12 -- 1 4 2009 9 4-1 4-2 4-3 4-4 4-5 4-6 4-7 4-8 4-9 4-10 c 2011 1/(13)
4--1 2009 9 3 x,, z v = (, b, c) v v 2 + b 2 + c 2 x,, z 1 i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) v 1 = ( 1, b 1, c 1 ), v 2 = ( 2, b 2, c 2 ) v 1 v 2 v 1 v 2 v 1 v 2 = v 1 v 2 cos θ = 1 2 + b 1 b 2 + c 1 c 2, i j k v 1 v 2 = 1 b 1 c 1 = (b 1 c 2 c 1 b 2, c 1 2 1 c 2, 1 b 2 b 1 2 ) 2 b 2 c 2 θ v 1 v 2 0 θ π v 1 v 2 v 1, v 2 0 = (0, 0, 0) v 1, v 2 v 1 v 2 v 1, v 2 v 1 v 2 sin θ v 1, v 2, v 1 v 2 v i = ( i, b i, c i ) i = 1, 2, 3 [v 1 v 2 v 3 ] v 1 v 3 1 b 1 c 1 [v 1 v 2 v 3 ] = v 1 (v 2 v 3 ) = 2 b 2 c 2 3 b 3 c 3 v 1 (v 2 v 3 ) v 1 (v 2 v 3 ) = (v 1 v 3 ) v 2 (v 1 v 2 ) v 3 c 2011 2/(13)
4--2 2009 9 nbl, = (,, ) f (x,, z) v(x,, z) = (u(x,, z), v(x,, z), w(x,, z)) f, grd f v, div v ( f, f, f ) + + w v, rot v ( w, w, ) 2 f, ( f ), f 2 f + 2 f 2 + 2 f 2 2 (x,, z) n = (n x, n, n z ) f (x,, z) f n f = n x + n f + n f z v(x,, z) (x,, z) v v(x,, z) v (x,, z) 1 2 v f (x,, z) k k 2 f (x,, z) ( f g) = ( f ) g + ( g) f, ( f A) = ( f ) A + f ( A), ( f A) = ( f ) A + f ( A), (A B) = (B ) A + (A ) B + A ( B) + B ( A), (A B) = B ( A) A ( B), (A B) = (B ) A (A ) B + A ( B) B ( A), ( f ) = 0, ( A) = 0, ( A) = ( A) 2 A c 2011 3/(13)
4--3 2009 9 e r, e θ (r, θ) x = r cos θ, = r sin θ f = f r e r + 1 f r θ e θ, 2 f = 1 ( f ) 1 2 f r + r r r r 2 θ, 2 A = R e r + Θ e θ A = 1 (rr) + 1 Θ r r r θ (r, θ, ϕ) x = r sin θ cos ϕ, = r sin θ sin ϕ, z = r cos θ f = f r e r + 1 f r θ e 1 f θ + r sin θ ϕ e ϕ, 2 f = 1 ( r 2 f ) 1 ( f ) 1 2 f + sin θ + r 2 r r r 2 sin θ θ θ r 2 sin 2 θ ϕ, 2 A = R e r + Θ e θ + Φ e ϕ A = 1 (r 2 R) 1 (Θ sin θ) 1 Φ + + r 2 r r sin θ θ r sin θ ϕ, 1 ( (Φ sin θ) A = Θ ) er + 1 ( 1 R r sin θ θ ϕ r sin θ ϕ (rφ) ) eθ + 1 ( (rθ) R ) eϕ r r r θ (r, θ, z) x = r cos θ, = r sin θ, z = z f = f r e r + 1 f r θ e θ + f e z, 2 f = 1 ( f ) 1 2 f r + r r r r 2 θ + 2 f 2, 2 A = R e r + Θ e θ + Z e z A = 1 (rr) + 1 Θ r r r θ + Z, A = ( 1 Z r θ Θ ) er + ( R Z ) eθ + 1 ( (rθ) R ) ez r r r θ θ r e θ e e r r z eϕ r θ x x ϕ e θ x z r θ e z eθ e r c 2011 4/(13)
4--4 2009 9 t r(t) = (x(t), (t), z(t)) r() r(t) s(t) s(t) = t t r (τ) dτ = x (τ) 2 + (τ) 2 + z (τ) 2 dτ s s r(s) = (x(s), (s), z(s)) s t(s) = r (s) = (x (s), (s), z (s)) n(s), b(s) t (s) = κ(s)n(s), b(s) = t(s) n(s) κ(s) t (s) n, b t, n, b τ(s) t (s) = κ(s)n(s), n (s) = κ(s)t(s) + τ(s)b(s), b (s) = τ(s)b(s) c 2011 5/(13)
4--5 2009 9 u, v r(u, v) = (x(u, v), (u, v), z(u, v)) dr = r r du + dv E = r 2 = ( ) 2 ( ) 2 ( ) 2, r + + F = r = + +, G = r 2 = ( ) 2 ( ) 2 ( ) 2 + + dr dr = E(du) 2 + 2Fdudv + G(dv) 2 n n = r r / r r L = 2 r 2 n, M = 2 r n, N = 2 r 2 n dr dn = L(du) 2 + 2Mdudv + N(dv) 2 L, M, N L = N = 1 EG F 2 1 EG F 2 2 x 2 2 x 2 2 2 2 2 2 z 2 2 z 2, M = 1 EG F 2 2 x P P P κ 1, κ 2 H = 1 2 (κ 1 + κ 2 ) K = κ 1 κ 2 H = 1 2 EN 2FM + GL LN M2, K = EG F 2 EG F 2 2 2 z, c 2011 6/(13)
4--6 2009 9 1 n k r 0 r k = (x k, k, z k ) f (r) k f k r n 1 k rk+1 f k (x k+1 x k ) f k k=0 f x f dx, z r n n 1 f k r k+1 r k f k=0 f ds t r(t) ( t b) t b b f dx = f (r(t))x (t)dt, f ds = f (r(t)) r (t) dt A(r) = (P(r), Q(r), R(r)) k A k n 1 A k (r k+1 r k ) A dr k=0 t(r) r b A dr = Pdx + Qd + Rdz = A tds = A(r(t)) r (t)dt 2 n k k f (r) k f k n f k k f d f (r) k=1 u, v r(u, v) u b, c v d d b f d = f (r(u, v)) r c r dudv c 2011 7/(13)
r n(r) A(r) A nd A d A nd = A d = d b c A(r(u, v)) ( r r ) dudv r n r 3 n k k ϕ(r) ϕ k n ϕ k k ϕd k=1 u, v, w u b, c v d, e w f r(u, v, w) = (x(u, v, w), (u, v, w), z(u, v, w)) f ϕd = e d b c ϕ(r(u, v, w)) J(u, v, w) du dv dw J (x,, z) J(u, v, w) = (u, v, w) = w w w c 2011 8/(13)
4--7 2009 9 1 x P(x, ), Q(x, ) ( Q P ) dxd = Pdx + Qd x 2 xz A(x,, z) = z n (P(x,, z), Q(x,, z), R(x,, z)) A d = A nd n x ( P + Q + R ) dxddz = Pddz + Qdzdx + Rdxd 3 f (x,, z), g(x,, z) ( f 2 g+ f g) d = f g n d, ( ( f 2 g g 2 g f ) d = f n g f ) d n f n n f c 2011 9/(13)
4 xz z n A = (P, Q, R) n ( A) nd = A t ds t x t ( R Q ) ( P ddz + R ) ( Q dzdx + P ) dxd = Pdx + Qd + Rdz c 2011 10/(13)
4--8 2009 9 dω = ω D D D n D D ω n 1 dω ω k ω = i 1<i 2< <i k f i1i 2 i k dx i1 dx i2 dx ik dx i dx j = dx j dx i dx i dx i = 0 f d f = f 1 dx 1 + f 2 dx 2 + + f n dx n dω dω = i 1<i 2< <i k d f i1i 2 i k dx i1 dx i2 dx ik D xz 2 1 ω ω = Pdx + Qd + Rdz ( R Q ) ( P d dz+ R ) ( Q dz dx+ P ) dx d = Pdx+ Qd+Rdz f dx d f dxd (x,, z) (x,, z) (u, v, w) dx d dz = du dv dw (u, v, w) c 2011 11/(13)
4--9 2009 9 P p i j (x 1, x 2, x 3 ) (e 1, e 2, e 3 ) ( x 1, x 2, x 3 ) (ẽ 1, ẽ 2, ẽ 3 ) x i = p i j x j, ẽ i = p i j e j A j=1 j=1 A = 1 e 1 + 2 e 2 + 3 e 3 = ã 1 ẽ 1 + ã 2 ẽ 2 + ã 3 ẽ 3 A i ã i ã i = p i j j j=1 T 11 T 12 T 13 T = T 21 T 22 T 23 T i j = p i p jb T b T 31 T 32 T,b=1 33 T 2 T i j T + T i j + i j ϕt i j 0 T T T = T v = (v 1, v 2, v 3 ) w = Tv ( ) w j = T i j v j w T j=1 T i j = T ji T T i j = T ji T Tv = λv v ( 0) T λ 27 T T i jk P T i jk = p i p jb p kc T bc T 3 n,b,c=1 2 1 c 2011 12/(13)
4--10 2009 9 1 P i r i 1 i N F i r i F i F i N r i F i = 0 i=1 2 3 t (x,, z) ρ(x,, z, t) v(x,, z, t) ρ + (ρv) = 0 t 3 3 v(x,, z, t) v tds 4 ρ(x,, z) D(x,, z) D = ρ Q D d = ρd = Q 5 ω = (ω 1, ω 2, ω 3 ) P r = (x 1, x 2, x 3 ) ρ M = (M 1, M 2, M 3 ) M = r (ω r) ρ d I i j = ( k=1 xk 2 ρ d) δ i j x i x j ρ d M i = I i j ω j I i j j=1 c 2011 13/(13)