Discussion:
Bliss's Theorem
John Koch
2003-10-11 22:21:20 UTC
Does anyone know of a statement of Bliss's Theorem on a website? If

In case there is more than one "Bliss's Theorem," it's the theorem
that it used to justify the arc length formula for parametric
equations being a Riemann integral, when the derivation doesn't lead
to a Riemann sum.

Thanks,

John
John Koch
2003-10-16 23:31:46 UTC
Anyone?
David C. Ullrich
2003-10-17 10:12:51 UTC
Post by John Koch
Does anyone know of a statement of Bliss's Theorem on a website? If
In case there is more than one "Bliss's Theorem," it's the theorem
that it used to justify the arc length formula for parametric
equations being a Riemann integral, when the derivation doesn't lead
to a Riemann sum.
I'm not at all clear on exactly what that last "when the derivation
doesn't lead to a Riemann sum" means... (is that one of the
hypotheses, or a comment on why the proof is not clear or what?)

If you could give a precise statement of what you want to prove
someone could show you how to prove it. _Is_ the question just
how to prove that the arclength is given by that integral?
(That's what it sounds like the question is, but I'd be surprised
to hear that that fact was someone's theorem... _if_ that's
the question then what are the hypotheses? In particular
are we assuming that the curve is continuously differentiable?)
Post by John Koch
Thanks,
John
************************

David C. Ullrich
John Koch
2003-10-17 20:53:02 UTC

particular. I had just heard that "Bliss's Theorem" was used to
justify some applications of elementary calculus where the value at
the ith subinterval of the partition is chosen conveniently rather
than arbitrarily, as in a Riemann Sum.

Thanks,

John
Post by David C. Ullrich
Post by John Koch
Does anyone know of a statement of Bliss's Theorem on a website? If
In case there is more than one "Bliss's Theorem," it's the theorem
that it used to justify the arc length formula for parametric
equations being a Riemann integral, when the derivation doesn't lead
to a Riemann sum.
I'm not at all clear on exactly what that last "when the derivation
doesn't lead to a Riemann sum" means... (is that one of the
hypotheses, or a comment on why the proof is not clear or what?)
If you could give a precise statement of what you want to prove
someone could show you how to prove it. _Is_ the question just
how to prove that the arclength is given by that integral?
(That's what it sounds like the question is, but I'd be surprised
to hear that that fact was someone's theorem... _if_ that's
the question then what are the hypotheses? In particular
are we assuming that the curve is continuously differentiable?)
Post by John Koch
Thanks,
John
************************
David C. Ullrich
David C. Ullrich
2003-10-17 21:58:40 UTC
Post by John Koch
particular. I had just heard that "Bliss's Theorem" was used to
justify some applications of elementary calculus where the value at
the ith subinterval of the partition is chosen conveniently rather
than arbitrarily, as in a Riemann Sum.
Right. I can't figure out what there is to be "justified" there:
If f is Riemann integrable then it follows from the definition
of the Riemann integral that you _can_ choose those points
conveneniently...
Post by John Koch
Thanks,
John
Post by David C. Ullrich
Post by John Koch
Does anyone know of a statement of Bliss's Theorem on a website? If
In case there is more than one "Bliss's Theorem," it's the theorem
that it used to justify the arc length formula for parametric
equations being a Riemann integral, when the derivation doesn't lead
to a Riemann sum.
I'm not at all clear on exactly what that last "when the derivation
doesn't lead to a Riemann sum" means... (is that one of the
hypotheses, or a comment on why the proof is not clear or what?)
If you could give a precise statement of what you want to prove
someone could show you how to prove it. _Is_ the question just
how to prove that the arclength is given by that integral?
(That's what it sounds like the question is, but I'd be surprised
to hear that that fact was someone's theorem... _if_ that's
the question then what are the hypotheses? In particular
are we assuming that the curve is continuously differentiable?)
Post by John Koch
Thanks,
John
************************
David C. Ullrich
************************

David C. Ullrich