Re: Thomas Covenant series

on 16/05/2006 11:46 am, Jonathan L Cunningham at spam@xxxxxxxxxxxxxxxxxxxx
wrote:

Tim S <Tim@xxxxxxxxxxxxxxxxxxxxxxxx> wrote:

on 15/05/2006 11:39 am, Jonathan L Cunningham at spam@xxxxxxxxxxxxxxxxxxxx
wrote:

Tim S <Tim@xxxxxxxxxxxxxxxxxxxxxxxx> wrote:

<some blather of little value>


Heh! My reaction at this point, as my eyes glazed over, was "you're
asking *me* to visualise?" In three dimensions at that!

That was my reaction too ...

"Eyes glazed over" is a strange metaphor. I presume it's derived
from the expression "glassy eyed".

I presume they both derive the same underlying image, of eyes as lifeless
and unmoving as little glass balls.

(Or is that too visual for you?) (That's a semi-serious question.)


How _do_ you intuit integration and differentiation?

I'm glad you didn't put the emphasis on "you". :-)

More introspection: I think of a differential coefficient as a
ratio, I think of integration as (a kind of) sum.

So, perhaps, the intuitive problem boils down to explaining why
the inverse of a ratio is a sum: it feels like: "the inverse of
two thirds is five: inverse(2/3) = 2+3"

That's very interesting.

I don't really expect to be able to help you with this, I'm just curious
about how you think (and shamelessly milking you for strange new ways to
think about things) but just in case...

Notice that before we add up, we multiply the each of the fractions dy/dx by
their denominator dx : Integral (f(x) dx) matching Sum(dy/dx . dx)

<Tim writhes in agony at the lies he is having to tell>

So we're just adding up the numerators. The ratio in differentiation is
chopping y up into pieces, each of size dx, and integration is putting them
back together again.

<Tim faints from pain>

If you don't mind indulging me a little more, how do you intuit slope, and
the relation between slope and differentiation? Likewise with integration
and area?


Metaphors about onion skins, oysters, bits of grit and pearls
become appropriate at this point.

Onion skins?

*If* (and it's a very weak
conjecture) the previous paragraph is my inner five-year-old
complaining[*] he doesn't understand calculus then the solution
is to redefine differentiation in terms he can understand.

Hence everybody's babble about rates, speeds, graphs and increments, just in
case any of those tickles your intuition.


Jonathan
[*] Not that anyone ever tried to teach me calculus at five, but
that patterns of thought persist, and get overlaid. More metaphors
about pearls, plus new metaphors about sedimentary rocks and
fossils become appropriate.

I tend to think of it as tools in a toolbox. And machine tools in a
workshop.

Tim

.

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