My book gives me this definition for contravariant vector:
Let an n-tuple of real numbers $a^1,a^2, \dots, a^n$ be associated with a point P of an n-dimensional Riemannian space with coordinates $u^1,u^2, \dots, u^n$. Furthermore let there be associated with P an n-tuple of real number $b^1,b^2, \dots, b^n$ with respect to any coordinate system $v^1,v^2, \dots, v^n$ which can be obtained from the coordinates $u^{\alpha}$ by an allowable transformation. If these numbers satisfy the relations:
$$ b^{\alpha} = \frac{\partial v^{\alpha}}{\partial u^{\beta}} a^{\beta} \qquad \alpha = 1, \dots, n $$
$$ a^{\alpha} = \frac{\partial u^{\alpha}}{\partial v^{\beta}} b^{\beta} \qquad \alpha = 1, \dots, n $$
We say that a contravariant vector is given.
Where the summation convention is intended. Now, because by ''allowable transformation'' the book intend that the function $u^{\alpha}(v^\beta)$ is of class $r \geqslant 1$ and the inverse transformation $v^{\alpha}(u^{\beta})$ exists and is of the same class, I want to carry the following example:
I want to prove by definition that, respect to cartesian coordinates the vector with coordinates $(1,1)$ is contravariant respect to a transformation in polar coordinates.
The transformation in polar coordinates is $C^{\infty}$ so is by definition an allowable transformation. I will then have :
$$ a^{\beta} = (1,1) \qquad b^{\alpha} = (\sqrt{2}, \frac{\pi}{4}) $$
And:
$$ \begin{cases} u^1 = v^1\cos v^2 \\ u^2 = v^1\sin v^2 \end{cases} \qquad \begin{cases} v^1 = \sqrt{(u^1)^2 + (u^2)^2}\\ v^2 = \tan^{-1} \left( \frac{u^2}{u^1} \right) \end{cases} $$
But, if I plug these information in the contravariant definition, for instance:
$$ a^{2} = \frac{\partial u^{2}}{\partial v^{\beta}} b^{\beta} \qquad \Rightarrow \qquad 1 = \sqrt{2}\sin v^2 + \frac{\pi}{4}v^1\cos v^2 $$
Which is false. So, where is my error ?
P.S. I already proved that for linear transformations, or rotations, the definition holds. My question is why doesnt'hold for a common transformation like this, which seems to have all the requisites.
You're getting confused between the coordinates themselves (non-linear, describe points) and the associated frames (linear, describe vectors). In non-linear coordinates (and necessarily in non-flat Riemannian spaces) we need to make the distinction between points and vectors.
When you convert $(1,1) \to (\sqrt 2, \pi/4)$, you're talking expressing a point by its $(r,\theta)$ coordinates rather than by its $(x,y)$ ones. If $b$ is meant to be a vector instead, then the components $b^\alpha$ are the numbers such that $b = \sum_\alpha b^\alpha e_\alpha$ for $e_\alpha = \partial {\bf x} / \partial v^\alpha$ the coordinate frame associated to $v$.