390 Quick
Answers 31 March
Thank
you for accommodating the video last class. It seemed to
work well. There was a nice amount of history discussed at
the conference, and an art historian told me … only in some
regions does Islamic art not portray sentient beings. I will
try to be more careful about that. Isfahan was … one of
those places. So, it fit with al-Khayami's story.
To be more explicit about GREAT Day rehearsals. You need to
schedule by Wednesday 2 April. You need one the next week,
and one the week after. You may do them _before_ this, not
after. If you do not do two rehearsals on this schedule, you
will not earn credit for your presentation.
Drafts
of papers due in one week. The sooner you get to me the
sooner you will get back. I will be giving an exam
on Thursday evening, so not in office hours. The room will
still be available if you wish. I will be available for
email consultation. Here's a radical idea - if you get your
draft done early and submit it by Thursday afternoon, I will
likely have time to read and give full feedback during the
exam. Remember, in general - earlier submission means
earlier feedback.
Today’s
a little different day - I will get ahead and preview some if 8.2
today. I will try to not spoil the stories.
Lecture
Reactions
Do not comment on this - but I think it's been mentioned in the
reading before - originally English didn't have silent letters in
it, everything you see in these old spellings was
pronounced. Pronunciation was different then.
Remember Mercator projection are used for navigating _by
compass_. This is not a straight path, and almost no one
would navigate that way today. They are also, I admit,
conformal, which means they preserve angles locally, and therefore
some ideas of shape. However, they are not the only
conformal maps, and there are others with better properties.
Multiplication
is hard. I want to multiply 8.7654321 and 2.3456789 so
instead I take log(8.7654321) and log(2.3456789) using tables. Then I add the results
(adding it easy). That gives log(8.7654321 x 2.3456789).
So I use a table backwards to see what number this is
the logarithm of and that tells me the product is.
There is a (unusually large) slide rule above the math
dept. office. If you want to know how to use one or what it
has to do with logarithms, come ask me.
Napier chose his unusual base to be very close to one because he
wanted the entries in his table to be close together and not
spread out. Bases close to one do this better.
The motivation for series, which are surely growing more and more
common, is to be able to apply calculus techniques to
non-polynomial functions. That, in fact, was the entire
motivation for them in Calc II, if you missed it.
One
reasonable answer to the Newton-Leibniz debate is
“neither”. Calculus was growing slowly over
centuries. We’ve seen ideas long ago, and there are
several who took very serious steps. Newton and Leibniz
were the first two to present a systematic theory. On the
other hand both still struggled with some of the details.
In
Barrow’s FTC, remember the top function is an area accumulation
function. The top function measures how much area the
bottom function has accumulated up to that point. To be
more precise R times the top function gives the area. This
was done because a length is not an area, so R times the top
function gives the area. As far as I know none of these
people called it ‘the fundamental theorem’. Comparisons
between Leibniz and Barrow’s proofs: Leibniz used
infinitesimals and that the integral of dx is x. Barrow
avoiding the subject by working more geometrically with
tangents. Regarding my claim that R = 1, there was no
scale on the graph nor units. We may use any units we
choose. I choose to use units of R. We've seen this
before.
We've seen a few times "quadrature" refers to the process of
finding area. Newton called derivatives "fluxions", and
variables "fluents".
It is a historical curiosity that limits do come last to the
calculus story, despite you discussing them first in class.
Reading
Reactions
Although
we are surely at the age of series work, questions of convergence
really haven't arisen yet.
I’m *not* going to get into physics here, but I do want you to see
that gravity is a curious thing … how does the sun control the
earth when they are so far apart. The moon’s position became
important for use in navigation tables, as explained in the
subsequent paragraphs after saying it became important. The
moon is the simplest example of a three-body problem. We can
assume that the earth is only affected by the sun’s gravity, and
we do fine. But, to analyse the moon we cannot ignore either
the sun (because it’s so big) or the earth (because it’s so
close).
The
English are avoiding work in the calculus from fallout from the
Newton-Leibniz controversy. Nationalistic isolation does
harm. Curiously, we (here now) tend to learn more
Leibniz-style calculus, but give Newton more credit.
The
college and the city (I think the college was actually first) in
California are named after Bishop Berkeley (and the element, for
that matter), but not for any significant reason (some people
liked something he said about westward expansion).
I
will present Euler’s form of deMoivre’s formula. We will
discuss Euler next time.
I
find it poetic justice that Maclaurin devised Cramer’s rule.
I’ve seen this in original sources (from a past student’s
project).
Simpson
is one of the first examples we see of statistics - we will see
more next time. Realising that errors cancel in means is a
step forward. This is the beginning of the distribution of
sample means.