Sergei Yakovenko's blog: on Math and Teaching

Monday, December 14, 2009

Lecture 7 (Dec 15, 2009)

Topology of the plane

  1. Functions of two variables as maps f\colon \mathbb R^2\to\mathbb R. Domains in the plane. Open and closed domains.
  2. Convergence of planar points: \lim_{n\to\infty}(x_n,y_n)=(X,Y)\in\mathbb R^2\iff\lim x_n=X\ \&\ \lim y_n=Y. Alternative description: any square Q_\varepsilon(X,Y)=\{|x-X|<\varepsilon,\ |y-Y|<\varepsilon\} contains almost all elements of the sequence.
  3. Limits of functions: f\colon U\to\mathbb R, Z=(X,Y)\in\mathbb R. We say that A=\lim_{(x,y)\to Z}f(x,y), if for any sequence of points \{(x_n,y_n)\} converging to Z, the sequence f(x_n,y_n) converges to A as n\to\infty. We say that f is continuous at Z, if A=f(Z).
  4. Exercise: f is continuous at Z, if the preimage f^{-1}(J), of any interval J=(A-\delta,A+\delta),\ \delta>0,ย contains the intersection Q_\varepsilon\cap U for some sufficiently small \varepsilon>0.
  5. Exercise: if f is continuous at all points of the rectangle \{x\in I,\ y\in J\}\subseteq\mathbb R^2, then for any y\in J the function f_y\colon I\to\mathbb R, defined by the formula f_y(x)=f(x,y) is continuous on I. Can one exchange the role of x and y? Formulate and think about the inverse statement.
  6. Exercise: Check that the functions f(x,y)=x\pm y and g(x,y)=xy are continuous. What can be said about the continuity of the function f(x,y)=y/x?
  7. Exercise: formulate and prove a theorem on continuity of the composite functions.
  8. Exercise: Give the definition of a continuous function f\colon I\to\mathbb R^2 for I\subseteq\mathbb R. Planar curves. ย Simple curves. Closed curves.
  9. Intermediate value theorem for curves. Connected sets, connected components. Jordan lemma.
  10. Rotation of a closed curve around a point. Continuity of the rotation number. Yet another “Intermediate value theorem” for functions of two variables.

Merry Hanukka!

4 Comments »

  1. I have a question about the difference between the term function and map, in which cases it is used? ๐Ÿ˜ฆ

    Also on the terms variety and manifold ๐Ÿ˜ฆ
    Regards. ๐Ÿ™‚

    Comment by Goethe — Thursday, August 12, 2010 @ 9:11 | Reply

  2. Dear Johann Wolfgang ๐Ÿ˜‰

    The terms “function” and “map” formally are completely interchangeable (another terms for the same is “application”, “mapping” etc.) However, in conjunction with some adjectives the usage becomes more restricted: you may say “vector-valued function” but never “vector-valued map”. Very loosely speaking, the word “function” is used when the source and the target sets have completely different nature (like function of matrix, function of several arguments), while “map” often assumes that these two sets have some common features, – like coordinate maps on a manifold etc. But the thumb rule should always be “read the definition first”…

    Variety and manifold are also almost interchangeable. In more advanced parts of Algebraic Geometry the term “variety” may denote “manifold with singularities”, while “manifold” usually associates with the “smooth manifold”, “topological manifold” e.a. Here the above thumb rule also applies completely ๐Ÿ˜‰

    Comment by Sergei Yakovenko — Thursday, August 12, 2010 @ 9:27 | Reply

  3. oh! thank you very much for the reply ๐Ÿ™‚

    Courses I’ve taken (in spanish), at least I do not remember that one teacher has made known the difference between the definition of function and map, in fact (at the risk of being wrong) in spanish, the management of both terms often used without distinction of any kind (in spanish).

    I am self reading a book on algebraic geometry, Shafarevich – BAG. In which the author handles both terms, function and map. So to translate this into spanish, it becomes a somewhat confusing, e.g. Regular map and regular function, clearly looking at the definitions, the picture is clarifying, but I was a somewhat confusing at first instance.

    Again, thank you very much for the reply.

    Comment by goethe — Friday, August 13, 2010 @ 7:52 | Reply

  4. In algebraic geometry one has to be especially careful, using the above thumb rule on every step. Sometimes the terms can be misleading: for instance, a “rational function”, the central notion of AG, is NOT A FUNCTION in the standard sense!! Even if you extend the set of values by the “infinite point” (i.e., replacing the affine complex line by the projective line a.k.a. the Riemann sphere), there will remain the “indeterminacy points” like the origin on the (x,y)-plane for the “rational function” y/x. Beware ๐Ÿ˜ฆ !

    Comment by Sergei Yakovenko — Friday, August 13, 2010 @ 11:14 | Reply


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