[math-fun] Draft of my July 2019 essay
This month I plan to publish a Mathematical Enchantments essay on the 14th, rather than the 17th, so that I can link to it in my third pitch for Christian's Big Internet Math-Off on the 15th. But as always comments are welcome at any time since it's easy to make edits in WordPress. The podcast episode I refer to can be heard at https://kpknudson.com/my-favorite-theorem/2019/7/10/episode-44-james-propp but the essay is designed to stand on its own. Thanks, Jim Propp
Oops, forgot to include the link to the essay! It's http://mathenchant.org/051-draft3.pdf Jim On Thu, Jul 11, 2019 at 1:57 PM James Propp <jamespropp@gmail.com> wrote:
This month I plan to publish a Mathematical Enchantments essay on the 14th, rather than the 17th, so that I can link to it in my third pitch for Christian's Big Internet Math-Off on the 15th. But as always comments are welcome at any time since it's easy to make edits in WordPress.
The podcast episode I refer to can be heard at
https://kpknudson.com/my-favorite-theorem/2019/7/10/episode-44-james-propp
but the essay is designed to stand on its own.
Thanks,
Jim Propp
Thanks for this Jim... I still have nightmares about delta-epsilon proofs, so it’s nice to see something less complicated for once. Compare and contrast with Liouville’s Theorem for entire holomorphic functions could make for a nice follow up. Though, if the game is on calculus in general, then Liouville’s other Theorem on phase volumes—that a Hamiltonian flow leaves volume a constant function, V(t)=V(0)— could raise even more interest. Every physicist encounters the n-dimensional cartesian case in statistical mechanics; however, I have searched and found almost nothing about the 2D case on phase sphere or phase hyperboloid, as opposed to the usual phase plane. (Or am I missing something?) More cheers for calculus! So much fun! —Brad
On Jul 11, 2019, at 12:58 PM, James Propp <jamespropp@gmail.com> wrote:
Oops, forgot to include the link to the essay! It's
http://mathenchant.org/051-draft3.pdf
Jim
On Thu, Jul 11, 2019 at 1:57 PM James Propp <jamespropp@gmail.com> wrote:
This month I plan to publish a Mathematical Enchantments essay on the 14th, rather than the 17th, so that I can link to it in my third pitch for Christian's Big Internet Math-Off on the 15th. But as always comments are welcome at any time since it's easy to make edits in WordPress.
The podcast episode I refer to can be heard at
https://kpknudson.com/my-favorite-theorem/2019/7/10/episode-44-james-propp
but the essay is designed to stand on its own.
Thanks,
Jim Propp
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
You don't seem to (unless I missed it) state the constant value theorem.
From context I assume its
if f'(x)=0 then f is constant but I never saw it stated. bill g. On Thu, Jul 11, 2019 at 3:01 PM <bradklee@gmail.com> wrote:
Thanks for this Jim... I still have nightmares about delta-epsilon proofs, so it’s nice to see something less complicated for once.
Compare and contrast with Liouville’s Theorem for entire holomorphic functions could make for a nice follow up.
Though, if the game is on calculus in general, then Liouville’s other Theorem on phase volumes—that a Hamiltonian flow leaves volume a constant function, V(t)=V(0)— could raise even more interest.
Every physicist encounters the n-dimensional cartesian case in statistical mechanics; however, I have searched and found almost nothing about the 2D case on phase sphere or phase hyperboloid, as opposed to the usual phase plane. (Or am I missing something?)
More cheers for calculus! So much fun!
—Brad
On Jul 11, 2019, at 12:58 PM, James Propp <jamespropp@gmail.com> wrote:
Oops, forgot to include the link to the essay! It's
http://mathenchant.org/051-draft3.pdf
Jim
On Thu, Jul 11, 2019 at 1:57 PM James Propp <jamespropp@gmail.com> wrote:
This month I plan to publish a Mathematical Enchantments essay on the 14th, rather than the 17th, so that I can link to it in my third pitch for Christian's Big Internet Math-Off on the 15th. But as always comments are welcome at any time since it's easy to make edits in WordPress.
The podcast episode I refer to can be heard at
https://kpknudson.com/my-favorite-theorem/2019/7/10/episode-44-james-propp
but the essay is designed to stand on its own.
Thanks,
Jim Propp
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Look in the section "The Celestial Orrery". The second-to-last paragraph in that section begins: Among differential equations, none is simpler than f'(t) = 0, expressing the relation that as the quantity t changes, the quantity f(t) doesn’t change at all; a basic step in solving many problems in math and physics is passing from the assertion that f'(t) = 0 for all t to the conclusion that there exists some c such that f(t) = c for all t. The Constant Value Theorem is what gives us the right to draw this conclusion. The Theorem says that if the function f doesn’t change (that is, if f'(t) = 0 for all t), then f is constant (that is, there exists c such that f(t) = c for all t). ... Tom William Gasarch writes:
You don't seem to (unless I missed it) state the constant value theorem. From context I assume its
if f'(x)=0 then f is constant
but I never saw it stated.
bill g.
On Thu, Jul 11, 2019 at 3:01 PM <bradklee@gmail.com> wrote:
Thanks for this Jim... I still have nightmares about delta-epsilon proofs, so it’s nice to see something less complicated for once.
Compare and contrast with Liouville’s Theorem for entire holomorphic functions could make for a nice follow up.
Though, if the game is on calculus in general, then Liouville’s other Theorem on phase volumes—that a Hamiltonian flow leaves volume a constant function, V(t)=V(0)— could raise even more interest.
Every physicist encounters the n-dimensional cartesian case in statistical mechanics; however, I have searched and found almost nothing about the 2D case on phase sphere or phase hyperboloid, as opposed to the usual phase plane. (Or am I missing something?)
More cheers for calculus! So much fun!
—Brad
On Jul 11, 2019, at 12:58 PM, James Propp <jamespropp@gmail.com> wrote:
Oops, forgot to include the link to the essay! It's
http://mathenchant.org/051-draft3.pdf
Jim
On Thu, Jul 11, 2019 at 1:57 PM James Propp <jamespropp@gmail.com> wrote:
This month I plan to publish a Mathematical Enchantments essay on the 14th, rather than the 17th, so that I can link to it in my third pitch for Christian's Big Internet Math-Off on the 15th. But as always comments are welcome at any time since it's easy to make edits in WordPress.
The podcast episode I refer to can be heard at
https://kpknudson.com/my-favorite-theorem/2019/7/10/episode-44-james-propp
but the essay is designed to stand on its own.
Thanks,
Jim Propp
How about d/dt f(t) = 0 implies d^n/dt^n f(t)=0, n>1, Implies f(t)=f(0)? Then f(dt)= f(0) + f’(0)*dt +(1/2)*f’’(0)*dt^2 + ... = f(0), and by induction on +dt, f(t)=f(0), thus the CFT is true. This type of “proof” shows up in undergrad Quantum, but I don’t know if it would be good enough for math? —Brad
On Jul 11, 2019, at 2:31 PM, William Gasarch <gasarch@cs.umd.edu> wrote:
You don't seem to (unless I missed it) state the constant value theorem. From context I assume its
if f'(x)=0 then f is constant
but I never saw it stated.
bill g.
On Thu, Jul 11, 2019 at 3:01 PM <bradklee@gmail.com> wrote:
Thanks for this Jim... I still have nightmares about delta-epsilon proofs, so it’s nice to see something less complicated for once.
Compare and contrast with Liouville’s Theorem for entire holomorphic functions could make for a nice follow up.
Though, if the game is on calculus in general, then Liouville’s other Theorem on phase volumes—that a Hamiltonian flow leaves volume a constant function, V(t)=V(0)— could raise even more interest.
Every physicist encounters the n-dimensional cartesian case in statistical mechanics; however, I have searched and found almost nothing about the 2D case on phase sphere or phase hyperboloid, as opposed to the usual phase plane. (Or am I missing something?)
More cheers for calculus! So much fun!
—Brad
On Jul 11, 2019, at 12:58 PM, James Propp <jamespropp@gmail.com> wrote:
Oops, forgot to include the link to the essay! It's
http://mathenchant.org/051-draft3.pdf
Jim
On Thu, Jul 11, 2019 at 1:57 PM James Propp <jamespropp@gmail.com> wrote:
This month I plan to publish a Mathematical Enchantments essay on the 14th, rather than the 17th, so that I can link to it in my third pitch for Christian's Big Internet Math-Off on the 15th. But as always comments are welcome at any time since it's easy to make edits in WordPress.
The podcast episode I refer to can be heard at
https://kpknudson.com/my-favorite-theorem/2019/7/10/episode-44-james-propp
but the essay is designed to stand on its own.
Thanks,
Jim Propp
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
-
bradklee＠gmail.com -
James Propp -
Tom Karzes -
William Gasarch