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A. Vector analysys I
A1.vector to form
in this package, a vector shoud be transformed to a form.
for example,[a,b,c]<---->a*Dx+b*Dy+c*Dz (so called work form)
[a,b,c]<---->a*Dy@Dz+b*Dz@Dx+c*Dx@Dy (so called fluid form)
vtof1,vtof2 need the scale_factor,so not to use f_star(),to use fstar_with_clf()
or under global environment ( see lorentz_example.txt)
(%i17) fstar_with_clf([x,y,z],[x,y,z],vtof1([a,b,c]));
(%o17) c Dz + b Dy + a Dx
(%i18) fstar_with_clf([x,y,z],[x,y,z],vtof2([a,b,c]));
(%o18) a Dy Dz - b Dx Dz + c Dx Dy
(%i19) fstar_with_clf([r,th,phi],[r*sin(phi)*cos(th),r*sin(phi)*sin(th),r*cos(phi)],
vtof1([a,b,c]));
(%o19) b Dth sin(phi) r + c Dphi r + a Dr
(%i20) fstar_with_clf([r,th,phi],[r*sin(phi)*cos(th),r*sin(phi)*sin(th),r*cos(phi)],
vtof2([a,b,c]));
2 b Dphi Dr
(%o20) a Dphi Dth sin(phi) r + c Dr Dth sin(phi) - ---------
sin(phi)
(%i21) fstar_with_clf([r,th,phi],[r*sin(phi)*cos(th),r*sin(phi)*sin(th),r*cos(phi)],
scale_factor);
(%o21) [1, sin(phi) r, r]
A2. grad,rot or curle,div,laplacian
grad(f) is d(f),if A is vector,a:vtof1(A) then rot(A) is h_st(d(a)),or nest2([h_st,d],a)
h_st() is hodge star operator,div(f) is h_st(d(h_st(f))),laplacian(f) is d(h_st(d(f)))
or-1(d(antid(a))+antid(d(a)))s ok generally
1.gradiant
(%i22) fstar_with_clf([x,y,z],[x,y,z],(depends(f,[x,y,z]),d(f)));
df df df
(%o22) Dz -- + Dy -- + Dx --
dz dy dx
(%i23) fstar_with_clf([r,th,phi],[r*sin(phi)*cos(th),r*sin(phi)*sin(th),r*cos(phi)]
,(depends(f,[r,th,phi]),d(f)));
df df df
(%o23) Dth --- + Dr -- + Dphi ----
dth dr dphi
2.rotate
(%i24) fstar_with_clf([x,y,z],[x,y,z],(depends([a,b,c],[x,y,z]),aa:vtof1([a,b,c]),
h_st(d(aa))));
db da dc da dc db
(%o24) -- Dz - -- Dz - -- Dy + -- Dy + -- Dx - -- Dx
dx dy dx dz dy dz
(%i25) format(%,%poly(Dx,Dy,Dz),factor);
db da dc da dc db
(%o25) (-- - --) Dz - (-- - --) Dy + (-- - --) Dx
dx dy dx dz dy dz
(%i29) fstar_with_clf([r,th,phi],[r*sin(phi)*cos(th),r*sin(phi)*sin(th),r*cos(phi)],
(depends([a,b,c],[r,th,phi]),aa:vtof1([a,b,c]),h_st(d(aa))))$
(%i30) format(%,%poly(Dr,Dth,Dphi),factor);
db da
Dphi (-- sin(phi) r + b sin(phi) - ---)
dr dth dc da
(%o30) --------------------------------------- - Dth sin(phi) (-- r + c - ----)
sin(phi) dr dphi
db dc
Dr (---- sin(phi) + b cos(phi) - ---)
dphi dth
- -------------------------------------
sin(phi) r
3.divergent
(%i31) fstar_with_clf([x,y,z],[x,y,z],(depends([a,b,c],[x,y,z]),aa:vtof1([a,b,c]),
nest2([d,h_st],aa)));
dc db da
(%o31) -- Dx Dy Dz + -- Dx Dy Dz + -- Dx Dy Dz
dz dy dx
(%i32) fstar_with_clf([x,y,z],[x,y,z],(depends([a,b,c],[x,y,z]),aa:vtof1([a,b,c]),
nest2([h_st,d,h_st],aa)));
dc db da
(%o32) -- + -- + --
dz dy dx
(%i33) fstar_with_clf([r,th,phi],[r*sin(phi)*cos(th),r*sin(phi)*sin(th),r*cos(phi)],
(depends([a,b,c],[r,th,phi]),aa:vtof1([a,b,c]),nest2([h_st,d,h_st],aa)));
db dc
--- ----
c cos(phi) dth dphi 2 a da
(%o33) ---------- + ---------- + ---- + --- + --
sin(phi) r sin(phi) r r r dr
4.laplacian
(%i34) fstar_with_clf([x,y,z],[x,y,z],(depends(f,[x,y,z]),nest2([d,h_st,d],f)));
2 2 2
d f d f d f
(%o34) Dx Dy Dz --- + Dx Dy Dz --- + Dx Dy Dz ---
2 2 2
dz dy dx
(%i36) fstar_with_clf([x,y,z],[x,y,z],(depends(f,[x,y,z]),nest2([h_st,d,h_st,d],f)));
2 2 2
d f d f d f
(%o36) --- + --- + ---
2 2 2
dz dy dx
(%i37) fstar_with_clf([r,th,phi],[r*sin(phi)*cos(th),r*sin(phi)*sin(th),r*cos(phi)],
(depends(f,[r,th,phi]),nest2([h_st,d,h_st,d],f)));
2 2
d f d f
df df ---- -----
2 -- ---- cos(phi) 2 2 2
dr dphi dth dphi d f
(%o37) ---- + ------------- + ------------ + ----- + ---
r 2 2 2 2 2
sin(phi) r sin (phi) r r dr
(%i38) fstar_with_clf([r,th,phi],[r*sin(phi)*cos(th),r*sin(phi)*sin(th),r*cos(phi)],
(depends(f,[r,th,phi]),nest2([d,h_st,d],f)));
2
d f 2 df
(%o38) Dphi Dr Dth --- sin(phi) r + 2 Dphi Dr Dth -- sin(phi) r
2 dr
dr
2
d f
Dphi Dr Dth ----
2 2
d f dth df
+ Dphi Dr Dth ----- sin(phi) + ---------------- + Dphi Dr Dth ---- cos(phi)
2 sin(phi) dphi
A3. Example for vorticity
problem: in R3 we think velocity field v=rot(A),div(A)=0,so A is vector potential.
prove voricity w=rot(v) satisfy A= -w
a solution: as restricted 2 spaces,A can be posed,a1*Dx+a2*Dy.
w=rot(rot(A)),then
(%i40) fstar_with_clf([x,y,z],[x,y,z],(depends([a1,a2],[x,y]),A:a1*Dx+a2*Dy,
nest2([h_st,d,h_st,d],A)));
2 2 2 2
d a2 d a1 d a2 d a1
(%o40) - ---- Dy + ----- Dy + ----- Dx - ---- Dx
2 dx dy dx dy 2
dx dy
from condition ,div(A)=0
(%i41) fstar_with_clf([x,y,z],[x,y,z],(depends([a1,a2],[x,y]),A:a1*Dx+a2*Dy,d(h_st(A))));
da2 da1
(%o78) --- Dx Dy Dz + --- Dx Dy Dz
dy dx
d(a2)/dy+d(a1)/dx=0 ,then d^2(a1)/dxdy= -d^2(a2)/dx^2,d^2(a2)/dxdy=-d^2(a1)/dy^2
A is -(d(antid(A))+antid(d(A))
(%i42) fstar_with_clf([x,y,z],[x,y,z],(depends([a1,a2],[x,y]),A:a1*Dx+a2*Dy,-(nest2([d,antid],A)+nest2([antid,d],A))));
2 2 2 2
d a2 d a2 d a1 d a1
(%o42) ---- Dy + ---- Dy + ---- Dx + ---- Dx
2 2 2 2
dy dx dy dx
then
A= -w
but in this case,old fashion style is good with A=grad(div(A)-rot(rot(A))
B.Changing integral variable more than two variables
B1.this is from (z)*(1-z)
changing integral variable more than two variables,such like
(%i55) f_star([u,v],(s:u/(v+1),t:u*v/(v+1),exp(-(s+t))*s^(-z)*t^(z-1)*d(s)@d(t)))$
(%i57) format(%o55,%poly(u,v),factor);
u v u
- ----- - -----
z - 1 v + 1 v + 1
Du Dv v %e
(%o57) ------------------------------
v + 1
this integral,v,[0,inf],u,[0,inf] is %pi/sin(%pi) (=beta(z,1-z))
B2. from entrance exam. of graduate course in japan
(1)integrate x+y+z=u,y+z=u*v,z=u*v*w,on 0<u<1,0<v<1,0<w<1 by pullback (x^2+y^2+z^2)dxdydz
(%i43) solve([x+y+z=u,y+z=u*v,z=u*v*w],[x,y,z]);
(%o43) [[x = u (1 - v), y = u v (1 - w), z = u v w]]
(%i45) f_star([u,v,w],(x:u*(1-v),y:u*v*(1-w),z:u*v*w,(x^2+y^2+z^2)*d(x)@d(y)@d(z)));
2 2 2 2 2 2 2 2 2
(%o45) Du Dv Dw u v (u v w + u v (1 - w) + u (1 - v) )
(%i46) defint(defint(defint(u^2*v*(u^2*v^2*w^2+u^2*v^2*(1-w)^2+u^2*(1-v)^2),w,0,1)
,v,0,1),u,0,1);
(%o46) 1/20
(%o47) kill(x,y,z);
slightly easy problem
(2) (d^2/dx^2+d^2/dy^2)(log(sqrt(x^2+y^2)))
(%i38) fstar_with_clf([x,y],[x,y],(d(h_st(d(log(sqrt(x^2+y^2)))))))$
(%i39) format(%,%poly(Dx,Dy),factor);
(%o39) 0
(3) Integrate (x*Dx@Dy) on x^2/a^2+y^2/b^2<=1,x>=0,y>=0
(%i40) fstar_with_clf([x,y],[x,y],(depends([a1,a2],[x,y]),d(a1*d(x)+a2*d(y))));
da2 da1
(%o40) --- Dx Dy - --- Dx Dy
dx dy
(%i41) fstar_with_clf([x,y],[x,y],(d(1/2*x^2*Dy)));
(%o41) Dx Dy x
1/2*x^2*Dy x=a*cos(th),y=b*sin(th) on boundary you may use line integrate.
B3. Integral
we must pay attention to effects of some singularity and of topology (De Rham cohomology )
this points is most difficult to calculate multivariable integral in CAS,I think.
see Frankel book "The Geometry of PHYSICS" p162 5.5 Finding Potential.
(Pontrjaguin index)
(%i74) f_star([phi,theta],(pu1:sin(phi)*cos(theta),pu2:sin(phi)*sin(theta),pu3:cos(phi),
pu1*d(pu2)@d(pu3)+pu2*d(pu3)@d(pu1)+pu3*d(pu1)@d(pu2)))$
(%i76) trigsimp(%o74);
(%o76) Dphi Dtheta sin(phi)
So this map induce (x,y)-->(pu1,pu2,pu3) IxI--->S2,boundary 1X1 -->one point.
%o76 is volume form in S2,if we can pull back IxI,we integrate this and say
this integral value is n*4pai. n is a integer.
(%i79) f_star([x,y],(depends([p1,p2,p3],[x,y]),p1*d(p2)@d(p3)+p2*d(p3)@d(p1)+p3*d(p1)@d(p2)));
dp2 dp3 dp2 dp3 dp1 dp3 dp1 dp3
(%o79) p1 (Dx Dy --- --- - Dx Dy --- ---) + p2 (Dx Dy --- --- - Dx Dy --- ---)
dx dy dy dx dy dx dx dy
dp1 dp2 dp1 dp2
+ (Dx Dy --- --- - Dx Dy --- ---) p3
dx dy dx dy
(%o79) is written by using vector,(p1,p2,p3).{(d/dx(p1,p2,p3)Xd/dy(p1,p2,p3))*Dx@Dy}
integrate with IxI,this is homotopy invariant.
generaly speaking,in exchaing coordinates, vector equations are not always invariant but differntial forms are invariant. so we can calculate forms with
some suitable or fit coordinate,then by pulling back it with general ones.
C.Integral factor (always possible locally,but not global)
locally necessary and sufficient condition is d(w)@w=0. so if f^-1 is integral factor
,d(w)=d(log|f|)@w . usually we use frobenius method(or theorem) but I propose a new method
with quantization.
(%i72) fstar_with_clf([x,y,z],[x,y,z],trans_toexact(y*z*d(x)+x*z*d(y)+d(z)));
2 2
Dz u1 y z + Dz u2 x z - Dx u3 y - Dy u3 x
(%o72) -------------------------------------------
2 2 2 2
u1 y z + u2 x z + u3
u1,u2,u3 is scale factor.
integral factor do'nt depends on metric,so u1->0,u2->0. or u2->0,u3->0 so on.
this is a quantization from clifford algebra to grassmann algebra.
(%i74) limit(limit(%o72,u1,0),u2,0);
(%o74) - Dx y - Dy x
(%i75) limit(limit(%o72,u3,0),u2,0);
Dz
(%o75) --
z
now w is y*z*d(x)+x*z*d(y)+d(z),by %o74 d(w)=(- Dx y - Dy x)@w,
d(log|f|)=-( Dx y + Dy x)=-d(x*y),f^(-1)=%e^(x*y) is integral factor.
by %o75 d(w)=Dz/z @w,d(log|f|)=Dz/z,f^(-1)=1/z is also integral factor.
but %e^(x*y)is better than 1/z. see Flanders P92,93.
D. Poission bracket (symplectic form,symplectic manifold)
for example symplectic manifold dim 6,it has 2form named to symplectic forms.
1. w is closed form 2. w^3==w@w@w is not 0 everywhere .
(if dim 2*m,w^mis not0)
(%i39) f_star([p1,q1,p2,q2,p3,q3],(omega:Dp1@Dq1+Dp2@Dq2+Dp3@Dq3,omega@omega@omega));
(%o39) - 6 Dp1 Dp2 Dp3 Dq1 Dq2 Dq3
(%i40) f_star([p1,q1,p2,q2,p3,q3],(omega:Dp1@Dq1+Dp2@Dq2+Dp3@Dq3,d(omega)));
(%o40) 0
so this omega is simplectic form.
Poission bracket is definded as coordinates free
inner(Xf,(inner(Xg,omega))),and inner(Xg,omega) is d(g).
(%i42) f_star([p1,p2,p3,q1,q2,q3],(depends([g],[p1,p2,p3,q1,q2,q3]),omega:Dp1@Dq1+Dp2@Dq2
+Dp3@Dq3,inner(diff(g,q1)*Dp1-diff(g,p1)*Dq1+diff(g,q2)*Dp2-diff(g,p2)*Dq2+diff(g,q3)*Dp3
-diff(g,p3)*Dq3,omega)));
dg dg dg dg dg dg
(%o42) Dq3 --- + Dq2 --- + Dq1 --- + Dp3 --- + Dp2 --- + Dp1 ---
dq3 dq2 dq1 dp3 dp2 dp1
because poission bracket is defined
by inner(Xf,d(g)) in this package p_braket().
(%i55) f_star([p1,q1,p2,q2,p3,q3],(depends([f,g],[p1,q1,p2,q2,p3,q3]),p_braket(f,g)));
df dg df dg df dg df dg df dg df dg
(%o55) --- --- + --- --- + --- --- - --- --- - --- --- - --- ---
dp3 dq3 dp2 dq2 dp1 dq1 dq3 dp3 dq2 dp2 dq1 dp1
Jacobi identity
(%i63) f_star([p1,q1,p2,q2,p3,q3],(depends([f,g,h],[p1,q1,p2,q2,p3,q3]),
p_braket(p_braket(f,g),h)+p_braket(p_braket(g,h),f)+p_braket(p_braket(h,f),g)))$
(%i64) ratsimp(%);
(%o64) 0
(%i67) f_star([p1,q1,p2,q2,p3,q3],(depends([f,g,h],[p1,q1,p2,q2,p3,q3]),
p_braket(f,g*h)-g*p_braket(f,h)-h*p_braket(f,g)))$
(%i68) ratsimp(%);
(%o68) 0
Is canonical? [p,q]--->[P,Q]
(%i69) f_star([p,q],(P:q*cot(p),Q:log(sin(p)/q),d(P)@d(Q)))$
(%i70) trigsimp(%);
(%o70) Dp Dq
DP@DQ=Dp@Dq show canonical coordinates each other.
if H is hamiltonian, d(q)/dt=p_braket(q,H),d(p)/dt=p_braket(p,H)
(%i71) f_star([p1,q1,p2,q2,p3,q3],(depends([f,g,h],[p1,q1,p2,q2,p3,q3]),
p_braket(f*g,h)-f*p_braket(g,h)-g*p_braket(f,h)))$ is 0
if p_braket(g,H)=0 and p_braket(f,H)=0 (f,g is first integral),then
p_braket(f*g,h)=0 so f*g is a first integral too.
from jacobi identity p_braket(f,g) is also first integral.
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