diff --git a/books/bookvol10.3.pamphlet b/books/bookvol10.3.pamphlet index e43566c..bfb6bb7 100644 --- a/books/bookvol10.3.pamphlet +++ b/books/bookvol10.3.pamphlet @@ -25431,7 +25431,7 @@ Dequeue(S:SetCategory): DequeueAggregate S with )set message test on )set message auto off )clear all ---S 1 of 36 +--S 1 of 100 coefRing := Integer --R --R @@ -25439,7 +25439,7 @@ coefRing := Integer --R Type: Domain --E 1 ---S 2 of 36 +--S 2 of 100 lv : List Symbol := [x,y,z] --R --R @@ -25447,7 +25447,7 @@ lv : List Symbol := [x,y,z] --R Type: List(Symbol) --E 2 ---S 3 of 36 +--S 3 of 100 der := DERHAM(coefRing,lv) --R --R @@ -25455,7 +25455,7 @@ der := DERHAM(coefRing,lv) --R Type: Domain --E 3 ---S 4 of 36 +--S 4 of 100 R := Expression coefRing --R --R @@ -25463,7 +25463,7 @@ R := Expression coefRing --R Type: Domain --E 4 ---S 5 of 36 +--S 5 of 100 f : R := x**2*y*z-5*x**3*y**2*z**5 --R --R @@ -25472,7 +25472,7 @@ f : R := x**2*y*z-5*x**3*y**2*z**5 --R Type: Expression(Integer) --E 5 ---S 6 of 36 +--S 6 of 100 g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2 --R --R @@ -25481,7 +25481,7 @@ g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2 --R Type: Expression(Integer) --E 6 ---S 7 of 36 +--S 7 of 100 h : R :=x*y*z-2*x**3*y*z**2 --R --R @@ -25490,7 +25490,7 @@ h : R :=x*y*z-2*x**3*y*z**2 --R Type: Expression(Integer) --E 7 ---S 8 of 36 +--S 8 of 100 dx : der := generator(1) --R --R @@ -25498,7 +25498,7 @@ dx : der := generator(1) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 8 ---S 9 of 36 +--S 9 of 100 dy : der := generator(2) --R --R @@ -25506,7 +25506,7 @@ dy : der := generator(2) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 9 ---S 10 of 36 +--S 10 of 100 dz : der := generator(3) --R --R @@ -25514,7 +25514,7 @@ dz : der := generator(3) --R Type: DeRhamComplex(Integer,[x,y,z]) --E 10 ---S 11 of 36 +--S 11 of 100 [dx,dy,dz] := [generator(i)$der for i in 1..3] --R --R @@ -25522,7 +25522,7 @@ dz : der := generator(3) --R Type: List(DeRhamComplex(Integer,[x,y,z])) --E 11 ---S 12 of 36 +--S 12 of 100 alpha : der := f*dx + g*dy + h*dz --R --R @@ -25535,7 +25535,7 @@ alpha : der := f*dx + g*dy + h*dz --R Type: DeRhamComplex(Integer,[x,y,z]) --E 12 ---S 13 of 36 +--S 13 of 100 beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy --R --R @@ -25543,7 +25543,7 @@ beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy --R Type: DeRhamComplex(Integer,[x,y,z]) --E 13 ---S 14 of 36 +--S 14 of 100 exteriorDifferential alpha --R --R @@ -25559,7 +25559,7 @@ exteriorDifferential alpha --R Type: DeRhamComplex(Integer,[x,y,z]) --E 14 ---S 15 of 36 +--S 15 of 100 exteriorDifferential % --R --R @@ -25567,7 +25567,7 @@ exteriorDifferential % --R Type: DeRhamComplex(Integer,[x,y,z]) --E 15 ---S 16 of 36 +--S 16 of 100 gamma := alpha * beta --R --R @@ -25580,15 +25580,16 @@ gamma := alpha * beta --R Type: DeRhamComplex(Integer,[x,y,z]) --E 16 ---S 17 of 36 -exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha * exteriorDifferential(beta)) +--S 17 of 100 +exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - _ + alpha * exteriorDifferential(beta)) --R --R --R (17) 0 --R Type: DeRhamComplex(Integer,[x,y,z]) --E 17 ---S 18 of 36 +--S 18 of 100 a : BOP := operator('a) --R --R @@ -25596,7 +25597,7 @@ a : BOP := operator('a) --R Type: BasicOperator --E 18 ---S 19 of 36 +--S 19 of 100 b : BOP := operator('b) --R --R @@ -25604,7 +25605,7 @@ b : BOP := operator('b) --R Type: BasicOperator --E 19 ---S 20 of 36 +--S 20 of 100 c : BOP := operator('c) --R --R @@ -25612,7 +25613,7 @@ c : BOP := operator('c) --R Type: BasicOperator --E 20 ---S 21 of 36 +--S 21 of 100 sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz --R --R @@ -25620,7 +25621,7 @@ sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz --R Type: DeRhamComplex(Integer,[x,y,z]) --E 21 ---S 22 of 36 +--S 22 of 100 theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz --R --R @@ -25628,7 +25629,7 @@ theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz --R Type: DeRhamComplex(Integer,[x,y,z]) --E 22 ---S 23 of 36 +--S 23 of 100 totalDifferential(a(x,y,z))$der --R --R @@ -25637,7 +25638,7 @@ totalDifferential(a(x,y,z))$der --R Type: DeRhamComplex(Integer,[x,y,z]) --E 23 ---S 24 of 36 +--S 24 of 100 exteriorDifferential sigma --R --R @@ -25650,7 +25651,7 @@ exteriorDifferential sigma --R Type: DeRhamComplex(Integer,[x,y,z]) --E 24 ---S 25 of 36 +--S 25 of 100 exteriorDifferential theta --R --R @@ -25659,7 +25660,7 @@ exteriorDifferential theta --R Type: DeRhamComplex(Integer,[x,y,z]) --E 25 ---S 26 of 36 +--S 26 of 100 one : der := 1 --R --R @@ -25667,7 +25668,7 @@ one : der := 1 --R Type: DeRhamComplex(Integer,[x,y,z]) --E 26 ---S 27 of 36 +--S 27 of 100 g1 : der := a([x,t,y,u,v,z,e]) * one --R --R @@ -25675,7 +25676,7 @@ g1 : der := a([x,t,y,u,v,z,e]) * one --R Type: DeRhamComplex(Integer,[x,y,z]) --E 27 ---S 28 of 36 +--S 28 of 100 h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one --R --R @@ -25683,7 +25684,7 @@ h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one --R Type: DeRhamComplex(Integer,[x,y,z]) --E 28 ---S 29 of 36 +--S 29 of 100 exteriorDifferential g1 --R --R @@ -25692,7 +25693,7 @@ exteriorDifferential g1 --R Type: DeRhamComplex(Integer,[x,y,z]) --E 29 ---S 30 of 36 +--S 30 of 100 exteriorDifferential h1 --R --R @@ -25713,7 +25714,7 @@ exteriorDifferential h1 --R Type: DeRhamComplex(Integer,[x,y,z]) --E 30 ---S 31 of 36 +--S 31 of 100 coefficient(gamma, dx*dy) --R --R @@ -25722,7 +25723,7 @@ coefficient(gamma, dx*dy) --R Type: Expression(Integer) --E 31 ---S 32 of 36 +--S 32 of 100 coefficient(gamma, one) --R --R @@ -25730,7 +25731,7 @@ coefficient(gamma, one) --R Type: Expression(Integer) --E 32 ---S 33 of 36 +--S 33 of 100 coefficient(g1,one) --R --R @@ -25738,7 +25739,7 @@ coefficient(g1,one) --R Type: Expression(Integer) --E 33 ---S 34 of 36 +--S 34 of 100 gamma := alpha * beta --R --R @@ -25751,7 +25752,7 @@ gamma := alpha * beta --R Type: DeRhamComplex(Integer,[x,y,z]) --E 34 ---S 35 of 36 +--S 35 of 100 t1:=generator(1)$der --R --R @@ -25759,7 +25760,7 @@ t1:=generator(1)$der --R Type: DeRhamComplex(Integer,[x,y,z]) --E 35 ---S 36 of 36 +--S 36 of 100 degree(t1) --R --R @@ -25767,6 +25768,657 @@ degree(t1) --R Type: PositiveInteger --E 36 +)clear all + + +--S 37 of 100 +X:=DERHAM(Integer,[x,y,z]) +--R +--R +--R (1) DeRhamComplex(Integer,[x,y,z]) +--R Type: Domain +--E 37 + +--S 38 of 100 +[dx,dy,dz]:=[generator(i)$X for i in 1..3] +--R +--R +--R (2) [dx,dy,dz] +--R Type: List(DeRhamComplex(Integer,[x,y,z])) +--E 38 + +--S 39 of 100 +f:BOP:=operator('f) +--R +--R +--R (3) f +--R Type: BasicOperator +--E 39 + +--S 40 of 100 +g:BOP:=operator('g) +--R +--R +--R (4) g +--R Type: BasicOperator +--E 40 + +--S 41 of 100 +h:BOP:=operator('h) +--R +--R +--R (5) h +--R Type: BasicOperator +--E 41 + +--S 42 of 100 +a:BOP:=operator('a) +--R +--R +--R (6) a +--R Type: BasicOperator +--E 42 + +--S 43 of 100 +b:BOP:=operator('b) +--R +--R +--R (7) b +--R Type: BasicOperator +--E 43 + +--S 44 of 100 +c:BOP:=operator('c) +--R +--R +--R (8) c +--R Type: BasicOperator +--E 44 + +--S 45 of 100 +sigma:=f(x,y,z)*dx + g(x,y,z)*dy + h(x,y,z)*dz +--R +--R +--R (9) h(x,y,z)dz + g(x,y,z)dy + f(x,y,z)dx +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 45 + +--S 46 of 100 +theta:=a(x,y,z)*dx*dy + b(x,y,z)*dx*dz + c(x,y,z)*dy*dz +--R +--R +--R (10) c(x,y,z)dy dz + b(x,y,z)dx dz + a(x,y,z)dx dy +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 46 + +--S 47 of 100 +d ==> exteriorDifferential +--R +--R Type: Void +--E 47 + +--S 48 of 100 +leadingCoefficient sigma +--R +--R +--R (12) h(x,y,z) +--R Type: Expression(Integer) +--E 48 + +--S 49 of 100 +leadingBasisTerm sigma +--R +--R +--R (13) dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 49 + +--S 50 of 100 +reductum sigma +--R +--R +--R (14) g(x,y,z)dy + f(x,y,z)dx +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 50 + +--S 51 of 100 +coefficient(sigma,dz) +--R +--R +--R (15) h(x,y,z) +--R Type: Expression(Integer) +--E 51 + +--S 52 of 100 +homogeneous? sigma +--R +--R +--R (16) true +--R Type: Boolean +--E 52 + +--S 53 of 100 +homogeneous? (sigma+theta) +--R +--R +--R (17) false +--R Type: Boolean +--E 53 + +--S 54 of 100 +retractable? sigma +--R +--R +--R (18) false +--R Type: Boolean +--E 54 + +--S 55 of 100 +retractable? (1::X) +--R +--R +--R (19) true +--R Type: Boolean +--E 55 + +--S 56 of 100 +[degree x for x in [sigma,theta,1::X]] +--R +--R +--R (20) [1,2,0] +--R Type: List(NonNegativeInteger) +--E 56 + +--S 57 of 100 +R:=Expression(Integer) +--R +--R +--R (21) Expression(Integer) +--R Type: Domain +--E 57 + +--S 58 of 100 +T:R->R +--R +--R Type: Void +--E 58 + +--S 59 of 100 +T(x)==x^2 +--R +--R Type: Void +--E 59 + +--S 60 of 100 +map(T,sigma) +--R +--R Compiling function T with type Expression(Integer) -> Expression( +--R Integer) +--R +--R 2 2 2 +--R (24) h(x,y,z) dz + g(x,y,z) dy + f(x,y,z) dx +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 60 + +--S 61 of 100 +exteriorDifferential sigma +--R +--R +--R (25) +--R (h (x,y,z) - g (x,y,z))dy dz + (h (x,y,z) - f (x,y,z))dx dz +--R ,2 ,3 ,1 ,3 +--R + +--R (g (x,y,z) - f (x,y,z))dx dy +--R ,1 ,2 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 61 + +--S 62 of 100 +d theta +--R +--R +--R (26) (c (x,y,z) - b (x,y,z) + a (x,y,z))dx dy dz +--R ,1 ,2 ,3 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 62 + +--S 63 of 100 +F:=operator 'F +--R +--R +--R (27) F +--R Type: BasicOperator +--E 63 + +--S 64 of 100 +nullForm:=F(x,y,z)*(1::X) +--R +--R +--R (28) F(x,y,z) +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 64 + +--S 65 of 100 +d nullForm +--R +--R +--R (29) F (x,y,z)dz + F (x,y,z)dy + F (x,y,z)dx +--R ,3 ,2 ,1 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 65 + +--S 66 of 100 +totalDifferential(x^2+y^2+sin(x)*z^2)$X +--R +--R +--R 2 +--R (30) 2z sin(x)dz + 2y dy + (z cos(x) + 2x)dx +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 66 + +--S 67 of 100 +d(d sigma) +--R +--R +--R (31) +--R h (x,y,z) - h (x,y,z) - g (x,y,z) + g (x,y,z) + f (x,y,z) +--R ,2,1 ,1,2 ,3,1 ,1,3 ,3,2 +--R + +--R - f (x,y,z) +--R ,2,3 +--R * +--R dx dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 67 + +--S 68 of 100 +d(d theta) +--R +--R +--R (32) 0 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 68 + +--S 69 of 100 +sigma*theta +--R +--R +--R (33) (a(x,y,z)h(x,y,z) - b(x,y,z)g(x,y,z) + c(x,y,z)f(x,y,z))dx dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 69 + +--S 70 of 100 +dx*dy+dy*dx +--R +--R +--R (34) 0 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 70 + +--S 71 of 100 +sigma+dz +--R +--R +--R (35) (h(x,y,z) + 1)dz + g(x,y,z)dy + f(x,y,z)dx +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 71 + +--S 72 of 100 +theta-(2/3)*sigma+%pi*dx +--R +--R +--R (36) +--R 2h(x,y,z) 2g(x,y,z) - 2f(x,y,z) + 3%pi +--R - --------- dz - --------- dy + c(x,y,z)dy dz + ------------------ dx +--R 3 3 3 +--R + +--R b(x,y,z)dx dz + a(x,y,z)dx dy +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 72 + +--S 73 of 100 +G:=diagonalMatrix([1,1,1]) +--R +--R +--R +1 0 0+ +--R | | +--R (37) |0 1 0| +--R | | +--R +0 0 1+ +--R Type: Matrix(Integer) +--E 73 + +--S 74 of 100 +dot(sigma,sigma,G) +--R +--R +--R 2 2 2 +--R (38) h(x,y,z) + g(x,y,z) + f(x,y,z) +--R Type: Expression(Integer) +--E 74 + +--S 75 of 100 +hodgeStar(sigma,G) +--R +--R +--R (39) h(x,y,z)dx dy - g(x,y,z)dx dz + f(x,y,z)dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 75 + +--S 76 of 100 +A:=dot(sigma,sigma,G)*sqrt(abs(determinant(G))) +--R +--R +--R 2 2 2 +--R (40) h(x,y,z) + g(x,y,z) + f(x,y,z) +--R Type: Expression(Integer) +--E 76 + +--S 77 of 100 +B:=sigma*hodgeStar(sigma,G) +--R +--R +--R 2 2 2 +--R (41) (h(x,y,z) + g(x,y,z) + f(x,y,z) )dx dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 77 + +--S 78 of 100 +test(A*dx*dy*dz=B) +--R +--R +--R (42) true +--R Type: Boolean +--E 78 + +--S 79 of 100 +hodgeStar(dx,G) +--R +--R +--R (43) dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 79 + +--S 80 of 100 +hodgeStar(dy,G) +--R +--R +--R (44) - dx dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 80 + +--S 81 of 100 +d hodgeStar(d nullForm,G) --- = Laplace(F) ? +--R +--R +--R (45) (F (x,y,z) + F (x,y,z) + F (x,y,z))dx dy dz +--R ,3,3 ,2,2 ,1,1 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 81 + +--S 82 of 100 +H:=diagonalMatrix([1+x^2,z^2+exp(-y^2),1/z^2]) +--R +--R +--R + 2 + +--R |x + 1 0 0 | +--R | | +--R | 2 | +--R | - y 2 | +--R (46) | 0 %e + z 0 | +--R | | +--R | 1| +--R | 0 0 --| +--R | 2| +--R + z + +--R Type: Matrix(Expression(Integer)) +--E 82 + +--S 83 of 100 +hodgeStar(dy,H) +--R +--R +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R | (x + 1)%e + (x + 1)z +--R |abs(---------------------------) +--R | 2 +--R \| z +--R (47) - ----------------------------------- dx dz +--R 2 +--R - y 2 +--R %e + z +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 83 + +--S 84 of 100 +hodgeStar(sigma,H) +--R +--R +--R (48) +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R 2 | (x + 1)%e + (x + 1)z +--R z h(x,y,z) |abs(---------------------------) dx dy +--R | 2 +--R \| z +--R + +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R | (x + 1)%e + (x + 1)z +--R g(x,y,z) |abs(---------------------------) +--R | 2 +--R \| z +--R - ------------------------------------------- dx dz +--R 2 +--R - y 2 +--R %e + z +--R + +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R | (x + 1)%e + (x + 1)z +--R f(x,y,z) |abs(---------------------------) +--R | 2 +--R \| z +--R ------------------------------------------- dy dz +--R 2 +--R x + 1 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 84 + +--S 85 of 100 +VolH:=sqrt(abs(determinant(H)))*dx*dy*dz +--R +--R +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R | (x + 1)%e + (x + 1)z +--R (49) |abs(---------------------------) dx dy dz +--R | 2 +--R \| z +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 85 + +--S 86 of 100 +AA:=dot(sigma,sigma,H)*VolH +--R +--R +--R (50) +--R 2 +--R 2 2 2 2 - y 2 4 2 +--R ((x + 1)z h(x,y,z) + f(x,y,z) )%e + (x + 1)z h(x,y,z) +--R + +--R 2 2 2 2 +--R (x + 1)g(x,y,z) + z f(x,y,z) +--R * +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R | (x + 1)%e + (x + 1)z +--R |abs(---------------------------) +--R | 2 +--R \| z +--R / +--R 2 +--R 2 - y 2 2 +--R (x + 1)%e + (x + 1)z +--R * +--R dx dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 86 + +--S 87 of 100 +BB:=sigma*hodgeStar(sigma,H) +--R +--R +--R (51) +--R 2 +--R 2 2 2 2 - y 2 4 2 +--R ((x + 1)z h(x,y,z) + f(x,y,z) )%e + (x + 1)z h(x,y,z) +--R + +--R 2 2 2 2 +--R (x + 1)g(x,y,z) + z f(x,y,z) +--R * +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R | (x + 1)%e + (x + 1)z +--R |abs(---------------------------) +--R | 2 +--R \| z +--R / +--R 2 +--R 2 - y 2 2 +--R (x + 1)%e + (x + 1)z +--R * +--R dx dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 87 + +--S 88 of 100 +test(AA=BB) +--R +--R +--R (52) true +--R Type: Boolean +--E 88 + +--S 89 of 100 +M:=DERHAM(Integer,[t,x,y,z]) +--R +--R +--R (53) DeRhamComplex(Integer,[t,x,y,z]) +--R Type: Domain +--E 89 + +--S 90 of 100 +[dt,dx,dy,dz]:=[generator(i)$M for i in 1..4] +--R +--R +--R (54) [dt,dx,dy,dz] +--R Type: List(DeRhamComplex(Integer,[t,x,y,z])) +--E 90 + +--S 91 of 100 +L:=diagonalMatrix([1,-1,-1,-1]) +--R +--R +--R +1 0 0 0 + +--R | | +--R |0 - 1 0 0 | +--R (55) | | +--R |0 0 - 1 0 | +--R | | +--R +0 0 0 - 1+ +--R Type: Matrix(Integer) +--E 91 + +--S 92 of 100 +hodgeStar(dt,L) +--R +--R +--R (56) - dx dy dz +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 92 + +--S 93 of 100 +hodgeStar(dx,L) +--R +--R +--R (57) - dt dy dz +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 93 + +--S 94 of 100 +hodgeStar(dy,L) +--R +--R +--R (58) dt dx dz +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 94 + +--S 95 of 100 +hodgeStar(dz,L) +--R +--R +--R (59) - dt dx dy +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 95 + +--S 96 of 100 +hodgeStar(dt*dz,L) +--R +--R +--R (60) - dx dy +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 96 + +--S 97 of 100 +P:=a(t,x,y,z)*(dx+dy+dz-dt) +--R +--R +--R (61) a(t,x,y,z)dz + a(t,x,y,z)dy + a(t,x,y,z)dx - a(t,x,y,z)dt +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 97 + +--S 98 of 100 +dot(P,P,L) +--R +--R +--R 2 +--R (62) - 2a(t,x,y,z) +--R Type: Expression(Integer) +--E 98 + +--S 99 of 100 +P*P +--R +--R +--R (63) 0 +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 99 + +--S 100 of 100 +d P +--R +--R +--R (64) +--R (- a (t,x,y,z) + a (t,x,y,z))dy dz + (- a (t,x,y,z) + a (t,x,y,z))dx dz +--R ,4 ,3 ,4 ,2 +--R + +--R (- a (t,x,y,z) + a (t,x,y,z))dx dy + (a (t,x,y,z) + a (t,x,y,z))dt dz +--R ,3 ,2 ,4 ,1 +--R + +--R (a (t,x,y,z) + a (t,x,y,z))dt dy + (a (t,x,y,z) + a (t,x,y,z))dt dx +--R ,3 ,1 ,2 ,1 +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 100 + )spool )lisp (bye) \end{chunk} @@ -25900,7 +26552,8 @@ that is, D satisfies: We try this for the one-forms alpha and beta. - exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha * exteriorDifferential(beta)) + exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - _ + alpha * exteriorDifferential(beta)) 0 Type: DeRhamComplex(Integer,[x,y,z]) @@ -26084,6 +26737,7 @@ DeRhamComplex(CoefRing,listIndVar:List Symbol): Export == Implement where NNI ==> NonNegativeInteger O ==> OutputForm R ==> Expression(CoefRing) + SMR ==> SquareMatrix(#listIndVar,R) Export == Join(LALG(R), RetractableTo(R)) with leadingCoefficient : % -> R @@ -26121,6 +26775,14 @@ DeRhamComplex(CoefRing,listIndVar:List Symbol): Export == Implement where ++ exteriorDifferential(df) returns the exterior ++ derivative (gradient, curl, divergence, ...) of ++ the differential form df. + dim : % -> NNI + ++ dimension of the underlying space + ++ that is, dim ExtAlg = 2^dim + hodgeStar : (%,SMR) -> % + ++ computes the Hodge dual of the differential form % with respect + ++ to the metric g. + dot : (%,%,SMR) -> R + ++ compute the inner product of two differential forms w.r.t. g Implement == ASY add Rep := ASY @@ -26137,24 +26799,21 @@ DeRhamComplex(CoefRing,listIndVar:List Symbol): Export == Implement where exteriorDifferential(x) == x = 0 => 0 - termDiff(leadingCoefficient(x)$Rep,leadingBasisTerm x) + exteriorDifferential(reductum x) + termDiff(leadingCoefficient(x)$Rep,leadingBasisTerm x) + _ + exteriorDifferential(reductum x) lv := [concat("d",string(liv))$String::Symbol for liv in listIndVar] displayList:EAB -> O displayList(x):O == le: L I := exponents(x)$EAB --- reduce(_*,[(lv.i)::O for i in 1..dim | le.i = 1])$L(O) --- reduce(_*,[(lv.i)::O for i in 1..dim | one?(le.i)])$L(O) reduce(_*,[(lv.i)::O for i in 1..dim | ((le.i) = 1)])$L(O) makeTerm:(R,EAB) -> O makeTerm(r,x) == -- we know that r ^= 0 x = Nul(dim)$EAB => r::O --- one? r => displayList(x) (r = 1) => displayList(x) --- r = 1 => displayList(x) r::O * displayList(x) terms : % -> List Record(k: EAB, c: R) @@ -26162,10 +26821,64 @@ DeRhamComplex(CoefRing,listIndVar:List Symbol): Export == Implement where -- it is the case that there are at least two terms in a a pretend List Record(k: EAB, c: R) + err1:="CoefRing has no IntegralDomain" + err2:="Not Implemented" + err3:="Degenerate metric" + + -- coord space dimension + dim(f) == dim + + -- flip 0->1, 1->0 + flip(b:ExtAlgBasis):ExtAlgBasis == + bl := b pretend List(NNI) + [(i+1) rem 2 for i in bl] pretend ExtAlgBasis + + -- list the positions of a's (a=0,1) in x + pos(x:EAB, a:NNI):List(NNI) == + y:= x pretend List(NNI) + [j for j in 1..#y | y.j=a] + + -- compute factors for hodgeStar + facs(cc:Record(k:EAB,c:R),g:SMR):R == + not CoefRing has IntegralDomain => error(err1) + not diagonal? g => error(err2) + G:=reduce("*",[g(j,j) for j in 1..dim]::List(R)) + G=0 => error(err3) + idx:=pos(cc.k,0) -- pos of 0 since already flipped + eps:=concat(pos(cc.k,1),pos(cc.k,0))::List(NNI) + dom:=[j for j in 1..dim]::List(NNI) + sgn:=sign(coercePreimagesImages([dom,eps])::Permutation(NNI))::R + if idx ^= [] then + fg:R:=sgn*reduce("*",[1/g(j,j) for j in idx]::List(R)) + else + fg:R:=sgn + fg*sqrt(abs(G))*cc.c + + -- export + hodgeStar(x,g) == + t:=terms(x) + s:=[copy(r) for r in t] -- we need a copy of x + for j in 1..#t repeat + s.j.k := flip(s.j.k) + s.j.c := facs(s.j,g) -- builtin g + s pretend % + + -- compute dot of singletons + dot1(r:Record(k:EAB,c:R),s:Record(k:EAB,c:R),g:SMR):R == + not CoefRing has IntegralDomain => error(err1) + test(r.k ^= s.k) => 0::R + idx := pos(r.k,1) + reduce("*",[1/g(j,j) for j in idx]::List(R))*r.c*s.c + + -- export + dot(x,y,g) == + tx:=terms(x) + ty:=terms(y) + reduce("+",[dot1(tx.j,ty.j,g) for j in 1..#tx]) + coerce(a):O == a = 0$Rep => 0$I::O ta := terms a --- reductum(a) = 0$Rep => makeTerm(leadingCoefficient a, a.first.k) null ta.rest => makeTerm(ta.first.c, ta.first.k) reduce(_+,[makeTerm(t.c,t.k) for t in ta])$L(O) diff --git a/changelog b/changelog index 9c8a661..69ce447 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,8 @@ +20141008 kxp src/axiom-website/patches.html 20141008.01.kxp.patch +20141008 kxp src/input/Makefile test new derham code +20141008 kxp src/input/derham2.input test new derham code +20141008 kxp books/bookvol10.3 extend derham with new code +20141008 kxp books/bookvol10.3 add code, tests to derham domain 20141006 kxp src/axiom-website/patches.html 20141006.01.kxp.patch 20141006 kxp books/bookvol10.3 DERHAM: fix signature of 'degree' 20141006 kxp Kurt Pagani diff --git a/patch b/patch index 554e7b0..1ffe3a3 100644 --- a/patch +++ b/patch @@ -1,3 +1,3 @@ -books/bookvol10.3 DERHAM: fix signature of 'degree' +books/bookvol10.3 DERHAM: add code for differential forms -Kurt Pagani found and posted a fix +Kurt Pagani posted additional code diff --git a/src/axiom-website/patches.html b/src/axiom-website/patches.html index 9a78b64..98c3e98 100644 --- a/src/axiom-website/patches.html +++ b/src/axiom-website/patches.html @@ -4672,6 +4672,8 @@ src/share/algebra/*.daase update databases
books/endpaper fix algebra hierarchy for OSAGP change
20141006.01.kxp.patch books/bookvol10.3 DERHAM: fix signature of 'degree'
+20141008.01.kxp.patch +books/bookvol10.3 DERHAM: add code for differential forms
diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 94ddfec..6b9ee5e 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -314,7 +314,7 @@ REGRESSTESTS= ackermann.regress \ danzwill.regress danzwill2.regress davenport.regress \ davis.regress \ decimal.regress defintef.regress defintrf.regress \ - derham.regress derivefail.regress \ + derham.regress derham2.regress derivefail.regress \ dfloat.regress dftrig.regress \ dhmatrix.regress \ dhtri.regress directproduct.regress distexpr.regress \ @@ -730,7 +730,7 @@ FILES= ${OUT}/ackermann.input \ ${OUT}/danzwill.input ${OUT}/danzwill2.input ${OUT}/davenport.input \ ${OUT}/davis.input ${OUT}/decimal.input ${OUT}/defs.input \ ${OUT}/defintef.input ${OUT}/defintrf.input ${OUT}/derham.input \ - ${OUT}/derivefail.input ${OUT}/de2re.input \ + ${OUT}/derham2.input ${OUT}/derivefail.input ${OUT}/de2re.input \ ${OUT}/dfloat.input ${OUT}/dftrig.input ${OUT}/dhmatrix.input \ ${OUT}/dhtri.input ${OUT}/directproduct.input \ ${OUT}/distexpr.input ${OUT}/divisor.input ${OUT}/donsimple.input \ @@ -1090,7 +1090,7 @@ DOCFILES= \ ${DOC}/de2re.input.dvi ${DOC}/decimal.input.dvi \ ${DOC}/defintef.input.dvi ${DOC}/defintrf.input.dvi \ ${DOC}/defs.input.dvi ${DOC}/derham.input.dvi \ - ${DOC}/derivefail.input.dvi \ + ${DOC}/derham2.input.dvi ${DOC}/derivefail.input.dvi \ ${DOC}/dfloat.input.dvi ${DOC}/dftrig.input.dvi \ ${DOC}/dhmatrix.input.dvi ${DOC}/dhtri.input.dvi \ ${DOC}/directproduct.input.dvi ${DOC}/distexpr.input.dvi \ diff --git a/src/input/derham2.input.pamphlet b/src/input/derham2.input.pamphlet new file mode 100644 index 0000000..d487d73 --- /dev/null +++ b/src/input/derham2.input.pamphlet @@ -0,0 +1,693 @@ +\documentclass{article} +\usepackage{amssymb} +\usepackage{axiom} +\setlength{\textwidth}{400pt} +\begin{document} +\title{\$SPAD/src/input derham2.input} +\author{Kurt Pagani} +\maketitle +\begin{abstract} +\end{abstract} +\eject +\begin{chunk}{*} +)set break resume +)sys rm -f derham2.output +)spool derham2.output +)set message test on +)set message auto off +)clear all + +\end{chunk} + +$X \sim \mathit{C}^\infty(\mathbb{R}^3,\oplus^3_{p=0}\wedge^p(\mathbb{R}^3))$ +\begin{chunk}{*} +--S 1 of 64 +X:=DERHAM(Integer,[x,y,z]) +--R +--R +--R (1) DeRhamComplex(Integer,[x,y,z]) +--R Type: Domain +--E 1 + +--S 2 of 64 +[dx,dy,dz]:=[generator(i)$X for i in 1..3] +--R +--R +--R (2) [dx,dy,dz] +--R Type: List(DeRhamComplex(Integer,[x,y,z])) +--E 2 + +--S 3 of 64 +f:BOP:=operator('f) +--R +--R +--R (3) f +--R Type: BasicOperator +--E 3 + +--S 4 of 64 +g:BOP:=operator('g) +--R +--R +--R (4) g +--R Type: BasicOperator +--E 4 + +--S 5 of 64 +h:BOP:=operator('h) +--R +--R +--R (5) h +--R Type: BasicOperator +--E 5 + +--S 6 of 64 +a:BOP:=operator('a) +--R +--R +--R (6) a +--R Type: BasicOperator +--E 6 + +--S 7 of 64 +b:BOP:=operator('b) +--R +--R +--R (7) b +--R Type: BasicOperator +--E 7 + +--S 8 of 64 +c:BOP:=operator('c) +--R +--R +--R (8) c +--R Type: BasicOperator +--E 8 + +--S 9 of 64 +sigma:=f(x,y,z)*dx + g(x,y,z)*dy + h(x,y,z)*dz +--R +--R +--R (9) h(x,y,z)dz + g(x,y,z)dy + f(x,y,z)dx +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 9 + +--S 10 of 64 +theta:=a(x,y,z)*dx*dy + b(x,y,z)*dx*dz + c(x,y,z)*dy*dz +--R +--R +--R (10) c(x,y,z)dy dz + b(x,y,z)dx dz + a(x,y,z)dx dy +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 10 + +--S 11 of 64 +d ==> exteriorDifferential +--R +--R Type: Void +--E 11 + +--S 12 of 64 +leadingCoefficient sigma +--R +--R +--R (12) h(x,y,z) +--R Type: Expression(Integer) +--E 12 + +--S 13 of 64 +leadingBasisTerm sigma +--R +--R +--R (13) dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 13 + +--S 14 of 64 +reductum sigma +--R +--R +--R (14) g(x,y,z)dy + f(x,y,z)dx +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 14 + +--S 15 of 64 +coefficient(sigma,dz) +--R +--R +--R (15) h(x,y,z) +--R Type: Expression(Integer) +--E 15 + +--S 16 of 64 +homogeneous? sigma +--R +--R +--R (16) true +--R Type: Boolean +--E 16 + +--S 17 of 64 +homogeneous? (sigma+theta) +--R +--R +--R (17) false +--R Type: Boolean +--E 17 + +--S 18 of 64 +retractable? sigma +--R +--R +--R (18) false +--R Type: Boolean +--E 18 + +--S 19 of 64 +retractable? (1::X) +--R +--R +--R (19) true +--R Type: Boolean +--E 19 + +--S 20 of 64 +[degree x for x in [sigma,theta,1::X]] +--R +--R +--R (20) [1,2,0] +--R Type: List(NonNegativeInteger) +--E 20 + +--S 21 of 64 +R:=Expression(Integer) +--R +--R +--R (21) Expression(Integer) +--R Type: Domain +--E 21 + +--S 22 of 64 +T:R->R +--R +--R Type: Void +--E 22 + +--S 23 of 64 +T(x)==x^2 +--R +--R Type: Void +--E 23 + +--S 24 of 64 +map(T,sigma) +--R +--R Compiling function T with type Expression(Integer) -> Expression( +--R Integer) +--R +--R 2 2 2 +--R (24) h(x,y,z) dz + g(x,y,z) dy + f(x,y,z) dx +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 24 + +--S 25 of 64 +exteriorDifferential sigma +--R +--R +--R (25) +--R (h (x,y,z) - g (x,y,z))dy dz + (h (x,y,z) - f (x,y,z))dx dz +--R ,2 ,3 ,1 ,3 +--R + +--R (g (x,y,z) - f (x,y,z))dx dy +--R ,1 ,2 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 25 + +--S 26 of 64 +d theta +--R +--R +--R (26) (c (x,y,z) - b (x,y,z) + a (x,y,z))dx dy dz +--R ,1 ,2 ,3 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 26 + +--S 27 of 64 +F:=operator 'F +--R +--R +--R (27) F +--R Type: BasicOperator +--E 27 + +--S 28 of 64 +nullForm:=F(x,y,z)*(1::X) +--R +--R +--R (28) F(x,y,z) +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 28 + +--S 29 of 64 +d nullForm +--R +--R +--R (29) F (x,y,z)dz + F (x,y,z)dy + F (x,y,z)dx +--R ,3 ,2 ,1 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 29 + +--S 30 of 64 +totalDifferential(x^2+y^2+sin(x)*z^2)$X +--R +--R +--R 2 +--R (30) 2z sin(x)dz + 2y dy + (z cos(x) + 2x)dx +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 30 + +--S 31 of 64 +d(d sigma) +--R +--R +--R (31) +--R h (x,y,z) - h (x,y,z) - g (x,y,z) + g (x,y,z) + f (x,y,z) +--R ,2,1 ,1,2 ,3,1 ,1,3 ,3,2 +--R + +--R - f (x,y,z) +--R ,2,3 +--R * +--R dx dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 31 + +--S 32 of 64 +d(d theta) +--R +--R +--R (32) 0 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 32 + +--S 33 of 64 +sigma*theta +--R +--R +--R (33) (a(x,y,z)h(x,y,z) - b(x,y,z)g(x,y,z) + c(x,y,z)f(x,y,z))dx dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 33 + +--S 34 of 64 +dx*dy+dy*dx +--R +--R +--R (34) 0 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 34 + +--S 35 of 64 +sigma+dz +--R +--R +--R (35) (h(x,y,z) + 1)dz + g(x,y,z)dy + f(x,y,z)dx +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 35 + +--S 36 of 64 +theta-(2/3)*sigma+%pi*dx +--R +--R +--R (36) +--R 2h(x,y,z) 2g(x,y,z) - 2f(x,y,z) + 3%pi +--R - --------- dz - --------- dy + c(x,y,z)dy dz + ------------------ dx +--R 3 3 3 +--R + +--R b(x,y,z)dx dz + a(x,y,z)dx dy +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 36 + +\end{chunk} + +Definition 1: a*hodgeStar(b,g) = dot(a,b,g)$\times$VolumeForm(g) + +$\alpha\land\star\beta=\langle\alpha,beta\rangle\eta,\quad +\eta=\sqrt{|g|}dx_1\land\ldots\land dx_n$ +\begin{chunk}{*} +--S 37 of 64 +G:=diagonalMatrix([1,1,1]) +--R +--R +--R +1 0 0+ +--R | | +--R (37) |0 1 0| +--R | | +--R +0 0 1+ +--R Type: Matrix(Integer) +--E 37 + +--S 38 of 64 +dot(sigma,sigma,G) +--R +--R +--R 2 2 2 +--R (38) h(x,y,z) + g(x,y,z) + f(x,y,z) +--R Type: Expression(Integer) +--E 38 + +--S 39 of 64 +hodgeStar(sigma,G) +--R +--R +--R (39) h(x,y,z)dx dy - g(x,y,z)dx dz + f(x,y,z)dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 39 + +--S 40 of 64 +A:=dot(sigma,sigma,G)*sqrt(abs(determinant(G))) +--R +--R +--R 2 2 2 +--R (40) h(x,y,z) + g(x,y,z) + f(x,y,z) +--R Type: Expression(Integer) +--E 40 + +--S 41 of 64 +B:=sigma*hodgeStar(sigma,G) +--R +--R +--R 2 2 2 +--R (41) (h(x,y,z) + g(x,y,z) + f(x,y,z) )dx dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 41 + +--S 42 of 64 +test(A*dx*dy*dz=B) +--R +--R +--R (42) true +--R Type: Boolean +--E 42 + +--S 43 of 64 +hodgeStar(dx,G) +--R +--R +--R (43) dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 43 + +--S 44 of 64 +hodgeStar(dy,G) +--R +--R +--R (44) - dx dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 44 + +--S 45 of 64 +d hodgeStar(d nullForm,G) --- = Laplace(F) ? +--R +--R +--R (45) (F (x,y,z) + F (x,y,z) + F (x,y,z))dx dy dz +--R ,3,3 ,2,2 ,1,1 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 45 + +--S 46 of 64 +H:=diagonalMatrix([1+x^2,z^2+exp(-y^2),1/z^2]) +--R +--R +--R + 2 + +--R |x + 1 0 0 | +--R | | +--R | 2 | +--R | - y 2 | +--R (46) | 0 %e + z 0 | +--R | | +--R | 1| +--R | 0 0 --| +--R | 2| +--R + z + +--R Type: Matrix(Expression(Integer)) +--E 46 + +--S 47 of 64 +hodgeStar(dy,H) +--R +--R +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R | (x + 1)%e + (x + 1)z +--R |abs(---------------------------) +--R | 2 +--R \| z +--R (47) - ----------------------------------- dx dz +--R 2 +--R - y 2 +--R %e + z +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 47 + +--S 48 of 64 +hodgeStar(sigma,H) +--R +--R +--R (48) +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R 2 | (x + 1)%e + (x + 1)z +--R z h(x,y,z) |abs(---------------------------) dx dy +--R | 2 +--R \| z +--R + +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R | (x + 1)%e + (x + 1)z +--R g(x,y,z) |abs(---------------------------) +--R | 2 +--R \| z +--R - ------------------------------------------- dx dz +--R 2 +--R - y 2 +--R %e + z +--R + +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R | (x + 1)%e + (x + 1)z +--R f(x,y,z) |abs(---------------------------) +--R | 2 +--R \| z +--R ------------------------------------------- dy dz +--R 2 +--R x + 1 +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 48 + +--S 49 of 64 +VolH:=sqrt(abs(determinant(H)))*dx*dy*dz +--R +--R +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R | (x + 1)%e + (x + 1)z +--R (49) |abs(---------------------------) dx dy dz +--R | 2 +--R \| z +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 49 + +--S 50 of 64 +AA:=dot(sigma,sigma,H)*VolH +--R +--R +--R (50) +--R 2 +--R 2 2 2 2 - y 2 4 2 +--R ((x + 1)z h(x,y,z) + f(x,y,z) )%e + (x + 1)z h(x,y,z) +--R + +--R 2 2 2 2 +--R (x + 1)g(x,y,z) + z f(x,y,z) +--R * +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R | (x + 1)%e + (x + 1)z +--R |abs(---------------------------) +--R | 2 +--R \| z +--R / +--R 2 +--R 2 - y 2 2 +--R (x + 1)%e + (x + 1)z +--R * +--R dx dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 50 + +--S 51 of 64 +BB:=sigma*hodgeStar(sigma,H) +--R +--R +--R (51) +--R 2 +--R 2 2 2 2 - y 2 4 2 +--R ((x + 1)z h(x,y,z) + f(x,y,z) )%e + (x + 1)z h(x,y,z) +--R + +--R 2 2 2 2 +--R (x + 1)g(x,y,z) + z f(x,y,z) +--R * +--R +--------------------------------+ +--R | 2 +--R | 2 - y 2 2 +--R | (x + 1)%e + (x + 1)z +--R |abs(---------------------------) +--R | 2 +--R \| z +--R / +--R 2 +--R 2 - y 2 2 +--R (x + 1)%e + (x + 1)z +--R * +--R dx dy dz +--R Type: DeRhamComplex(Integer,[x,y,z]) +--E 51 + +--S 52 of 64 +test(AA=BB) +--R +--R +--R (52) true +--R Type: Boolean +--E 52 + +\end{chunk} +{\bf Minkowski spacetime} with metric signature $(+---)$ and +coordinates (t,x,y,z) + +\noindent +Checking {\tt http://en.wikipedia.org/wiki/Hodge\_dual} +\begin{chunk}{*} +--S 53 of 64 +M:=DERHAM(Integer,[t,x,y,z]) +--R +--R +--R (53) DeRhamComplex(Integer,[t,x,y,z]) +--R Type: Domain +--E 53 + +--S 54 of 64 +[dt,dx,dy,dz]:=[generator(i)$M for i in 1..4] +--R +--R +--R (54) [dt,dx,dy,dz] +--R Type: List(DeRhamComplex(Integer,[t,x,y,z])) +--E 54 + +--S 55 of 64 +L:=diagonalMatrix([1,-1,-1,-1]) +--R +--R +--R +1 0 0 0 + +--R | | +--R |0 - 1 0 0 | +--R (55) | | +--R |0 0 - 1 0 | +--R | | +--R +0 0 0 - 1+ +--R Type: Matrix(Integer) +--E 55 + +--S 56 of 64 +hodgeStar(dt,L) +--R +--R +--R (56) - dx dy dz +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 56 + +--S 57 of 64 +hodgeStar(dx,L) +--R +--R +--R (57) - dt dy dz +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 57 + +--S 58 of 64 +hodgeStar(dy,L) +--R +--R +--R (58) dt dx dz +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 58 + +--S 59 of 64 +hodgeStar(dz,L) +--R +--R +--R (59) - dt dx dy +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 59 + +--S 60 of 64 +hodgeStar(dt*dz,L) +--R +--R +--R (60) - dx dy +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 60 + +--S 61 of 64 +P:=a(t,x,y,z)*(dx+dy+dz-dt) +--R +--R +--R (61) a(t,x,y,z)dz + a(t,x,y,z)dy + a(t,x,y,z)dx - a(t,x,y,z)dt +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 61 + +--S 62 of 64 +dot(P,P,L) +--R +--R +--R 2 +--R (62) - 2a(t,x,y,z) +--R Type: Expression(Integer) +--E 62 + +--S 63 of 64 +P*P +--R +--R +--R (63) 0 +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 63 + +--S 64 of 64 +d P +--R +--R +--R (64) +--R (- a (t,x,y,z) + a (t,x,y,z))dy dz + (- a (t,x,y,z) + a (t,x,y,z))dx dz +--R ,4 ,3 ,4 ,2 +--R + +--R (- a (t,x,y,z) + a (t,x,y,z))dx dy + (a (t,x,y,z) + a (t,x,y,z))dt dz +--R ,3 ,2 ,4 ,1 +--R + +--R (a (t,x,y,z) + a (t,x,y,z))dt dy + (a (t,x,y,z) + a (t,x,y,z))dt dx +--R ,3 ,1 ,2 ,1 +--R Type: DeRhamComplex(Integer,[t,x,y,z]) +--E 64 +)spool +)lisp (bye) + +\end{chunk} +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document}