Groups | Search | Server Info | Keyboard shortcuts | Login | Register [http] [https] [nntp] [nntps]

Groups > rec.puzzles > #28038

Re: Four more "oldies"

Show all headers | View raw

David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote or quoted:
>On Mon, 24 Aug 2026 07:40:44 -0000 (UTC), David Entwistle wrote:
>>That's an interesting question.
>AI produces some astonishing gibberish in response to this question...

  Might be a bit spoilerish:

  AI was ok here. It explained to me that when one gets higher up to
  the north pole, the same vertical increment gives a surface increment
  that is larger because the surface inclination has decreased.
  But on the other hand, the total surface gets smaller because its
  circumference is reduced. And these two effects cancel, so only one
  sine effect remains. So, this was the approach of Archimedes.
  (That's what the chatbot explained to me.)

  Now, I always wanted to better understand just what a "surface
  element" is. So here are my thoughts:

  The boring general approach of today's calculus uses surface elements.
  A surface element is something like the square with the corners
  (0, 0), (0, 1), (1, 0), and (1, 1). The surface of this square is 1.
  So if we call the height dy and the width dx, it's dx ^ dy = 1,
  where the product "^" means "the surface of a rectangle with these
  sides".

  Now, assume a new y coordinate y' = 2 y. The new coordinates of the
  square are now (0, 0)', (0, 2)', (1, 0)', and (1, 2)'. Now, dx=1, but
  dy'=2, so the product is 2. But the surface has not increased, just
  the coordinates have changed, so this product does not give the real
  surface anymore. I call such a product the "formal surface", because
  it "formally" gives us the surfaces in the new coordinate system
  "formally" applying the rule "width * height" for a rectangle, but
  it is not really the surface because it contains a coordinate effect.
  So I also call this formal surface the /coordinate surface/.

  On the other hand y = y' / 2, and dy = dy' / 2, so the same surface 
  element dx ^ dy is now written dx ^( dy' / 2 )= (1/2) dx ^ dy'.

  Just summing the surface of the area with the coordinates (0, 0)',
  (0, 2)', (1, 0)', and (1, 2)' gives twice the surfaces but the new 
  surface element transforms this back to the surface area in the 
  original coordinates muliplying it by (1/2), and so we get the 
  same area as before.

  The first coordinate system is distinguished: When it is used, the
  surface of a rectangle is the product of its width and height with 
  no correction needed, because it has the special surface element 
  dx ^ dy = 1 dx ^ dy with "1" as the scaling factor, and scaling by
  one is no scaling at all. I call such a coordinate system the 
  "reference coordinate system". The surface of a rectangle in the
  reference coordinate system is the "real surface" of a rectangle,
  not just its formal surface.

  So, to measure the surface in any coordinate system, we can take
  two steps:
  - get the formal surface in that coordinate system
  - use the surface element to convert that formal surface into the
    real surface.

  In an R^3, a Cartesian coordinate system with the orthonormal basis
  vectors, each of length 1 (think, "one meter") is our reference
  coordinate system. To get the real surface of any rectangle in this
  system, we can just multiply its coordinate-width by its coordinate-
  height.

  But for a sphere embedded in an R^3 we prefer /spherical coordinates/.
  However, as the spherical coordinates are not the reference coordi-
  nates, to get the real surface of any object, we need to multiply
  by an appropriate surface element that converts the spherical 
  coordinate surface back into a real surface.

  When t is the polar angle/colatitude, p is the azimuthal angle and
  r is the radius, the surface element for spherical coordinates is

dA = r^2 sin t dt ^ dp.

  Qualitatively this makes sense, because the same dp corresponds
  to a larger real surface when r is larger. So to convert the 
  coordinate surface of spherical coordinates to a real surface,
  we need to multiply them with this surface element.

  The area of any surface of the sphere that is delimited by
  ranges [p0, p1] and [t0, t1] of p and t values can then be
  calculated as the surface integral

p1     t1
 /     /
 |     |   r^2 sin t dt ^ dp,
 /     /
p=p0  t=t0

  where we use the surface element "r^2 sin t dt ^ dp" to correct
  the effect of using the spherical coordinates, so as to get
  the real surface of this area.

  A "line element" does something similar, but for a length instead
  of an area.

Back to rec.puzzles | Previous | Next — Previous in thread | Next in thread | Find similar | Unroll thread

Thread

Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-23 17:44 +0000
  Re: Four more "oldies" David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-08-24 07:40 +0000
    Re: Four more "oldies" David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> - 2026-08-25 07:27 +0000
      Re: Four more "oldies" ram@zedat.fu-berlin.de (Stefan Ram) - 2026-08-25 12:33 +0000
  Re: Four more "oldies" richard@cogsci.ed.ac.uk (Richard Tobin) - 2026-08-24 09:45 +0000
    Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-24 10:33 +0000
    Re: Four more "oldies" Phil Carmody <pc+usenet@asdf.org> - 2026-08-27 22:45 +0300
  Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-24 16:25 +0100
    Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-24 17:04 +0000
      Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-25 00:41 +0100
      Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-25 01:08 +0100
        Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-26 07:08 +0000
          Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-26 23:18 +0100
            Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-27 00:48 +0100
            Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-27 10:42 +0000
              Re: Four more "oldies" Mike Terry <news.dead.person.stones@darjeeling.plus.com> - 2026-08-28 04:55 +0100
  Re: Four more "oldies" Charlie Roberts <croberts@gmail.com> - 2026-08-24 13:27 -0400
  Re: Four more "oldies" James Dow Allen <user4353@newsgrouper.org.invalid> - 2026-08-29 00:04 +0000

csiph-web