1 ベクトル解析 - suzuka-ct.ac.jp

1 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.)))

2 = A(t).. A(t).. dA(t) da1 (t) da2 (t) da3 (t). =( , , ), dt dt dt dt 2. Z Z Z Z.. A(t)dt =( a1 (t)dt, a2 (t)dt, a3 (t)dt).. + B(t)). d(A(t) . dA(t) . dB(t). (1) = +. dt dt dt . d( (t)A(t)) d (t) . dA(t). (2) = A(t) + (t). dt dt dt B(t)). d(A(t) . dA(t) . (3) = B(t) dB(t). + A(t). dt dt dt . d(A(t) B(t)) . dA(t) . (4) = dB(t). B(t) + A(t). dt dt dt . d|A(t)|2 . dA(t) . (5) =2 A(t). dt dt Z Z Z.. [1] (A(t) + B(t))dt = A(t)dt + B(t)dt Z Z. (t)dt = (t)A(t). [2] (t)A (t)A(t)dt . Z Z. B (t)dt = A(t). [3] A(t) B(t). A (t) B(t)dt . Z Z.. [4] A(t) B (t)dt = A(t) B(t) A (t) B(t)dt . P 2 u, v 1 . r r = r(u, v) . u, v . A u, v . A = (a1 (u, v), a2 (u, v), a3 (u, v)) . = A(u, A v) A(u, v) .. 3.. 2 r = r(t) = (x(t), y(t), z(t)) P (t = a) . Q(t = b) s . Z b Z bp . s= | r (t)|dt = (x (t))2 + (y (t))2 + (z (t))2 dt a a . ds p = | r (t)| = (x (t))2 + (y (t))2 + (z (t))2. dt r = r(t) P d rdt(t) = (x (t), y (t), z (t)).

3 P d rdt(t) .. 3 r = r(t) P t . d . r(t) d . r(t). t = dt dt d r(t) dt d r(t). = ds = =. d . r(t). | dt | dt dt ds ds r(t) = (3 cos t, 3 sin t, 4t) .. d r(t). = ( 3 sin t, 3 cos t, 4). dt . q d r(t). | | = 9(sin2 t + cos2 t) + 16 = 5, dt t , t = 1 ( 3 sin t, 3 cos t, 4). 5.. Z t Z t . s= | r (t)|dt = 5|dt = 5t. 0 0. J. 4.. z = f (x, y) S P .. OP = r r = (x, y, f (x, y)) S .. P OP = r u, v .. r = (x(u, v), y(u, v), z(u, v)) S . P (u = c, v = d) 2 u, v u( . r = r(c, v) resp. r(u, d) v . P 1 v . r v (c,d). resp. r u (c,d). v . P v u r u . r (c,d) r(c,d) (c,d). r v (c,d). v u . v u . P .. 4 r = r(u, v) n .. r . r u v n r . r | u v |.. 5 P (u, v) D . r = r(u, v) S . Z Z. r r S= | |dudv D u v a n, S . a .. r(u, v) = (u, v, a2 u2 v 2 ).. r u r v = (1, 0, ), = (0, 1, ). u a u v v 2 2 2 a u2 v 2. 2. 5. , r r u v = ( , , 1). u v a u v 2 2 2 a u2 v 2. 2.. r r a | |= . u v a u2 v 2. 2.)

4 1 r r u v a 2 u2 v 2. n . r . r u = ( , , ). | u v | v a a a . Z Z Z Z. r r a S= | |dudv = 8 dudv D u v a u2 v 2. 2. D. D = {(u, v); u2 + v 2 a2 , u 0, v 0}.. u = r cos , v = r sin , . D = {(r, ); 0 , 0 r a}. 2.. u v .. |J| = =r u r vr . Z Z. a S=8 dudv D a2 u2 v 2. Z Z. a =8 rdrd . D a2 r2. Z Z a 2 a =8 rdrd . 0 0 a2 r2. Z .. 2. r=a = 8a [ a2 r2 ]r=0 d = 4 a2. 0. J. 6.. V . A . (x, y, z) ( x , y , z ) . (x, y, z) grad( ) .. ( = ( x , y , z ) .. = ( , , ) = ( , , ) = grad( ). x y z x y z . y, z) (x, y, z) A(x, A(x, y, z) =. A(x, y, z) (x, y, z).. grad . 6 (x, y, z) grad( ) (x, y, z) = c . (x, y, z) = c n . 1. n = . | |.. p r(2) log r ( 1r ) r = x2 + y 2 + z 2.. y, z) = (a(x, y, z), b(x, y, z), c(x, y, z)) . A(x, a x b + y c + z A(x, y, z) divA . 7.. = a b c . div A + + = A. x y z . = ( x , y , z ) . ( , , ) (a(x, y, z), b(x, y, z), c(x, y, z)). A x y z . x a . ( A A. = a +b +c.))

5 X y z f (x, y, z) . A = A . A(x, y, z) div A .. r = (x, y, z) . div r = 1+1+1 = 3.. ( rr )(2) ( r r2 ) ( r r3 ) . p r = (x, y, z), r = x2 + y 2 + z 2.. (x, y, z) = ( , , ) . x y z div . 2 2 2 . = ( , , ) ( , , )= + + 2. x y z x y z x2 y 2 z . 2 2 2. = + +. x2 y 2 z 2.. div(grad(f )) = f . f = 0.. 8. r(2) log r ( 1r ).. A(x, y, z) = (a(x, y, z), b(x, y, z), c(x, y, z)) . c ( y b a z c b , z x , x a rotA. ) A . y A A.. =( . A , , ) (a(x, y, z), b(x, y, z), c(x, y, z)). x y z . = ( c b, a c, b a). y z z x x y c b a c b a =( , , ). y z z x x y . = A. rotA .. i j k .. = ( y z , z x rotA. x , y . ) = x . b c c a a z . b a y b c .. y, z) (x, y, z) . 7 A(x, y, z) = rotA. A(x, = 0 . rotA = A = ( ) = 0 .. A(x, y, z) B(x, y, z) .. A = rotB A(x, y, z) .. B(x, y, z) . y, z) B(x, 8 A(x, y, z) .. A(x, y, z) = rotB(= B) div A =0.. = ( B). div A = 0 .. 9.. P Q C : r(t) = (x(t), y(t), z(t))(a.)

6 T b) C . f (x, y, z) . Z b f (x(t), y(t), z(t))dt a t C . Z. f dt C.. s C : r(t) = (x(t), y(t), z(t))(a . t b) . Z Z b ds f ds = f (x(t), y(t), z(t)) dt C a dt Z b p = f (x(t), y(t), z(t)) (x (t))2 + (y (t))2 + (z (t))2 dt a C .. f = x2 2yz + y C1 , C2 . (1). C1 : (1, 1, 0) (1, 1, 1).. (2). C2 : (0, 0, 0) (1, 1, 1).. (1)C1 . x = 1, y = 1, z = s, 0 s 1, 10.. f = x2 2yz + y = 12 2 1 s + 1 = 2 2s C1 . Z 1. (2 2s)ds = [2s s2 ]s=1. s=0 = 1. 0.. (2)C2 . s s s . x = ,y = ,z = ,0 s 3. 3 3 3.. s s s s s2 s f = x2 2yz + y = ( )2 2 + = + , 3 3 3 3 3 3. C2 . Z . 3. s2 s s3 s2 . ( + )ds = [ + ]s= s=0. 3. 0 3 3 9 2 3. 3 2 . 3 3 3 3 3. = + = =. 9 2 3 2 3 6. C2 . ds 2 . x = t, y = t, z = t, 0 t 1, = 1 + 12 + 12 = 3. dt . f = x2 2yz + y = t2 2t t + t = t2 + t C2 . Z 1 . 2. t3 t2 1 3 3 3. ( t + t) 3dt = 3[ + ]0 = =. 0 3 2 2 3 6. J. C : r(t) = (x(t), y(t), z(t))(a t b) . y, z) = (a(x, y, z), b(x, y, z), c(x, y, z)).

7 A(x, C C : r(s) = (x(s), y(s), z(s))( . 11. t ) C . t A. t = r (s) A t .. Z. A tds C. A C . C t C : r(t) = (x(t), y(t), z(t))(a t b) .. t = d r = d r dt ds dt ds . (i)C : r(t) = (x(t), y(t), z(t))(a t b) . Z Z b Z b tds = d r dt d r dt =. A A ds = A. C a dt ds a dt Z b dx(t) dy(t) dz(t). (a(x(t), y(t), z(t)) +b(x(t), y(t), z(t)) +c(x(t), y(t), z(t)) )dt a dt dt dt (ii)C : r(s) = (x(s), y(s), z(s))( s ) . Z. tds =. A. C. Z . dx(s) dy(s) dz(s). (a(x(s), y(s), z(s)) +b(x(s), y(s), z(s)) +c(x(s), y(s), z(s)) )ds ds ds ds .. = (x + 2y z, 2x + y, y 2z). A.. C; (1, 0, 1) (1, 1, 1) .. 12. C . x = 1, y = s, z = 1, 0 s 1.. t = ( dx , dy , dz ) = (0, 1, 0), ds ds ds = (1 + 2s 1, 2 + s, s 2), A. t = 2 + s, A. Z Z 1. tds = s2 1 5. A (2 + s)ds = [2s + ] =. C 0 2 0 2. J.. C : r(t) = (cos t, sin t, t)(0 t ).. = 2xz . = ( y, x, z). A.. ( . x = cos t, y = sin t, z = t . = 2xz = 2t cos t, = ( y, x, z) = ( sin t, cos t, t), A.))

8 R (t) = ( sin t, cot t, 1). Z t ds s(t) = | r (t)|dt, = | r (t)|. a dt p .. s (t) = sin2 t + cos2 t + 1 = 2, Z Z . ds = 2t cos t 2dt C 0. 13. Z .. = 2 2{[t sin t]0 sin tdt}. 0.. = 2 2 [cot s] 0 = 4 2. Z Z. tds =. A r dt A. C C. Z . = (sin2 t + cos2 t + t)dt 0. Z . = (1 + t)dt 0. t2 2. = [t + ]0 = +. 2 2. J. C; r = (a cos t, a sin t, bt)(0 t 2 ) =. 2. x + z A = (x, y, z) . C P (t = 0) P (T = t) .. C . A C .. S r = r(u, v) (u, v) u v . D S = (x(u, v), y(u, v), z(u, v)) =. (u, v) . Z Z. r r (u, v)| |dudv D u v S . Z Z. dS. S.. 14. S z = z(x, y) . s 2. 2. r r z z | |= + + 1, u v x y = (x, y, z(x, y)).. Z Z. r r (u, v)| |dudv =. D u v s 2 2. Z Z. z z (x, y, z(x, y)) + + 1dxdy D x y . S A(u, v) = (a(x, y, z), b(x, y, z), b(x, y, z)), x =. x(u, v), y = y(u, v), z = z(u, v) . Z Z Z Z. ndS =. A ( r r )dudv A. S D u v . S A.. = (0, 0, z). A.. S : x2 + y 2 + z 2 = 1.. S r . p r = (x, y, 1 x2 y 2 ).

9 R x r y = (1, 0, p ), = (1, 0, p ). x 1 x2 y 2 y 1 x2 y 2.. r r x y = ( p , p , 1). x y 1 x2 y 2 1 x2 y 2. 15.. D = {(x, y); x2 + y 2 1}.. Z Z Z Z. ndS =. A n| r r |dxdy A. S D x y Z Z . r . r y x r r = A r r | |dxdy D | x y | x y Z Z. = A ( r r )dxdy x y Z ZD. = zdxdy Z DZ p =2 1 x2 y 2 dxdy D. Z Z 1 . 2. =8 1 r2 rdrd . 0 0. 1 3. = 4 [ (1 r2 ) 2 ]r=1. r=0. 3. 4 . =. 3. J.. S : r = (sin u cos v, sin u sin v, cos u), (0 u , 0 v 2 ). 2.. = (xz, yz, 0). A.. ( . x = sin u cos v, y = sin u sin v, z = cos u, . = (xz, yz, 0) = cos u(sin u cos v, sin u sin v, 0), A.. r = (cos u cos v, cos u sin v, sin u), u 16. r = ( sin u sin v, sin u cos v, 0), v . r r = sin u(sin u cos v, sin u sin v, cos u), u v . ( r r ) = sin3 u cos v, A. u v . Z Z Z Z. ndS =. A ( r r )dudv A. D D u v Z 2 Z . 2 . = sin3 u cos vdudv =. 0 0 2. J. S . x = sin u cos v, y = sin u sin v, z = cos u . x2 + y 2 + z 2 = sin2 u cos2 v + sin2 u sin2 v + cos2 u = sin2 u(cos2 v + sin2 v) + cos2 u = sin2 u + cos2 u =1.)

10 O(0, 0, 0), 1 .. x = r sin u cos v, y = r sin u sin v, z = r cos u P (x, y, z) (r, u, v) . r O P u OP z . v P x y . H OH x .. r > 0, < u < , 0 < v < 2 . 2 2.. 17. S; r = (a cos u sin v, a sin u sin v, a cos v)(0 v . 1. u 2 ) = (x2 +y 2 ) 2 A. ,0 = (x, y, z). 2.. S n . S . A S . = (x + y, x + y, 2z). S; 2x + 2y + z = 2 A.. S r = r(x, y) .. A S . =. C; r = (cos t, sin t, 1), (0 t 2 ) A. (2z, z 3, xy) . C n . A C . RR. C S S rot(A) . ndS .. 9 S V S n . A . Z Z Z Z Z.. div AdV = ndS. A. V S.. = gradf . A. 10 f f n . Z Z Z Z Z. f f dV = dS. V S n 18. f f = 0 . Z Z. f dS = 0. S n .. Z Z Z Z Z. f (i) (g f + g f )dV = g dS. n Z Z VZ Z ZS. f g (ii) (g f f g)dV = (g f )dS. V S n n . Z Z Z. (fx gy )dxdy = (gdx + f dy). D C.. Z Z I I. ndS =. rotA tds =. A d r A. S C C. S C S n . S A S . C S rotA . C A . S; r = (cos u, sin u, v), (0 u 2 , 0 v 1) . A = (z, x, y) . S V O V . R. S r r3 ndS = 0 n S.