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Here, we do just enough topology so as to be able to talk about smooth manifolds. We begin with n-dimensional Euclidean space.
(R is the set of real numbers.) Thus, E1 is just the real line, E2 is the Euclidean plane, and E3 is 3dimensional Euclidean space. Why?
The magnitude, or norm, ||y|| of y = (y1, y2, . . . , yn) in En is defined to be
which we think of as its distance from the origin. Let me see some examples.
The distance between two points y = (y1, y2, ... , yn) and z = (z1, z2, ... , zn) in En is defined as ||z y||:
Distance between y and z = ||z y|| = ((z1 y1)2 + (z2 y2)2 + . . . + (zn yn)2)1/2
An example, please.
The properties of the norm are summed up in the following result.
Proposition 1.1 (Properties of the norm)
The norm satisfies the following:
(a) ||y|| 0, and ||y|| = 0 if and only if y = 0 (positive definite)
The proof of Proposition 1.1 is an exercise which may require reference to a linear algebra text (see "inner products").
Definition 1.2 A Subset U of En is called open if, for every y in U, all points of En within some positive distance r of y are also in U. (The size of r may depend on the point y chosen.) Let me see a picture.
Intuitively, an open set is a solid region minus its boundary. If we include the boundary, we get a closed set, which formally is defined as the complement of an open set.
(a) If a En, then the open ball with center a and radius r is the set of all points in En whose distance from a is less than r.
Open balls are open sets: If x B(a, r), then, with s = r ||xa||, one has B(x, s) B(a, r); that is, all points within a distance s of x are still inside B(a, r).
(b) En is open.
(c) is open.
(d) Unions of open sets are open.
(e) Open sets are unions of open balls. (Why is that?)
Definition 1.4 Now let M Es. A subset U M is called open in M (or relatively open) if, for every y in U, all points of M within some positive distance r of y are also in U.
In the following diagram, M is the hemisphere, Es is three-dimensional space (yellow) and U is the small "patch" on M (excluding its boundary). Notice that U is not open in Es, since there are points in Es arbitrarily close to U that lie outside U. However, it is open in M, since given any point y in U, all points of M within a small enough distance from y are still in U.
(a) Open balls in M If M Es, m M, and r > 0, define
For example, if M is the surface of the earth, m is the center of Honolulu and r = 100 miles, then BM(m, r) consists of all points on the surface of the earth less than 100 miles from central Honolulu. However, points above the surface--even one inch above cantral Honolulu--are not in BM(m, r).
and so BM(m, r) is open in M.
(b) M is open in M.
(c) is open in M.
(d) Unions of open sets in M are open in M.
(e) Open sets in M are unions of open balls in M.
Parametric Paths and Surfaces in E3
From now on, the three coordinates of 3-space will be referred to as y1, y2, and y3.
Definition 1.6 A smooth path in E3 is a set of three smooth (infinitely differentiable) real-valued functions of a single real variable t:
The variable t is called the parameter of the curve. If the vector (dy1/dt, dy2/dt, dy3/dt) nowhere zero, we speak of a non-singular path.
(a) Instead of writing y1 = y1(t), y2 = y2(t), y3 = y3(t), we shall simply write yi = yi(t).
(b) Since there is nothing special about three dimensions, we define a smooth path in En in exactly the same way: as a collection of smooth functions yi = yi(t), where this time i goes from 1 to n.
(a) Straight lines in E3
(b) Helix in E3
Definition 1.8 A smooth surface immersed in E3 is a collection of three smooth real-valued functions of two variables x1 and x2 (notice that x finally makes a debut).
y1 = y1(x1, x2)
yi = yi(x1, x2) (i = 1, 2, 3).
Note that holding x1 constant gives a smooth path, with different constants yielding different paths. Similarly, holding x2 constant gives another batch of paths that intersect the first ones. (See the picture.)
We also require that the 32 matrix whose ij entry is yi/xj has rank two. We call x1 and x2 the parameters or local coordinates.
(a) Planes in E3
We can paramaterize the plane through the point (p1, p2, p3) and parallel to the (independent) vectors (a1, a2, a3), (b1, b2, b3) by
(b) The paraboloid y3 = y12 + y22 can be paramaterized by setting
Note (x2)2 means x2 squared, and not x4. (Yes, I know the notation is strange, but that's the tradition...)
(c) The unit sphere y12 + y22 + y32 = 1, using spherical polar coordinates.
x1 and x2 are the usual polar coordinates (the angles shown in the figure).
(d) The ellipsoid
(e) The Jacobean matrix for spherical polar coordinates (Example (c)) is the matrix
Exercise Show that J has rank 2 everywhere except x1 = n (n an integer).
(f) The torus with radii a > b:
Question The parametric equations of a surface show us how to obtain a point on the surface once we know the two local coordinates (parameters). In other words, we have specified a function E2E3. How do we obtain the local coordinates from the Cartesian coordinates y1, y2, y3?
Answer We need to solve for the local coordinates xi as functions of yj. For instance, in the case of a sphere, we get
This allows us to give each point on much of the sphere two unique coordinates, x1, and x2. There is a problem with continuity when y2 = 0, since then x1 switches from 0 to 2. There is also a problem at the poles (y1 = y2 = 0), since then the above functions are not even defined. Thus, we restrict to the portion of the sphere given by
0 < x1 < 2, 0 < x2 < ,
which is an open subset U of the sphere. (Think of it as the surface of the earth with the Greenwich Meridian removed. Let me see a picture of U.) We call x1 and x2 the coordinate functions. They are functions
We can put them together to obtain a single function x: UE2 given by
x(y1, y2, y3) = (x1(y1, y2, y3), x2(y1, y2, y3))
where x1 and x2 are the functions specified by the above formulas (*), as a chart.
Definition 1.10 A chart of a surface S is a pair of functions
which specify each of the local coordinates (parameters) x1 and x2 as smooth functions of a general point (global or ambient coordinates) (y1, y2, y3) on the surface.
Question Why are these functions called a chart?
Answer The chart above assigns to each point on the sphere (away from the meridian) two coordinates. So, we can think of it as giving a two-dimensional map of the surface of the sphere, just like a geographic chart.
Question Our chart for the sphere is very nice, but is only appears to chart a portion of the sphere. What about the missing meridian?
Answer We can use another chart to get those by using different paramaterization that places the poles on the equator. (Diagram in class.)
In general, we chart an entire manifold M by "covering" it with open sets U which become the domains of coordinate charts.
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