.8) A a A a A = A A (1.9) A = i + j + k A = A = 1 2 +1 2 +1 2 = 3 1
4 1. ( 8 ) 1.4 A = A x i + A y j + A z k x, y, z α, β, γ cos α, cosβ, cosγ A direction cosine A (1.10) A x = A cos α, A y = A cos β, A z = A cos γ (1.10) z A z A cos g A g a b y O A y A cos b x A x A cos a 1.4 A 1.1 A = i +2j +3k (1) A = A, (2) (1) A = A = 1 2 +2 2 +3 2 = 14 (2) cos α = A x A = 1, cos β = A y 14 A = 2, cos γ = A z 14 A = 3 14 1.1 cos α, cosβ, cosγ cos 2 α +cos 2 β +cos 2 γ =1 Q&A Q1.1: A1.1: x y z right handed systemx y z left handed system Q1.2: A A A1.2: A A
Q1.3: 1.1 5 A1.3: A A = 0 A A 0 A A =0 0 0 Q1.4: A1.4: 1 A a A A A = A a A =(1/A)A = A/A Q1.5: A1.5: 1.4 A x, y, z α, β, γ A x = A cos α, A y = A cos β,a z = A cos γ l =cosα, m =cosβ,n =cosγ (l, m, n) A l = A x /A, m = A y /A, n = A z /A 2 1.5(a) A 1 α, β cos α>0, cos β>0 (b) B 2 α β cos α<0, cos β>0 B x B cos α A cos b y b A a A cos a x B B cos a b y B cos b (a) (b) 1.5 a x
6 1. 1.1.2 ( 1 ) A p pa p>0a p A p<0a p A p =0 0 (0)A = 0 p = 1 ( 1)A = A A 1.6 A A 1.6 A A = A x i + A y j + A z k pa A p (1.11) pa = p(a x i + A y j + A z k)=(pa x )i +(pa y )j +(pa z )k (1.11) ( 2 ) A, B law of parallelogram A B A + B 1.7 A B B A O B A + B A B A B = A +( B) A ( B) A = A x i + A y j + A z k, B = B x i + B y j + B z k A ± B A, B (1.12) B A B O A B A B 1.7
109 19 20 2 59 30 11, 62 13 15 56, 62, 103 59, 101 56 98 87, 95, 127 84, 100 13, 118 24 13, 119 1 6 112 112 74 74 66 67 111 111 70 107 70, 101 70 30 40 41 124 68 86 25 62 60, 105 114 68 1 26 18 19 7 39 42 40, 62 82, 104, 129 8, 10, 11, 80, 82, 128, 130 11 42, 50, 52, 58, 62, 84, 100, 107 32, 68, 72 75 28, 67 3 41, 47, 49, 58 70, 72, 73, 88 70 72 30, 45, 120 117 86 86, 88 3
174 10, 117 13, 118 37 54 8 35, 69, 92 26 40 42, 44, 45, 54, 62, 107 117 52 82, 84, 100 44 44, 124 28 44 44, 45 13, 24, 118 7, 54 8 40 C curl 56 D divergence 48 G gradient 40 75, 79 48, 55, 95 51, 93 40 40 30 33 37 6 13 12 18 1 2 1 2 26 20 11 39 62 L Laplacian 53 N nabla 40 R rotation 56 30, 119 54 42, 126 4 75 92 58, 107 28 80, 81, 88 13, 17 75, 79 76 114 115 13 53 53 53 50 51 50, 97 U u 74 V v 74
1965 1965 1974 1977 1979 1989 2008 1989 1991 1991 1996 1998 2007 Elementary Vector Analysis c Takeo Maruyama, Nozomu Ishii 2007 2007 4 25 1 2011 4 20 5 112 0011 4 46 10 CORONA PUBLISHING CO., LTD. Tokyo Japan 00140 8 14844033941 3131 :// ISBN 978 4 339 06093 5 Printed in Japan