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I Transformation of covariant vector components

  1. Jul 29, 2017 #1
    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. jcsd
  3. Jul 29, 2017 #2

    Orodruin

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    Apply the chain rule to the definition of the basis.
     
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