Toroidal Coordinates - Scale factors

[proven echo]

Guys, I need help with this problem. I've already tried to do the derivatives (directly), summarizing the expressions in the sinh and cosh relations, I've tried to do it using complex variables and nothing. I always come across an outcome that I can't get out of.

[dreamy forge]

Can you ask for a helper in your own channel

[blazing pilot]

What's a scaling factor?

[proven echo]

https://mathworld.wolfram.com/ScaleFactor.html

[proven echo]

I didn't know, how do I make my own channel?

[dreamy forge]

You basically already have one

You can @ helpers here yknow

[blazing pilot]

This is your channel

[blazing pilot]

Show how you did the derivatives directly

[proven echo]

One of the ways I tried to do it was like this. but something is missing.

[rain merlin]

you're super close

hold on gimme 2 sec I made a mistake XD

ok I got it

so note that from the last line, if we combine the first and last terms, then we have $\sinh^2{\tau}\sin^2{\sigma}+1 = \cosh^2{\tau}\sin^2{\sigma$

[iron sable]

lgkoo
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[rain merlin]

@proven echo see if that's enough to help you do the rest? Lmk if you need more hints

[proven echo]

I will analyze it here.

[rain merlin]

wait no @proven echo I'm sorry this is absolute bullshit from me XD. It should be $\sinh^2{\tau}\sin^2{\sigma}+ \sin^2{\sigma} = \cosh^2{\tau}\sin^2{\sigma$.

[iron sable]

lgkoo
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[rain merlin]

The right thing to do should be

from the last line, if we write the $\cos^2{\sigma} = 1 - \sin^2{\sigma}$ in the second term, then we have $\sinh^2{\tau}\sin^2{\sigma} + \cosh^2{\tau}(1-\sin^2{\sigma}) - 2\cosh{\tau}\cos{\sigma} +1 = \sinh^2{\tau}\sin^2{\sigma} - \cosh^2{\tau}\sin^2{\sigma} + \cosh^2{\tau} - 2\cosh{\tau}cos{\sigma} +1$

[iron sable]

lgkoo

[rain merlin]

which should now correctly simplify down to $(\cosh{\tau} - \cos{\sigma})^2$ after a bit more trig manipulation and factorisation

[iron sable]

lgkoo

[proven echo]

and now?

[rain merlin]

in this picture, from second to third line, you forgot the $\cosh^2{\tau}$ when you expanded $\cosh^2{\tau}(1-\sin^2{\sigma})$

[iron sable]

lgkoo

[proven echo]

true. I'll put it here.

Perfect! 😁 😁 😁

Now all we need to do is find the other two scaling factors. I'll try here.