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I know that by definition the transition between basis vectors $ e_{a}$ of one coordinate system to basis vectors $e_{a^\prime}$ of another coordinate systems is performed via the next expression: $$e_{a^\prime}=\Lambda^b_{a^\prime}e_b$$ $$\Lambda^b_{a^\prime}=\frac{\partial x^{b}}{\partial x^{a^\prime}}$$ $$e_b=e_a$$ But when I want to find basis vectors of spherical coordinate system via cartesian coordinate system by the definition above I come across the next problem:

The spherical system coordinates expressed as $x=rsin{\theta}cos{\phi}, y=rsin{\theta}sin{\phi}, z=rcos{\theta}$, so I am performing the steps which are shown in definition:$$\partial r=\frac{\partial x}{\partial r}\partial x +\frac{\partial y}{\partial r}\partial y + \frac{\partial z}{\partial r}\partial z $$ But according to my book (Relativity demystified) this is not correct and should be done in vice-versa way namely: $$\partial r=\frac{\partial r}{\partial x}\partial x +\frac{\partial r}{\partial y}\partial y + \frac{\partial r}{\partial z}\partial z $$ But I see no sense in their action and how they got to the correct answer. Is it misprint or my misunderstanding?

This is really a mistake. Relativity Demystified, example 2-5.

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Yes now I am sure that is is a mistake. Second one actually. They already made 2 serious misprints (example 2-4, example 2-5). So, I hope this info will be useful for someone.

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    $\begingroup$ I confim that this is a mistake. $\endgroup$ – Blazej Aug 12 '16 at 7:56

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