1 ベクトル解析 - suzuka-ct.ac.jp

1 .. 2 a b ( a, b)( a b .. ( a, b) = | a|| b| cos , a b .. (1)( a, b) = ( b, a), . (2)( a, b + c) = ( a, b) + ( a, c), ( . (3)m( a, b) = (m a, b), m . a = (a1 , a2 , a3 ), b = (b1 , b2 , b3 ) .. (4)( a, b) = a1 b1 + a2 b2 + a3 b3. (1),(2) ( i, i) = ( j, j) = ( k, k) = 1, ( i, j) = ( j, k) =. ( k, i) = 0 .. (i) a b ( a, b) = 0. (ii)( a, a) = | a|2. (iii)|( a, b)| | a|| b| .. 2 a b a b( [ a, b] .. 1 (i) a b a b = 0 (ii) a = 0 b = 0 . a b = 0 (iii)0 2 a b . a b . 1. | a|| b| sin a b . a b a b .. (1) a b = b a, a a = 0. (2) a ( b + c) = a b + a c ( a + b) c = a c + b c . , . (3)(m a) b = m( a b) = a (m b), m . a = (a1 , a2 , a3 ), b = (b1 , b2 , b3 ) .. (4) a b = (a2 b3 a3 b2 , a3 b1 a1 b3 , a1 b2 a2 b1 ).. a a a a a a . 2 3 3 1 1 2 . = ( , , ). b2 b3 b3 b1 b1 b2 . (1),(2) i j = k, j k = i, k i = j .. i j k .. (4) a b = a1 a2 a3 .. b1 b2 b3 .. P t 1 . r r = r(t) . t . A t . A = (a1 (t), a2 (t), a3 (t)) A.)))

1.3 スカラー場とベクトル場 1.3.1 スカラー場の勾配 空間のある領域にスカラー関数が定義されているときに、それを、ス

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