What Does the Definition of a Contravariant Vector Mean in Tensor Analysis?

[HasuChObe]

So I'm looking at Schaum's outlines for Tensors and the definition of a Contravariant vector is
$$\bar{T}^i=T^r\frac{\partial\bar{x}^i}{\partial x^r}$$
Where $\bar{x}^i$ and $x^r$ denote components of 2 different coordinates (the superscript does not mean 'to the power of') and $T^i$ and $T^r$ are contravariant tensors of order 1 (aka, a vector).

Lets say you have some 2-D vector ${\bf v}$. It can be described as
$${\bf v}=\bar{T}^1\hat{\bar{e}}_1+\bar{T}^2\hat{\bar{e}}_2=T^1\hat{e}_1+T^2 \hat{e}_2$$
The vector ${\bf v}$ is the same length, but the basis for each vector may be different. If the operation from $(\hat{e}_1,\hat{e}_2)\rightarrow(\hat{\bar{e}}_1,\hat{\bar{e}}_2)$ performs elongation, then $(T^1,T^2)\rightarrow(\hat{T}^1,\hat{T}^2)$ will shrink (and vice versa) to preserve the shape of ${\bf v}$. In this case, $(T^1,T^2)$ are said to be contravariant vectors because they grow contrary to the direction that the basis grows in. However, the definition I found in Schaum's outlines seem to say the opposite.

For example, if
$$\bar{T}^i=2,\,T^r=1,\,\frac{\partial\bar{x}^i}{\partial x^r}=2$$
Does that not say that going from unbarred to barred coordinates, the vector components are growing and so is the coordinate system? I must be confusing myself.

 

[TSC]

The last equation is unjustified.

 

[HasuChObe]

Can you elaborate?

 

[Matterwave]

Let's just look at a 1 dimensional example (your example), take $\bar{x}^1=2x^1$ So that $\frac{\partial \bar{x}^1}{\partial x^1}=2$. Then your example says if $T^1=1$ then $\bar{T}^1=2$. Let's just assume the space is flat so things are easy.

But what does $\bar{x}^1=2x^1$ mean? At the point where the old $x^1=1$ the new $\bar{x}^1=2$. So, if you think about it a bit, you will realize that your coordinates have actually shrank since you need a larger number to describe the same distance. In other words, it would be like if you switched from meters to half-meter measurements. Where the old coordinate says 1 meter, your new coordinate says 2 half meters. Therefore, your old vector was 1 meter long, your new vector is 2 half meters long. The length of the vector has remained invariant. In this example, the two vectors both point to a distance 1 meter from the origin!

 

[TSC]

Good explanation!