Re: Geometry with differential.
- From: David C. Ullrich <dullrich@xxxxxxxxxxx>
- Date: Thu, 10 Jul 2008 05:19:24 -0500
On Thu, 10 Jul 2008 01:34:48 -0700 (PDT), mina_world@xxxxxxxxxxx
wrote:
Hello teacher~
Curve a(t) = ( sin(3t).cos(t) , sin(3t).sin(t) , 0 )
I want to show that a(t) is a regular curve.
Namely, a'(t) =/= 0 for all t in R.
a'(t) = ( 3.cos(3t).cos(t) - sin(3t).sin(t) , 3.cos(3t).sin(t) +
sin(3t).cos(t) , 0 )
Can you show it without computer ?
This can't be hard. First let's throw away the 0 at
the end and say a(t) = ( sin(3t).cos(t) , sin(3t).sin(t)).
Now things are simpler if we look at a(t) as a
complex number instead of an ordered pair
of reals (x + iy = (x,y)):
a(t) = sin(3t) exp(it).
So
a'(t) = (3 cos(3t) + i sin(3t)) exp(it).
So a'(t) = 0 implies cos(3t) = sin(3t) = 0...
Namely,
For each t in R,
[3.cos(3t).cos(t) - sin(3t).sin(t)] =/= 0
or [3.cos(3t).sin(t) + sin(3t).cos(t)] =/= 0.
For reference,
sin(3t) = 3.sin(t) - 4.{sin(t)}^3
cos(3t) = 4.{cos(t)}^3 - 3.cos(t)
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.
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