There is currently some work in progress aiming to implement such
transformations:
https://github.com/sympy/sympy/pull/17507
Kalevi Suominen
On Sunday, September 1, 2019 at 8:32:56 PM UTC+3, Sebastián Pérez wrote:
>
> 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.
>
>
> ================================================================================================================================================================
> 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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