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I am not sure the curl of a vector field is well defined on a 2-dimensional manifold. The reference you mention actually deals with vector fields in $\mathbb{R}^3$. In dimension 2, one could define the curl by taking the Hodge dual of the exterior derivative of the 1-form that is associated to the vector field by metric, i.e. define $\mathrm{curl}\ v = *(\mathrm{d}v^\flat)$; this would lead a scalar field (not a vector field!) and I am not sure this is very useful.

I am not sure the curl of a vector field is well defined on a 2-dimensional manifold. The reference you mention actually deals with vector fields in $\mathbb{R}^3$. In dimension 2, one could define the curl by taking the Hodge dual of the exterior derivative of the 1-form that is associated to the vector field by metric, i.e. define $\mathrm{curl}\ v = *(\mathrm{d}v^\flat)$; this would lead a scalar field (not a vector field!) and I am not sure don't know if this is very useful.

def curl(v):
    if dim(v.domain()) == 2:
        g = v.domain().metric()
        return v.down(g).exterior_derivative().hodge_dual(g)
    else:
        return v.curl()

v = E.vector_field(name='v')
v[:] = [-x, y^2]
cv = curl(v)
print(cv)
cv.expr()

Scalar field on the 2-dimensional Riemannian manifold E
0

v[:] = -y , x
curl(v).expr()

2