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## Main Question or Discussion Point

Hi! I´m trying to get an intuition for these concepts and was just playing at home.

My thought was to start with a 2-Dimensional ON-coordinatesystem, the xy-plane and do the following:

1. Study the vector with the coordinate (1,1) in this system. Now its cov. and con. coordinates is of course in the start the same.

2. Start by imagine that we decrease the angle between the positive part och y and x axis. This means that we get a oblique system and we have a diffrence between cov. and con.

3. What should we see? Well, Take the covariant coordinates x = r cos (v) etc.. and differentiate them with respect to the angle, dv.

4. Do the same thing for the contravariant coordinates (here I tried to do it by expressing them through the elements of the transformation matrix). If the length is invariant and can be expressed by x_i x^i. We should see that the derivatives cancel each other so that:

x_i * (d x^i / dv) = x^i * (d x_i / dv) since (d [x^i x_i] / dv) = 0.

Now to the questions!

Do you see any wrong with my arguments here?

Is there any easier way to express the contravariant coordinates through the cartesian x and y?

Can you show that x_i x^i is invariant when we decrease the angle v in some much easier way?

My thought was to start with a 2-Dimensional ON-coordinatesystem, the xy-plane and do the following:

1. Study the vector with the coordinate (1,1) in this system. Now its cov. and con. coordinates is of course in the start the same.

2. Start by imagine that we decrease the angle between the positive part och y and x axis. This means that we get a oblique system and we have a diffrence between cov. and con.

3. What should we see? Well, Take the covariant coordinates x = r cos (v) etc.. and differentiate them with respect to the angle, dv.

4. Do the same thing for the contravariant coordinates (here I tried to do it by expressing them through the elements of the transformation matrix). If the length is invariant and can be expressed by x_i x^i. We should see that the derivatives cancel each other so that:

x_i * (d x^i / dv) = x^i * (d x_i / dv) since (d [x^i x_i] / dv) = 0.

Now to the questions!

Do you see any wrong with my arguments here?

Is there any easier way to express the contravariant coordinates through the cartesian x and y?

Can you show that x_i x^i is invariant when we decrease the angle v in some much easier way?

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