直交座標系の回転

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1 b T.Koama

2 x l x, Lx i ij j j xi i i i, x L T L L, L ± x L T xax axx, ( a a ) i, j ij i j ij ji λ λ + λ + + λ i i i x L T T T x ( L) L T xax T ( T L T ) A( L) T ( LAL T ) T ( L AL) λ ii L AL Λ λi i axx ij i j λ xi i, j i T x ( AλE) x

3 x L ii i i i T λ λ ( Λ λe) L ( A λe) L ( Λ λe ) L ( A λe) L L A λe L A λe Λ λ E a λ a a λ λ a a λ a λ λ ( λ λ)( λ λ) ( λ λ) a a a λ λ λ λ i λ A λe λ i λ Ax λ x x A λ (x,) ax + hx + b ax + h a h x + + ( + ) + ( + ) ( ) ( ) hx + b h b ax hx b x ax h hx b x x ( x ', ') ( x ', ')

4 αx' α x' αx' + β' x' αx+ ' β' ( x' ' ) ( x' ' ) β' β ' α, β a λ h h b λ ( a )( b ) h ( a b) ab h λ λ λ + λ + h > α h < α α, β > ( ax + bx + c, xx c / a, x + x b / a,) αβ ab h αx ' + β' x' ' + ( / α) ( / β) S S π π α β αβ π ab h x z a b c + + lx + m + z l + m +

5 z ( lx + m ) x z x ( lx + m ) a b c a b c x a b c ( lx m lmx ) l lm m + x x a c + + c + b c x S ' l m lm + a c + b c c c + al c + bm lm ac bc c 4 4 ( c + al)( c + bm) ablm 4 4 abc c + ( al + bm) c + ablm ablm 4 4 abc al + bm + c abc 4 4 π π abc S ' al + bm + c al + bm + c abc lx + m + z x S S' S π abc al + bm + c π abc al + bm + c 4

6 x z a b c + + lx + m + z l + m + z ( lx + m ) x z x ( lx + m ) a b c a b c x a b c { lx m lmx lx ( m ) } l lm m l m + x x x a c + + c + + b c c c c x x ( x, ) x x x x l lm m a c c b c + ( x x ) + ( x x )( ) + + ( ) l m ( x x ) ( ) + c c c 5

7 l lm l + x a c c c m lm m + x b c c c l lm l + x a c c c lm m m + c b c c ( x, ) x l bc al + bm + c al + bm + c abc m ac al + bm + c al + bm + c abc a l b m l lm + a c c l m lm a l + b m + c + a c + b c lm m c abc + c b c l lm c c l m lm m l m m c + b c + c c bc + c b c l l + a c c lm m c c l m l lm m + a c + c c c a c 6

8 l m lm c + al c + bm lm + + a c b c c ac bc c 4 4 ( c + al)( c + bm) ablm c + ( al + bm) c + ablm ablm abc abc al + bm + c abc 4 4 l m lm m l( c b m ) lm l c b c c c bc c bc l m l lm c + a l ml m + + m + a c c c c ac c ac l lm m l m a c c b c c c c + x + x x + + lm l lm lm m l m x x + x x + x + + c c c c c c c c l m x + + c c c ( lx + m + ) c la l mb m + c al + bm + c al + bm + c a l b m + a l + b m + c c al + bm + c al + bm + c 7

9 l lm l + x a c c c a c c c l lm l + x l lm l + x x x a c c c m lm m + x b c c c b c c c b m lm m + x m + c lm m x c c l lm m a c c b c + ( x x ) + ( x x )( ) + + ( ) l m ( x x ) ( ) + c c c l lm m + x + x a c c b c al + bm + c a c c + + b c a l + b m + c l lm m a l b m c + x + x + + l lm m t + x t x t a c + + c + b c t al + bm + c al + bm + c S ' l m lm a l b m c + + t + + t t t a c b c c abc π π abc S ' al + bm + c t al + bm + c abc t 8

10 lx + m + z x π abc S S' t a l + b m + c πabc πabc a l b m c S / t a l + b m + c ( al + bm + c) ( + + ) ( θ, ϕ ) l siθ cosϕ m siθ siϕ cosθ a a b aq c ap K( θ, ϕ) si θ cos ϕ q si θ si ϕ p cos + + θ al + bm + c al + aqm + a p a si θ cos ϕ + a q si θ si ϕ + a p cos θ a (si θ cos ϕ + q si θ si ϕ + p cos θ) ak( θϕ, ) S πabc( a l + b m + c ) πa pq{ a K ( θ, ϕ) } π pq{ a K ( θ, ϕ) } / / ( al + bm + c) { ak ( θϕ, )} K( θϕ, ) x, z, aaqap,, 9

11 x z a a q a p + + x asiθ cosϕ aqsiθ siϕ z apcosθ θ h z h x ϕ h h K( θ, ϕ ) K( θ, ϕ) si θ cos ϕ q si θ si ϕ p cos + + θ A πqp( ) A K ( θϕ, ) ak( θϕ, ) > A A θ, ϕ h K(, ) si cos q si si p cos θϕ θ ϕ+ θ ϕ+ θ K ( θϕ, ) si θcos ϕ+ q si θsi ϕ+ p cos θ ak( θ, ϕ) a si θ cos ϕ + aq si θ si ϕ + ap cos θ ak( θϕ, ) x + + z a K ( θϕ, ) x + + z v A v v

12 θ ( h) c( r) exp( ihr) dr exp{ ih( + r' )} ddr' r r' r' exp( ih) ddr' r' { r' } exp( ih) d d exp( ih) Ad h v r' h h, h r' θ πqp ih Ad h d K ( θ ) ( h ) exp( ) ( ) cos( ) cos( h) d si( h ) h h d h + h h cos( ) si( ) cos( ) si( ) h h h ( ) cos( h) d ( ) cos( h) d si( h) si( h) cos( h ) si( h ) h + h h h 4 si( h ) cos( h ) h h 4si( [ h ) h cos( h )] h θ ( h) [ si( h) hcos( h) ] qp [ si( h) hcos( h) ] 4π qp θ ( h ) V K h K h ( θϕ, ) ( θϕ, ) ( ) 4 V π

13 q K( θ, p) si θ cos ϕ + si θ si ϕ + p cos θ si θ + p cos θ ak( θ, p) [ si( h) hcos( h) ] p θ ( h) V K p h ( θ, ) ( ) p q K a θ ( h) V [ si( ah) ahcos( ah) ] ( ah )

> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3

> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3 13 2 13.0 2 ( ) ( ) 2 13.1 ( ) ax 2 + bx + c > 0 ( a, b, c ) ( ) 275 > > 2 2 13.3 x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D >

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