Convertion from and to spherical - cartesian

1. Jun 9, 2012

M. next

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

Last edited by a moderator: May 6, 2017
2. Jun 9, 2012

tiny-tim

Hi M. next!
It's just algebra (plus standard trigonometric identities) …

for example, the r' component of the RHSs is r'(sin2θcos2∅ + sin2θsin2∅ + cos2θ) = r'

3. Jun 9, 2012

M. next

Hmm.. I mean how do we get r' from (x', y', z')?

4. Jun 9, 2012

M. next

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