Hi everybody, I hope you are doing well.
I am trying to use sympy to make and transform unitary vectors from
spherical coordinates to cartesian coordinates but I have really not known
how to do that.
As you can see in the code below; I would like to be able to transform
e_r,e_theta, e_psi into its cartesian form.
For instance from *e_r* to *(**x/sqrt(x**2+y**2+z**2),
y/sqrt(x**2+y**2+z**2), z/sqrt(x**2+y**2+z**2)**)* (its cartesian form).
Thanks for your time.
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import sympy as sy
#Variables of the spherical coordinates
r=sy.Symbol('r', real=True, positive=True)
theta=sy.Symbol('theta')
psi=sy.Symbol('psi')
#Relationship Cartesian and spherical variables
x = r*sy.sin(theta)*sy.cos(psi)
y = r*sy.sin(theta)*sy.sin(psi)
z = r*sy.cos(theta)
#Derivatives
dx_dr=sy.diff(x,r)
dy_dr=sy.diff(y,r)
dz_dr=sy.diff(z,r)
dx_dtheta=sy.diff(x,theta)
dy_dtheta=sy.diff(y,theta)
dz_dtheta=sy.diff(z,theta)
dx_dpsi=sy.diff(x,psi)
dy_dpsi=sy.diff(y,psi)
dz_dpsi=sy.diff(z,psi)
#Norm
n_r=sy.sqrt(dx_dr**2+dy_dr**2+dz_dr**2)
n_theta=sy.sqrt(dx_dtheta**2+dy_dtheta**2+dz_dtheta**2)
n_psi=sy.sqrt(dx_dpsi**2+dy_dpsi**2+dz_dpsi**2)
#Scale factors
h1=sy.simplify(n_r) #h1=1
h2=sy.simplify(n_theta) #h2=r
h3=sy.refine(sy.simplify(n_psi), sy.Q.positive(sy.sin(theta)))
#h3=r*sin(theta)
#Unit vectors
def simplify(fe,d1,d2,d3):
'''
fe=Scale factors
d#=Respective Derivatives
'''
e1=sy.simplify((1/fe)*d1)
e2=sy.simplify((1/fe)*d2)
e3=sy.simplify((1/fe)*d3)
return (e1,e2,e3)
e_r=*simplify*(h1,dx_dr,dy_dr,dz_dr)
#(sin(theta)*cos(psi), sin(psi)*sin(theta), cos(theta))
e_theta=*simplify*(h2,dx_dtheta,dy_dtheta,dz_dtheta)
#(cos(psi)*cos(theta), sin(psi)*cos(theta), -sin(theta))
e_psi=*simplify*(h3,dx_dpsi,dy_dpsi,dz_dpsi)
#(-sin(psi), cos(psi), 0)
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