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Hello,

I'm studying Helmholtz equation in cylindrical coordinates. I'm using a solution of the scalar Helmholtz equation to build a vector field that could be a solution of the vector Helmholtz equation.

Here is my code with an orthonormal basis in a 3D manifold with cylindrical coordinates. Normally, SVH should be equal to 0 (actually, it is), but Sage doesn't simplify. I've used the simplify function, but it doesn't work. Can someone help me?

%display latex

from sage.manifolds.operators import *

E = Manifold(3,coordinates='cylindrical', name='E') # 3D Manifold in cylindrical coord

Ecyl.<r,ph,z> = E.chart(r'r:(0,+oo) ph:(0,2*pi):\phi z')

k = var('k')

assume(k>=0)

to_orthonormal = E.automorphism_field() # Orthonormal basis q

to_orthonormal[0,0] = 1

to_orthonormal[1,1] = 1/r

to_orthonormal[2,2] = 1

q = Ecyl.frame().new_frame(to_orthonormal, 'q')

E.set_default_frame(q)

eta = E.metric(name='eta', latex_name=r'\eta') #Minkowski metric

eta[0,0] = 1

eta[1,1] = 1

eta[2,2] = 1

h = E.scalar_field(bessel_J(0,sqrt(k^2)r)cos(0z)cos(0*ph), name='h') #Helmholtz equation solution in cylindrical coord

k2 = -h.laplacian(eta)*h^(-1) # Determination of k^2

H = h(rq[0]+z*q[2]) #Definition of a vector field H such that O is a vectorial Helmholtz equation solution

O = H.curl(metric=eta)

SHV = O.laplacian(eta) + k2*O

SHV.display()

Sage solution : -1/4((2bessel_J(2, kr)bessel_J(1, kr) - bessel_J(3, kr)bessel_J(0, kr) + bessel_J(1, kr)bessel_J(0, kr))k^3r^2 - 2(2bessel_J(1, kr)^2 - bessel_J(2, kr)bessel_J(0, kr) + bessel_J(0, kr)^2)k^2r + 4kbessel_J(1, kr)bessel_J(0, kr))z/(r^2bessel_J(0, kr))

Hello,

I'm studying Helmholtz equation in cylindrical coordinates. I'm using a solution of the scalar Helmholtz equation to build a vector field that could be a solution of the vector Helmholtz equation.

Here is my code with an orthonormal basis in a 3D manifold with cylindrical coordinates. Normally, SVH should be equal to 0 (actually, it is), but Sage doesn't simplify. I've used the simplify function, but it doesn't work. Can someone help me?

%display latex
 
latex

from sage.manifolds.operators import * 
 
 E = Manifold(3,coordinates='cylindrical', name='E')  # 3D Manifold in cylindrical coord
 
coord

Ecyl.<r,ph,z> = E.chart(r'r:(0,+oo) ph:(0,2*pi):\phi z')
 
z')

k = var('k')
 
assume(k>=0)
 
var('k')

assume(k>=0)

to_orthonormal = E.automorphism_field()  # Orthonormal basis q
 
q

to_orthonormal[0,0] = 1
 
1

to_orthonormal[1,1] = 1/r
 
1/r

to_orthonormal[2,2] = 1
 
1

q = Ecyl.frame().new_frame(to_orthonormal, 'q')
 
E.set_default_frame(q)
 
'q')

E.set_default_frame(q)

eta = E.metric(name='eta', latex_name=r'\eta') #Minkowski metric
 
metric

eta[0,0] = 1
 
1

eta[1,1] = 1
 
1

eta[2,2] = 1
 
1

h = E.scalar_field(bessel_J(0,sqrt(k^2)r)cos(0z)cos(0*ph), E.scalar_field(bessel_J(0,sqrt(k^2)*r)*cos(0*z)*cos(0*ph), name='h')  #Helmholtz equation solution in cylindrical coord
 
coord

k2 = -h.laplacian(eta)*h^(-1) # Determination of k^2
 
k^2

H = h(rq[0]+z*q[2]) h*(r*q[0]+z*q[2])  #Definition of a vector field H such that O is a vectorial Helmholtz equation solution
 
solution

O = H.curl(metric=eta)
 
H.curl(metric=eta)

SHV = O.laplacian(eta) + k2*O
 
SHV.display()k2*O

SHV.display()

-1/4*((2*bessel_J(2, k*r)*bessel_J(1, k*r) - bessel_J(3, kr)bessel_J(0, kr) k*r)*bessel_J(0, k*r) + bessel_J(1, kr)bessel_J(0, kr))k^3r^2 k*r)*bessel_J(0, k*r))*k^3*r^2 - 2(2bessel_J(1, kr)^2 2*(2*bessel_J(1, k*r)^2 - bessel_J(2, kr)k*r)*bessel_J(0, k*r) + bessel_J(0, kr) + bessel_J(0, kr)^2)k^2r + 4kbessel_J(1, kr)bessel_J(0, kr))z/(r^2bessel_J(0, kr))
k*r)^2)*k^2*r + 4*k*bessel_J(1, k*r)*bessel_J(0, k*r))*z/(r^2*bessel_J(0, k*r))