I'm reading "An Introduction to Tensors for Students of Physics and Engineering" (Kolecki 2002) and am stuck on page 18 (page 22 in the PDF). The author claims that, in a 3-dimensional Euclidean space, with two bases $u, v, w$ and $x, y, z$, we can define a contravariant basis set as $$\mathbf{e^{(1)}} = \frac{\partial{\mathbf{r}}}{\partial{u}},\ \mathbf{e^{(2)}} = \frac{\partial{\mathbf{r}}}{\partial{v}},\ \mathbf{e^{(3)}} = \frac{\partial{\mathbf{r}}}{\partial{w}}$$ where $\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$. Also, we can define a covariant basis set $$ \mathbf{e_{(1)}} = \mathbf{\nabla}u,\ \mathbf{e_{(2)}} = \mathbf{\nabla}v,\ \mathbf{e_{(3)}} = \mathbf{\nabla}w$$
Since these are covariant and contravariant vector sets, by definition we should have $\mathbf{e^{(1)}} \cdot \mathbf{e_{(1)}} = 1$, and the paper states that this is true. But I get a different result than the author when I try to do the proof: $$\begin{align} \mathbf{e^{(1)}} \cdot \mathbf{e_{(1)}} = \frac{\partial{\mathbf{r}}}{\partial{u}} \cdot \mathbf{\nabla}u & = \left( \frac{\partial}{\partial{u}} \left( x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\right) \right) \cdot \left(\left( \frac{\partial}{\partial{x}} \mathbf{i} + \frac{\partial}{\partial{y}} \mathbf{j} + \frac{\partial}{\partial z} \mathbf{k} \right) u \right) \\ & = \left( \frac{\partial{x}}{\partial{u}} \frac{\partial{u}}{\partial{x}} \right) + \left( \frac{\partial{y}}{\partial{u}} \frac{\partial{u}}{\partial{y}}\right) + \left( \frac{\partial{z}}{\partial{u}} \frac{\partial{u}}{\partial{z}}\right) \\ &= 1 + 1 + 1 = 3\end{align}$$
In the paper, the author's first step in the proof is the same as the second line in mine, but he then gets $...=\partial{u}/\partial{u} = 1$ somehow. I don't believe the author is wrong, but I don't see how he gets that result. Where have I made a mistake?