1 | initial version |

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
```

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