Re: vector fields and coordinate transforms

[Wikipedia on vector fields]
[...]
(*) V_y := @x/@y V

is the expression for the same vector field in the new
coordinates.
[...]
Reference:
[1] <http://en.wikipedia.org/wiki/Vector_field> (accessed 29.11.06,
10:06 CET).

[Pouya D. Tafti]
[...]
I naively think of them as first order
differential operators; i.e. something such:

V = sum_j Vx_j @_xj

where @_xj represents partial differentiation w.r.t. xj, and Vx_j's
are (scalar-valued) functions. We have

Vx_j = V xj.

So, given another coordinate system y, I would write the i-th
component of V in the new coordinate system as

Vy_i = V yi = sum_j Vx_j @yi/@xj

which is in conflict with (*), as I get @y/@x instead of @x/@y.
Where is the mistake?

[lapiz <pec01170@xxxxxxxxx>]
We have (@_x1,@_x2, ... ,@_xn) (@y/@x) = (@_y1,@_y2, ... ,@_yn) .

Sorry, but I don't think I get it. Why is this not

(@_x1,...,@_xn) (@x/@y) = (@_y1,...,@_yn)

or, equivalently,

(@./@x) (@x/@y) = (@./@y) ?

This latter equation results in the same conflict.

[lapiz <pec01170@xxxxxxxxx>]
So, "V_y := @x/@y V" means a representation of V_y by (@_x1,@_x2, ...
,@_xn).
I think your conflict is Vy_i = V yi.
"Vy_i" (yi component of V) is not "V yi" (V orientation differential of
yi).

Why not? What is wrong with the following derivation?

V yi = (sum_j Vy_j @_yj) yi = Vy_i.

Thanks,
--
Pouya D. Tafti
p dot d dot tafti at ieee dot org
.

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