[Python] Sympy vector module: Change of vector field coordinate system with CoordSys3D

I want a python script to express a given 2D/3D (cartesian) vector field in polar/spherical coordinates, including transformed basis vectors. That is, transform the vector field v = vx i + vy j (+ vz k) according to these formulas:

• x = r cos(theta)
• y = r sin(theta)
• i = cos(theta) i_r - sin(theta) i_theta
• j = sin(theta) i_r + cos(theta) i_theta

As far as I can tell from documentation, Sympy's vector module has functionality to do this with CoordSys3D, but I cannot make it work.

As a workaround, I have tried the following in the 2D:

C= vec.CoordSys3D('C')
P= vec.CoordSys3D('P')

def polar(v):
substitutions = [(x, r*sp.cos(theta)) ,(y, r * sp.sin(theta)), (C.i, sp.cos(theta)*P.i - sp.sin(theta)*P.j), (C.j, sp.sin(theta)*P.i + sp.cos(theta)*P.j)]
v_pol = sp.simplify(v.subs(substitutions))
return v_pol


This doesn't work, and I'm not sure I understand why. Feel like it ought to be a fairly simple problem to solve.

Any tips appreciated.

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