Orthogonal coordinate system

Hi friends,

Show that the following co-ordinate systems are orthogonal and find the corresponding $\left | \frac{\delta r}{\delta u} \right |, \left | \frac{\delta r}{\delta v} \right |, \left | \frac{\delta r}{\delta w} \right |$.

ii. Prolate spheroidal co-ordinates:

$x = a.sinhu.sinv.cosw, y = a.sinhu.sinv.sinw z = a.coshu.cosv$

with $a$ constant. (included full stops to facilitate reading)


I had no problems with part i (parabolic coordinates). I'm unsure how to partially differentiate these x, y and z terms. the sinh and cosh terms have confused me.


Thanks !

coshu differentiates to sinhu
sinhu differentiates to coshu

Henceforth

$\left | \frac{\delta r}{\delta u} \right | = (a.coshu.sinv.cosw,a.coshu.sinv.sinw,-a.sinhu.cosv)$

$[br] \left | \frac{\delta r}{\delta v} \right | = (a.sinhu.cosv.cosw,a.sinhu.cosv.sinw,-a.coshu.sinv)$

and

$\left | \frac{\delta r}{\delta u} \right | \cdot \left | \frac{\delta r}{\delta v} \right |$

equals

$a^2.sinhu.coshu.sinv.cosv.cos^2w+a^2.sinu.coshu.sinv.cosv.sin^2w+a^2.sinhu.coshu.cosv.sinv$

which should equal zero because the system is orthogonal. Unless i've done something wrong. How could i show this to equal zero?

I do not remember pro late.

Are the parametrics correct for the system?

$\left | \frac{\delta r}{\delta u} \right | \cdot \left | \frac{\delta r}{\delta v} \right |$

equals

$a^2.sinhu.coshu.sinv.cosv.cos^2w+a^2.sinu.coshu.sinv.cosv.sin^2w+a^2.sinhu.coshu.cosv.sinv$

There was a typo in the first post.


$x = a.sinhu.sinv.cosw$

$y = a.sinhu.sinv.sinw$

$z = a.coshu.cosv$

There was a typo in the first post.


$x = a.sinhu.sinv.cosw$

$y = a.sinhu.sinv.sinw$

$z = a.coshu.cosv$

it works

factorize out of the first component

then cos²w +sin²w and then you get exactly the same as the third but there is a minus in the third component so it dots to zero

Check your sign for the last component, remember, derivative of cosh u is sinh u, not -sinh u.

~snip~

Note the first two terms combine nicely. After doing that and fixing the sign error you should be done.

Thanks a lot !