I am trying to understand the Wikipedia article about covariance and contravariance of vectors. It says the following:
A contravariant vector has components that "transform as the coordinates do" under changes of coordinates (and so inversely to the transformation of the reference axes), including rotation and dilation.
That seems intuitive enough. For example, if I have a set of cartesian axes in the plane and obtain a new set by rotating them an angle $\theta$ about the origin, then, in the new coordinates, it would seem as though all points in the plane have rotated the opposite way.
By contrast, a covariant vector has components that change oppositely to the coordinates or, equivalently, transform like the reference axes. For instance, the components of the gradient vector of a function $$\nabla f = \frac{\partial f}{\partial x^1}\hat x^1 + \frac{\partial f}{\partial x^2}\hat x^2 + \frac{\partial f}{\partial x^3}\hat x^3$$ transform like the reference axes themselves.
This part I cannot quite visualize. Let's take the simple example of a topographic map, with $x$, $y$ describing eastness and northness, and $f(x,y)$ describing elevation. We are standing at the origin, on the slope of a hill that rises to our north, so, say $\nabla f = (0,1)$ here. Now we define new (primed) coordinate axes rotated by $\theta = \pi/2$, so that $x'$, $y'$ denote northness and westness, respectively. Obviously we still have the hill to our north, so $(\nabla f)' = (1,0)$, where the prime denotes representation in the primed coordinate basis. Hence it seems like $\nabla f$ "transforms as the coordinates do", contradicting the quoted statement from Wikipedia.
Is the statement from Wikipedia misleading, or am I making a mistake?
