Comments on the lecture of 20/11/06

(Preliminary version, may be slightly revised later.)

At the beginning of the lecture I wrote an extra exercise about connected
sets, which can be useful for solving other exercises.

This same exercise now appears as "Exercise 0" in an updated
version of the page about connected sets that I put on my private
Hedva 2m website on 16/11/06.

Some of you seemed to be confused about connected sets because you thought
that the curve in the set joining a pair of points in the set has to be,
davka, a straight line segment ("keta yashar"). It does not have to be
straight. For example, the circle ("ma'agal" NOT "igul") x^2+y^2=1
in R^2 is a connected set, and NONE of the curves in it joining pairs
of points in it can ever be straight.

(There is another kind of set, called CONVEX SET ("kvutsa k'moora")
which is not an official topic in this course, where each two points in the
set have to be joined by straight line segment which is completely contained
in the set.)

From one question during the break, I understood that at least one person
was confused because I first used the letter E to denote a set which is a curve.
Then later I used the same letter E to denote a set which we are
checking to see if it is connected. Sorry, but we only have a limited
number of letters, and many jobs for them to do, so it can happen that the
same letter will appear in different tasks. I agree that it would have been
better if I used a different letter. Some letters which are often
"traditionally" used to denote curves are C and L and the Greek capital
Gamma (which looks like F with the middle horizontal line removed, and
was used in Exercise 0).


Graphs and level curves of functions of two variables.

Please look at the four examples of functions and their graphs which are
discussed on pages 1 to 3 of my old hoveret at

http://www.math.technion.ac.il/courses/104011/targ/104011ta2.ps

I already noticed a couple of small, not serious, misprints on those pages.
In general, in case of doubts you can look at the original handwritten version
of this same material (which could also sometimes have some misprints)
in the pdf file at

http://www.math.technion.ac.il/courses/Math2H/Hoveret1.pdf

=============================

PAGE NUMBERING IN THE PDF FILES OF MY OLD HOVERET.

Since I may from time to time refer you to other things in that pdf file
and in the other pdf files which contain other parts of the same old
hoveret, we have to discuss the embarrasing (for me) topic of the page
numbers in those hovrot.

Because of strange historical reasons, as the hoveret was rewritten, I
had to simultaneously use several different systems of page numbering.
Sometimes page numbers appear in circles, other page numbers appear in
squares, or in triangles, and other page numbers appear at the bottom of
the page preceded by an aleph or a hey. The best thing I think is to now
IGNORE all these crazy numbers. When I want to refer you to a page in
these pdf files, I will use the PDF PAGE NUMBER. This means you open the
pdf file with Acrobat reader or whatever, and you use the right computer
command (Shift Ctrl L) to type the page that you want. Or you go to the
window at the bottom of the screen which indicates the page numbers ("4
of 91") and you type the page number that you want.
Hopefully, if you printed a hard copies of these pdf files your printer
put the same page numbers on your pages.

EXAMPLE: The discussion of those four examples of functions and their graphs
is on PDF pages 4 to 7 of the file

http://www.math.technion.ac.il/courses/Math2H/Hoveret1.pdf

(These correspond to pages 1 (in a triangle) to 4 (in a triangle), but
that should be irrelevant.)

====================================

What are the level curves of those four functions?


For some general information about using the pdf files of my old
hoveret, or hovrot, (which topics in them are relevant or not relevant
for Hedva 2m) see also

http://www.math.technion.ac.il/~mcwikel/h2m/hoveret-o05.html


Earlier in the lecture of 20/11/06 I spoke about and defined the tangent
vector to a curve.
Suppose for example when n=3, that a curve C is given by the parametric
representation

(u(t),v(t),w(t)) for t in the interval [a,b], where u, v and w are
differentiable functions with continuous derivative at every point of [a,b]
(one sided derivatives at the end points).

Let P be any point on C. So there is some constant number, call it r, in [a,b]
such that P=(u(r),v(r),w(r)).
Then the tangent to C at P is the line parallel to the vector
[u'(r),v'(r),w'(r)]

As I mentioned in the lecture, there can sometimes be some problems with this
definition, but we will usually find ways to overcome them. For example it
could happen that the curve goes through P more than once. Then there is
more than one way to choose the number r and we may have more than one
tangent to C at P. It could also sometimes happen that the vector
[u'(r),v'(r),w'(r)] is [0,0,0], very embarrassing!
In that case we will try to choose a different parametric representation
for the same curve.

For the moment, let us avoid these problems by deciding that we will
only consider curves C which have a parametric representation such that
the vector [u'(r),v'(r),w'(r)] is non zero for every point r in [a,b]
and such that, for each point P on C, there is only one choice of r in [a,b]
which gives that point.

I gave an intuitive justification for choosing the tangent vector to C
at P to be [u'(r),v'(r),w'(r)] by saying:
Suppose that the parameter t represents time (in general t does not have
to represent anything at all) and suppose that a particle moves along C so
that at time t it is at the point (u(t),v(t),w(t)).
Then at any given time t in [a,b] the vector [u'(t),v'(t),w'(t)]
is the velocity vector of that particle, and you would expect it to
be pointing along the tangent to the path C of the particle.

I said that there is another motivation for our definition of tangent
vector. It is a bit more "geometrical" and less "physical".
You can find it in the discussion which begins on PDF page 76
of the file

http://www.math.technion.ac.il/courses/Math2H/Hoveret2a.pdf

In particular, look at the paragraph at the bottom of PDF page 80, which
appears again, more completely about 1/4 of the way down on the next
page PDF 81. (There was a bit of a mixup here when my students and I
copied the version that you see here from an earlier version of the same
hoveret.)


End of commments.