Riley Hobson and Bence define covariant and contravariant bases in the following fashion for a position vector $$\textbf{r}(u_1, u_2, u_3)$$:(adsbygoogle = window.adsbygoogle || []).push({});

$$\textbf{e}_i = \frac{\partial \textbf{r}}{\partial u^{i}} $$

And

$$ \textbf{e}^i = \nabla u^{i} $$

In the primed co-ordinate system the equations become

$$\textbf{e}^{'}_{i} = \frac{\partial \textbf{r}}{\partial u^{'i}} $$

And

$$ \textbf{e}^{'i} = \nabla u^{'i} $$

From the chain rule we have that

$$\frac{\partial u^j}{\partial x} = \frac{\partial u^j}{\partial u^{'i}} \frac{\partial u^{'i}}{\partial x}$$

The next step (which I do not understand) is this:

$$ \textbf{e}^j = \frac{\partial u^j}{\partial u^{'i}} \textbf{e}^{'i}$$

How does this last step follow from the previous one? Thank you.

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# I Transformation of covariant vector components

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