Using polar coordinates with variables $r$ and $\theta$.
Let $\vec{r}$ be the position vector.
Consider $\nabla \theta \cdot \frac{d\vec{r}}{d\theta}$. This is the dot product of the gradient normal to level curves holding $\theta$ constant and the derivative of the position vector with respect to $\theta$.
This expands to $\nabla \theta \cdot \frac{d\vec{r}}{d\theta} = \left < \frac{\partial \theta}{\partial x}, \frac{\partial \theta}{\partial y} \right > \cdot \left < \frac{\partial x}{\partial \theta}, \frac{\partial y}{\partial \theta} \right > = \frac{\partial \theta}{\partial x} \frac{\partial x}{\partial \theta} + \frac{\partial \theta}{\partial y} \frac{\partial y}{\partial \theta}$
This sums to 1, not 2.
Why?