Bug in math for surface computations

[unalmis]

Let $R$ be a position vector and $x^i$ be the coordinates of a curvilinear system and $y^j$ those of an orthonormal system.
Then $\hat{y}^j$ can be defined unambiguously

$$\hat{y}^j = \frac{\partial R}{\partial y^j} / \Vert \frac{\partial R}{\partial y^j} \Vert $$

Now, the relation below does not follow (hence the $\neq$)

$$\frac{\partial R}{\partial x^i} \cdot \hat{y}^j \neq \frac{\partial y^j}{\partial x^i}$$

However, DESC assumed it does for all surface computations. (Note that a sufficient condition for the relation to hold is if the basis vectors associated with $y^j$ for all $j$ are orthonormal and constant, which is not true in a cylindrical system.)

After fixing this, here are some of the quantities that significantly differ once corrected.

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.geometry.surface.FourierRZToroidalSurface. Name: phi_z.
Mismatched elements: 225 / 225 (100%)
Max absolute difference: 10.2
Max relative difference: 0.91071429
 x: array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,...
 y: array([11.2     , 11.113545, 10.869131, 10.509017, 10.095472,  9.7     ,
        9.390983,  9.221852,  9.221852,  9.390983,  9.7     , 10.095472,
       10.509017, 10.869131, 11.113545, 11.182709, 11.096255, 10.85184 ,...

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.geometry.surface.FourierRZToroidalSurface. Name: e_phi|r,t.
Mismatched elements: 660 / 675 (97.8%)
Max absolute difference: 10.2
Max relative difference: 10.2
 x: array([[ 0.      , 11.2     , -0.2     ],
       [ 0.      , 11.113545, -0.2     ],
       [ 0.      , 10.869131, -0.2     ],...
 y: array([[ 0.      ,  1.      , -0.017857],
       [ 0.      ,  1.      , -0.017996],
       [ 0.      ,  1.      , -0.018401],...

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.geometry.surface.FourierRZToroidalSurface. Name: e_phi.
Mismatched elements: 660 / 675 (97.8%)
Max absolute difference: 10.2
Max relative difference: 10.2
 x: array([[ 0.      , 11.2     , -0.2     ],
       [ 0.      , 11.113545, -0.2     ],
       [ 0.      , 10.869131, -0.2     ],...
 y: array([[ 0.      ,  1.      , -0.017857],
       [ 0.      ,  1.      , -0.017996],
       [ 0.      ,  1.      , -0.018401],...

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.geometry.surface.ZernikeRZToroidalSection. Name: phi_z.
Mismatched elements: 120 / 120 (100%)
Max absolute difference: 1.
Max relative difference: inf
 x: array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,...
 y: array([0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
       0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
       0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,...

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.geometry.surface.ZernikeRZToroidalSection. Name: e_phi|r,t.
x and y nan location mismatch:
 x: array([[0., 0., 0.],
       [0., 0., 0.],
       [0., 0., 0.],...
 y: array([[nan, nan, nan],
       [nan, nan, nan],
       [nan, nan, nan],...

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.geometry.surface.ZernikeRZToroidalSection. Name: e_theta|r,p.
x and y nan location mismatch:
 x: array([[ 0.,  0.,  0.],
       [ 0.,  0.,  0.],
       [ 0.,  0.,  0.],...
 y: array([[nan, nan, nan],
       [nan, nan, nan],
       [nan, nan, nan],...

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.geometry.surface.ZernikeRZToroidalSection. Name: e_phi.
x and y nan location mismatch:
 x: array([[0., 0., 0.],
       [0., 0., 0.],
       [0., 0., 0.],...
 y: array([[nan, nan, nan],
       [nan, nan, nan],
       [nan, nan, nan],...

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.magnetic_fields._current_potential.CurrentPotentialField. Name: phi_z.
Mismatched elements: 225 / 225 (100%)
Max absolute difference: 10.2
Max relative difference: 0.91071429
 x: array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,...
 y: array([11.2     , 11.113545, 10.869131, 10.509017, 10.095472,  9.7     ,
        9.390983,  9.221852,  9.221852,  9.390983,  9.7     , 10.095472,
       10.509017, 10.869131, 11.113545, 11.182709, 11.096255, 10.85184 ,...

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.magnetic_fields._current_potential.CurrentPotentialField. Name: e_phi|r,t.
Mismatched elements: 660 / 675 (97.8%)
Max absolute difference: 10.2
Max relative difference: 10.2
 x: array([[ 0.      , 11.2     , -0.2     ],
       [ 0.      , 11.113545, -0.2     ],
       [ 0.      , 10.869131, -0.2     ],...
 y: array([[ 0.      ,  1.      , -0.017857],
       [ 0.      ,  1.      , -0.017996],
       [ 0.      ,  1.      , -0.018401],...

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.magnetic_fields._current_potential.CurrentPotentialField. Name: e_phi.
Mismatched elements: 660 / 675 (97.8%)
Max absolute difference: 10.2
Max relative difference: 10.2
 x: array([[ 0.      , 11.2     , -0.2     ],
       [ 0.      , 11.113545, -0.2     ],
       [ 0.      , 10.869131, -0.2     ],...
 y: array([[ 0.      ,  1.      , -0.017857],
       [ 0.      ,  1.      , -0.017996],
       [ 0.      ,  1.      , -0.018401],...

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.magnetic_fields._current_potential.FourierCurrentPotentialField. Name: phi_z.
Mismatched elements: 225 / 225 (100%)
Max absolute difference: 10.2
Max relative difference: 0.91071429
 x: array([1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,
       1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1., 1.,...
 y: array([11.2     , 11.113545, 10.869131, 10.509017, 10.095472,  9.7     ,
        9.390983,  9.221852,  9.221852,  9.390983,  9.7     , 10.095472,
       10.509017, 10.869131, 11.113545, 11.182709, 11.096255, 10.85184 ,...

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.magnetic_fields._current_potential.FourierCurrentPotentialField. Name: e_phi|r,t.
Mismatched elements: 660 / 675 (97.8%)
Max absolute difference: 10.2
Max relative difference: 10.2
 x: array([[ 0.      , 11.2     , -0.2     ],
       [ 0.      , 11.113545, -0.2     ],
       [ 0.      , 10.869131, -0.2     ],...
 y: array([[ 0.      ,  1.      , -0.017857],
       [ 0.      ,  1.      , -0.017996],
       [ 0.      ,  1.      , -0.018401],...

Not equal to tolerance rtol=1e-10, atol=1e-10
Parameterization: desc.magnetic_fields._current_potential.FourierCurrentPotentialField. Name: e_phi.
Mismatched elements: 660 / 675 (97.8%)
Max absolute difference: 10.2
Max relative difference: 10.2
 x: array([[ 0.      , 11.2     , -0.2     ],
       [ 0.      , 11.113545, -0.2     ],
       [ 0.      , 10.869131, -0.2     ],...
 y: array([[ 0.      ,  1.      , -0.017857],
       [ 0.      ,  1.      , -0.017996],
       [ 0.      ,  1.      , -0.018401],...

[dpanici]

phi_z it should be just 1 right? Are the x arrays the corrected (on your PR) values and the y array the current values in master? EDIT: ah I see that they are, ok good catch