# I Transformation of covariant vector components

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1. Jul 29, 2017

### saadhusayn

Riley Hobson and Bence define covariant and contravariant bases in the following fashion for a position vector $$\textbf{r}(u_1, u_2, u_3)$$:

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

2. Jul 29, 2017

### Orodruin

Staff Emeritus
Apply the chain rule to the definition of the basis.

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