# Convertion from and to spherical - cartesian

I googled it, and it says:
$\dot{x}$=$\dot{r}$sinθcos∅ + (rcosθcos∅)$\dot{θ}$ - (rsinθsin∅)$\dot{∅}$
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and so on for $\dot{y}$ & $\dot{z}$

And then they wrote "We will also need the inverse transformation obtained by solving the equations above w.r.t $\dot{r}$, $\dot{θ}$, and $\dot{∅}$

for example they got:
$\dot{r}$=sinθcos∅$\dot{x}$+sinθsin∅$\dot{y}$ + cosθ$\dot{z}$
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and so on, my question how did they deduced the latter from the former equations?

Here's the link if you want to see them clearer:
http://www.physics.sc.edu/~yar/Phys701_2009/homework/hw9_solutions.pdf [Broken]

Just what's the procedure?
Thanks

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tiny-tim
Homework Helper
Hi M. next! … for example they got:
$\dot{r}$=sinθcos∅$\dot{x}$+sinθsin∅$\dot{y}$ + cosθ$\dot{z}$

Just what's the procedure?

It's just algebra (plus standard trigonometric identities) …

for example, the r' component of the RHSs is r'(sin2θcos2∅ + sin2θsin2∅ + cos2θ) = r' Hmm.. I mean how do we get r' from (x', y', z')?

I solved three equations with three unknowns and it worked but after so much effort! Thanks anyways Tim.