396(1).wxm 1 / 14. (%i3) dr_nabla1(psi) := dx*diff(psi,x) + dy*diff(psi,y) + dz*diff(psi,z); d. d Y +dz d

Transcription

1 396(1).wxm 1 / 14 (%i1) kill(all); (%o0) done 1 Definitions (%i1) cross(a,b) := [a[]*b[3] - a[3]*b[], a[3]*b[1] - a[1]*b[3], a[1]*b[] - a[]*b[1]]; (%o1) cross( a,b ):=[a b 3 a 3 b,a 3 b 1 a 1 b 3,a 1 b a b 1 ] (%i) dr: [,, d]; (%o) [,,d] 1.1 Components of Eq.(14) (%i3) dr_nabla1(psi) := *diff(psi,x) + *diff(psi,y) + d*diff(psi,); d (%o3) dr_nabla1( ):= d X + d d Y +d d d (%i4) dr_nabla(psi) := (block [fh], fh: dr_nabla1(psi), *diff(fh,x) + *diff(fh,y) + d*diff(fh,) ); (%o4) dr_nabla( ):= d block fh,fh:dr_nabla1( ), d X fh + d d Y fh +d d d fh (%i5) dr_nabla3(psi) := (block [fh], fh: dr_nabla(psi), *diff(fh,x) + *diff(fh,y) + d*diff(fh,) ); (%o5) dr_nabla3( ):= d block fh,fh:dr_nabla( ), d X fh + d d Y fh +d d d fh (%i6) dr_nabla4(psi) := (block [fh], fh: dr_nabla3(psi), *diff(fh,x) + *diff(fh,y) + d*diff(fh,) ); (%o6) dr_nabla4( ):= d block fh,fh:dr_nabla3( ), d X fh + d d Y fh +d d d fh (%i7) dr_nabla5(psi) := (block [fh], fh: dr_nabla4(psi), *diff(fh,x) + *diff(fh,y) + d*diff(fh,) ); (%o7) dr_nabla5( ):= d block fh,fh:dr_nabla4( ), d X fh + d d Y fh +d d d fh

2 396(1).wxm / 14 (%i8) dr_nabla6(psi) := (block [fh], fh: dr_nabla5(psi), *diff(fh,x) + *diff(fh,y) + d*diff(fh,) ); (%o8) dr_nabla6( ):= d block fh,fh:dr_nabla5( ), d X fh + d d Y fh +d d d fh Averaging functions (%i9) /* Function for averaging delta r terms in an expression */ averagem(f, dr) := block([arglist, s0, den, i, j, res, l, el], /* make list of all summands in the formula */ arglist: [], s0: string(0), arglist: args(expand(f)), /* check if dr[i] appears in any denominator */ for j:1 thru length(arglist) do ( den: denom(arglist[j]), /*print ("DEN:", den),*/ for i:1 thru length(dr) do ( if not freeof(dr[i],den) then print(i, "Denominator error!") )), /* remove all list elements containing occurences of dr[i], dr[i]^3, for j:1 thru length(arglist) do ( for i:1 thru length(dr) do ( if not equal(string(coeff(arglist[j], dr[i], 1)), s0) or not equal(string(coeff(arglist[j], dr[i], 3)), s0) or not equal(string(coeff(arglist[j], dr[i], 5)), s0) then arglist[j]: 0 /*print("l(", dr[i], ") ", arglist[j])*/ )), res: 0, /* construct result expression */ l: length(arglist), for j:1 thru l do ( el: pop(arglist), res: res + el /*print(j,el,res)*/ ), res /*print(res)*/ )$

3 396(1).wxm 3 / 14 (%i10) /* Function for replacing deltax^ etc. to delta r^ terms in cartesian coordinates */ replacer(f,dri) := block([f1, i, dr, len], f1: f, len: length(dri), for i:1 thru len do ( f1: ratsubst(1/len*dr[av]^, dri[i]^, f1) ), factorsum(f1) )$ 3 General formulas for Taylor expansion with aver 3.1 1st order (%i11) depends(f, [X,Y,]); (%o11) [f( X,Y, )] (%i1) dr_nabla1(f); d (%o1) d d f + d d Y f + d d X f (%i14) f1: ratsimp(averagem(dr_nabla1(f), dr)); (%o13) 0 (%o14) 0 3. nd order (%i15) depends(f, [X,Y,]); (%o15) [f( X,Y, )] (%i16) Df: 1/!*dr_nabla(f); (%o16) (d d d d f + d d f +d d Y d d X f +d d d Y d f + d Y d f + d d X d Y f d d X d f + d d X d f + d d X d Y f )/ +

4 396(1).wxm 4 / 14 (%i18) f1: ratsimp(averagem(df, dr)); d d d f (%o17) + d d Y f + d d X f (%o18) dr av d d f rd order d d Y f + (%i19) depends(f, [X,Y,]); (%o19) [f( X,Y, )] 6 d d X f (%i0) Df3: 1/3!*dr_nabla3(f); (%o0) (d (d d d 3 f + d Y d f + f + d X d d Y d f +d d Y d f + d X d Y d f + d X d f +d d X d f + d X d Y d f )+ ( d f +d Y 3 d Y d f + d X d Y f +d d Y d f +d d Y d f + d X d Y d f + d X d f + Y d X d f +d Y d X d Y d f )+ ( d f +d X 3 d X d f + d X d f +d Y d X d f +d d X d f + d X d Y d f + d X d f + Y d X d Y f +d d X d Y d f ))/ 6 (%i) f1: ratsimp(averagem(df3, dr)); (%o1) 0 (%o) th order

5 396(1).wxm 5 / 14 (%i3) depends(f, [X,Y,]); (%o3) [f( X,Y, )]

6 396(1).wxm 6 / 14 (%i4) Df4: 1/4!*dr_nabla4(f); (%o4) (d (d (d d d 4 f + d Y d 3 f + d X d 3 f + d Y d f +d d Y d f + 3 d X d Y d f + d X d f +d d X d f + 3 d X d Y d f )+ ( d Y 3 d f +d d Y d f + d X d Y d f +d d Y d f +d d Y d f + 3 d X d Y d f + d X d Y d f + d X d Y d f +d d X d Y d f )+ ( f +d f + f +d d X 3 d d X d d X d Y d d X d f +d d X d f + 3 d X d Y d f + f + f +d f ))+ ( ( d X d Y d d X d Y d d X d Y d d Y 4 f +d d Y 3 d f + d X d Y 3 f +d d Y 3 d f +d d Y d f + d X d Y d f + d X d Y f + d X d Y 3 f +d d X d Y d f )+d ( d Y 3 d f +d d Y d f + d X d Y d f +d d Y d f +d d Y d f + 3 d X d Y d f + d X d Y d f + d X d Y d f +d d X d Y d f )+ ( d X 3 d f + Y d X d Y f +d d X d Y d f + d X d f + Y d X d f +d Y 3 d X d Y d f +d d X f + d Y d d X d Y f +d f ))+ ( ( d d X d Y d d X 4 f +d d X 3 d f + d X 3 d f +d Y f +d f + f + d X 3 d d X d d X d Y d

7 396(1).wxm 7 / 14 (%i6) f1: ratsimp(averagem(df4, dr)); (%o5) (d 4 d f 4 +4 d f +6 Y 4 d d f +6 X 4 d d X d f +6 (%o6) dr4 av d 4 f th order d4 d Y f +6 4 (%i7) depends(f, [X,Y,]); (%o7) [f( X,Y, )] (%i8) Df5: 1/5!*dr_nabla5(f)$ d Y d f +4 f )/ 4 d X d Y d Y d f + d X d f + 16 (%i30) f1: ratsimp(averagem(df5, dr)); (%o9) 0 (%o30) th order (%i31) depends(f, [X,Y,]); (%o31) [f( X,Y, )] (%i3) Df6: 1/6!*dr_nabla6(f)$ d X d Y f + d4 d X 4 f

8 396(1).wxm 8 / 14 (%i34) f1: ratsimp(averagem(df6, dr)); (%o33) (d 6 d 6 d f 6 +6 d 6 d f +15 Y 6 4 d d 6 f +15 d Y 4 d d 4 d 6 d Y d f 4 +6 d 6 d f +15 X 6 4 d d 6 d X 4 d f d 6 d X 4 d f +15 Y d 4 d 6 d X d f d 6 f +90 d X d Y4 d d 6 f )/ 70 d X d Y d (%o34) (dr 6 av ( d6 d 6 f + d 6 d Y 4 d f + d 6 f +90 d X 6 d6 d Y d 4 f + d 6 d6 d Y 6 f +15 d6 d X 4 d f + f ))/ d X d Y d d6 d X 4 d Y f + d6 d X d 4 f + 4 Vector potential of magnetic dipole (%i35) m: [m_x, m_y, m_]; (%o35) [m X,m Y,m ] d6 d X d Y 4 f + (%i36) r: [X, Y, ]; (%o36) [X,Y,] (%i37) dr: [,, d]; (%o37) [,,d] (%i38) r: r.r; (%o38) +Y +X (%i39) A: cross(m, r)/r^(3/); (%o39) [ m Y Y m, 3/ X m m X, 3/ (%i40) [m_x, m_y, m_]: [0,0,1]; (%o40) [0,0,1] (%i41) A: ev(a); (%o41) [ Y, 3/ X,0] 3/ Y m X X m Y ] 3/

9 396(1).wxm 9 / DA, nd order (%i43) DAX: 1/!*dr_nabla(A[1]); DAY: 1/!*dr_nabla(A[]); 3 d 15 Y d (%o4) ( ( + 5/ 9 Y 15 Y 3 A 3 X 15 X Y + )+d 5/ 7/ 5/ 7/ 3 Y d 15 Y d ( 5/ 3 15 Y A 7/ 5/ 7/ 15 X Y 15 X Y d / )+ ( 3 X A / / Y 5/ 7 15 X Y ))/ 7/ 5 15 X Y 7/ (%o43) ( ( 15 X Y 7 3 X d ( ) / + / + 3 d ( ) / + 15 X d ( ) / 3 Y ( ) +Y +X 5 +Y +X 7 +Y +X 15 X 3 9 X A / / )+d ( 7 15 X d ( ) / X Y ( ) / + +Y +X 5 +Y +X 7 +Y +X 15 X 3 A 15 X Y d )+ ( 7/ 5/ 7/ + 15 X Y A))/ / / / 15 X Y Y 5 / + 3 X 5/

10 396(1).wxm 10 / 14 (%i47) f1: ratsimp(averagem(dax, dr)); f1: ratsimp(averagem(day, dr)); p (%o44) ( +Y +X (( 1 Y 3 Y 3 3 X Y) d +( 9 Y +6 Y 3 9 X Y) +( 3 Y 3 Y 3 +1 X Y) ))/( 8 +( 8 Y +8 X ) 6 + ( 1 Y 4 +4 X Y +1 X 4 ) 4 +( ) +1 X 4 Y 4 +8 X 6 Y + X 8 ) (%o45) 0 p 8 Y 6 +4 X Y 4 +4 X 4 Y +8 X 6 + Y 8 +8 X Y 6 (%o46) ( +Y +X (( 1 X 3 X Y 3 X 3 ) d +( 3 X +1 X Y 3 X 3 ) +( 9 X 9 X Y +6 X 3 ) ))/( 8 +( 8 Y +8 X ) 6 + ( 1 Y 4 +4 X Y +1 X 4 ) 4 +( ) +1 X 4 Y 4 +8 X 6 Y + X 8 ) (%o47) 0 4. DA, 4th order (%i49) DA4X: 1/4!*dr_nabla4(A[1])$ DA4Y: 1/4!*dr_nabla4(A[])$ (%i53) f1: ratsimp(averagem(da4x, dr))$ f1: ratsimp(averagem(da4y, dr))$ 8 Y 6 +4 X Y 4 +4 X 4 Y +8 X 6 + Y 8 +8 X Y 6 (%o51) 35 Y ( Y 3 X +Y 4 5 X Y +3 X 4) dr4 av 18 11/ (%o53) 35 X ( Y 5 X +3 Y 4 5 X Y +X 4) dr4 av 18 11/ 4.3 DA, 6th order (%i55) DA6X: 1/6!*dr_nabla6(A[1])$ DA6Y: 1/6!*dr_nabla6(A[])$ (%i59) f1: ratsimp(averagem(da6x, dr))$ f1: ratsimp(averagem(da6y, dr))$ (%o57) (7 Y ( Y 4 75 X 4 1 Y X Y 75 X 4 + Y 6 1 X Y 4 15 X 4 Y +8 X 6 ) dr av +Y +X 15/ ) 6 )/(9 ( ) (%o59) (7 X ( Y 4 15 X 4 75 Y X Y 1 X 4 +8 Y 6 15 X Y 4 1 X 4 Y + X 6 ) dr 6 av )/(9 15/ )

11 396(1).wxm 11 / 14 5 Plots magnetic vector potential (%i60) A_XY: [ev(a[1], [=0]), ev(a[], [=0])]; (%o60) [ Y, ( Y +X ) 3/ X ] ( Y +X ) 3/ (%i61) plotdf(a_xy, [X,Y], [X,-3,3], [Y,-3,3])$ (%i6) A_X: [ev(a[1], [Y=1]), ev(a[3], [Y=1])]; (%o6) [ 1,0] ( +X +1) 3/ (%i63) plotdf(a_x, [X,], [X,-3,3], [,-3,3])$ 6 Coulomb potential (scalar) (%i64) r: [X, Y, ]; (%o64) [X,Y,] (%i65) dr: [,, d]; (%o65) [,,d] (%i66) r: r.r; (%o66) +Y +X (%i67) U: -1/sqrt(r); 1 (%o67) p +Y +X 6.1 DU, nd order (%i68) DU: 1/!*dr_nabla(U); 0 (%o68) (d d 3 d 3 Y 3 X A 3/ 5/ 5/ + 3 Y d 3 Y 3 X Y + A 5/ 3/ 5/ + A)/ / 3 X d 5/ 3 X Y + 5/ 3/ 3 X 5

12 396(1).wxm 1 / 14 (%i70) f1: ratsimp(averagem(du, dr)); (%o69) p +Y +X Y + X ( Y X ) d +( + Y X ) +( ) 6 +( 6 Y +6 X ) 4 +( ) (%o70) 0 6. DU, 4th order (%i71) DU4: 1/4!*dr_nabla4(U)$ 6 Y 4 +1 X Y +6 X 4 + Y 6 +6 X Y 4 +6 X 4 Y + X 6 (%i73) f1: ratsimp(averagem(du4, dr)); f: DU4Rav: replacer(f1,dr); (%o7) ( 8 4 +( 4 Y 4 X ) +3 Y 4 +6 X Y +3 X 4 d 4 +( 4 4 +( 16 Y 18 X ) 4 Y 4 18 X Y +6 X ( 16 X 18 Y ) +6 Y 4 18 X Y 4 X 4 ) d ( 6 X 4 Y ) +8 Y 4 4 X Y +3 X ( 18 Y 18 X ) 4 Y X Y 4 X 4 + p 3 4 +( 6 Y 4 X ) +3 Y 4 4 X Y +8 X 4 4 )/( +Y +X (8 8 + ( 3 Y +3 X ) 6 +( 48 Y X Y +48 X 4 ) 4 + ( 3 Y X Y X 4 Y +3 X 6 ) +8 Y 8 +3 X Y X 4 Y 4 +3 X 6 Y +8 X 8 )) (%o73) 7 ( 4 3 Y 3 X +Y 4 3 X Y +X 4) dr4 av 18 9/ 6.3 DU, 6th order (%i74) DU6: 1/6!*dr_nabla6(U)$ (%i76) f1: ratsimp(averagem(du6, dr))$ f: DU6Rav: replacer(f1,dr); (%o76) (( 6 15 Y 4 15 X 4 15 Y X Y 15 X 4 + Y 6 15 X Y 4 15 X 4 Y + X 6 ) dr 6 av )/(9 13/ ) 7 Plots Coulomb potential (%i77) assume(x>0,y>0,>0); (%o77) [X>0,Y>0,>0]

13 396(1).wxm 13 / 14 (%i81) UX: ev(u, [Y=0, =0]); DU4X: ev(du4rav/dr[av]^4*dra^4, [dra=., Y=0, =0]); DU6X: ev(du6rav/dr[av]^6*dra^6, [dra=., Y=0, =0]); Utot: UX+DU4X+DU6X; (%o78) 1 X (%o79) X 5 (%o80) X 7 (%o81) 1 X X X 7 (%i8) wxplotd([ux,du4x,du6x], [X,0,1], [y,-30,30], [legend, "UX", "D4UX", "D6UX"]); plotd: expression evaluates to non-numeric value somewhere in plotting range plotd: some values were clipped. plotd: expression evaluates to non-numeric value somewhere in plotting range plotd: some values were clipped. plotd: expression evaluates to non-numeric value somewhere in plotting range plotd: some values were clipped. (%t8) (%o8)

14 396(1).wxm 14 / 14 (%i83) wxplotd([ux,utot], [X,0,1], [y,-30,0], [legend, "UX", "U_{tot}"]); plotd: expression evaluates to non-numeric value somewhere in plotting range plotd: some values were clipped. plotd: expression evaluates to non-numeric value somewhere in plotting range plotd: some values were clipped. (%t83) (%o83)