Re: How to define a vector without coordinate system?

I would like to add something to may previous message.
In the literature, particularly related to manifolds there are lots of
papers in which we can read
that it very good that the definition of vector on manifold is
"coordinate free", that the concept of manifold is "coordinate
free" and lots of other similar statements which more or less say
that it is possible do some analytical calculation on "coordinate
free" objects.

Recently I have met several researchers which explicitly claim that we
can do some calculations using vectors without introducing coordinate
systems. I repeat CALCULATIONS not drawing.

I think there is a little misunderstanding in that matter.
If we are using some vectors in calculations, then that vectors have to
be defined in some way.
It is not sufficient to write just the bold letter F and say that F is
a vector.
That is not enough. The vector have to have some meaning.
We can define vector F using several different coordinate systems i.e.
coordinate system basis (contravariant and covariant coordinates) or we
can define vector in a basis which is nor related to any covariant and
contravariant basis vectors.

However I cannot imagine the situation in which we are doing some
calculation on vector F and that vector is described using some
"magic" (sorry for that word) "coordinate free" methodology
without specifying any coordinate system in order to describe the
vector F.

Lets return to the problem of vector differential equations on
manifolds.
We can write for example.

Div(grad(T))=f

In this particular case we have some "coordinate free" differential
operators.
Now we have the following question.
According to my knowledge we can write this equation in this form
because we can express this equation in all possible coordinate
systems.
That equation is "coordinate free" only in that sense.
We cannot go one step ached, i.e. we cannot say that there is something
like Div(grad(T))=f
which have some meaning without any coordinate system.

Lets return to my original question.
We can define vector F using contravariant and covariant coordinates.
We can use also basis which are neither contravariant nor covariant.
However we are always using expression in the form

F=\sum\limits_i F^i e_i

After that some wise man can say:
well the vector F can be defined in all possible coordinate systems
then actually the vector F is coordinate independent. Next man will
call that vector "coordinate free" and neglect all that knowledge.
Finally we can forger about original definition of our vector because
we have lots of coordinate free theorems (for example scalar product of
two perpendicular vectors is equal to zero etc.). Finally we have nice
"coordinate free" theory full of different statements.

Unfortunately after some time somebody have to use vector F in order to
solve some example which is related to physics.
Now we have very basic problem we have to have at least one vector in
order to use all these nice theorems about vectors and vector spaces
(there theorems are originally based on the concept of coordinates but
at this point everybody forget about that).

Then how to do that?
What is the source of vectors?
We cannot buy vectors in a shop :).
In my opinion we have to return to the beginning and use some
coordinate system.
I do not see any other possibility.
The whole "coordinate free" theory which do not use coordinate
systems is just "on top" of that basic problem.

Then I repeat my question again.
Does exist any reasonable way to define vector without coordinate
system?
I think that talking about vectors without coordinate system is
meaningless.

Regards,

Andrzej

.

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