Trandformed Field

[Sebastián Pérez]

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)

[Kalevi Suominen]

There is currently some work in progress aiming to implement such transformations:

https://github.com/sympy/sympy/pull/17507

Kalevi Suominen