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The result of t[e_uv,:] cannot be the same as t[:] with a mere substitution x=(u+v)/2 and y=(u-v)/2 because it gives the components of t in a different vector frame: the frame is e_uv for t[e_uv,:], whereas it is the default frame on M for t[:], which is e_xy. In other words, the components returned by t[e_uv,:] are those of the following expansion:

sage: t.display(e_uv)
t = (1/2*u + 1/2) d/du*du + (1/2*u - 1/2) d/du*dv + (1/2*v + 1/2) d/dv*du + (1/2*v - 1/2) d/dv*dv

whereas the components returned by t[:] are those of an expansion on a different basis:

sage: t.display()
t = x d/dx*dx + d/dx*dy + y d/dy*dx

Same thing for the vector w:

sage: w[:]
[-y, x]

agrees with

sage: w.display()
w = -y d/dx + x d/dy

while

sage: w[e_uv,:]
[v, -u]

agrees with

sage: w.display(e_uv)
w = v d/du - u d/dv