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We now generalize the ideas discussed in Lecture 1.
Definition 2.1 An open cover of M E_{s} is a collection {U_{a}} of open sets in M such that M = _{a}U_{a}. 
Examples
(a) E_{s} can be covered by open balls.
(b) E_{s} can be covered by the single (open) set E_{s}.
(c) The unit sphere in E_{s} can be covered by the collection {U_{1}, U_{2}} where
Definition 2.2 A subset M of E_{s} is called an ndimensional smooth manifold if we are given a collection
where:
(b) Each x_{a}^{r} is a smooth (what does that mean?) realvalued function defined on U (that is, x_{a}^{r}: U_{a}E_{1}), called the th coordinate, such that the map
x(u) = (x_{a}^{1}(u), x_{a}^{2}(u), . . . , x_{a}^{n}(u)) The tuple (U_{a}; x_{a}^{1}, x_{a}^{2}, . . ., x_{a}^{n}) is called a local chart of M. The collection of all charts is called a smooth atlas of M. Further, U_{a} is called a coordinate neighborhood. (c) If (U, x^{i}), and (V, ^{j}) are two local charts of M, and if UV , then we can write
with inverse
for each i and k, where all functions in sight are smooth. These functions are called the changeofcoordinates transformations.

By the way, we call the "big" space E_{s} in which the manifold M is embedded the ambient space.
Notes
1. Always think of the x^{i} as the local coordinates (or parameters) of the manifold. We can paramaterize each of the open sets U by using the inverse function x^{1} of x, which assigns to each point in some neighborhood of E_{n} a corresponding point in the manifold. Let me see an example.
2. Condition (c) implies that
det 
 0,  and 
det 
 0  , 
since the associated matrices must be invertible.
3. The ambient space need not be present in the general theory of manifolds; that is, it is possible to define a smooth manifold M without any reference to an ambient space at all  see any text on differential topology or differential geometry.
4. More terminology: We shall sometimes refer to the x^{i} as the local coordinates, and to the y^{j} as the ambient coordinates. Thus, a point in an ndimensional manifold M in E_{s} has n local coordinates, but s ambient coordinates.
Examples 2.3
(a) E_{n} is an ndimensional manifold, with the single identity chart defined by
One has
= 

x  = 

Notice the symmetry between x and . Also notice that these changeofcoordinate functions are only defined when 0, . Further,
Note that, in terms of complex numbers, we can write, for a point p = e^{iz} S^{1},
(c) Generalized Polar Coordinates Let us take M = S^{n}, the unit nsphere,
with coordinates (x^{1}, x^{2}, . . . , x^{n}) with
In the homework, you will be asked to obtain the associated chart by solving for the x^{i}. Note that if the sphere has radius r, then we can multiply all the above expressions by r, getting
(d) The torus T = S^{1}S^{1}, with the following four charts:
(e) The cylinder (exercise)
(f) S^{n}, with (again) stereographic projection, is an nmanifold; the two charts are given as follows. Let P be the point (0, 0, . . , 0, 1) and let Q be the point (0, 0, . . . , 0, 1). Then define two charts (S^{n}P, x^{i}) and (S^{n}Q, ^{i}) as follows. (See the figure.)
If (y_{1}, y_{2}, . . . , y_{n}, y_{n+1}) is a point in S^{n}, let






. . .  . . .  


We can invert these maps (that is, solve for the global coordinates y_{i} in terms of the local coordinates x_{i} and _{i}) as follows:
Let r^{2} = _{i }x^{i}x^{i}, and ^{2} = _{i }^{i}^{i}. Then:






. . .  . . .  





The changeofcoordinate maps are therefore:
x^{1}  = 

= 

= 


x^{2}  = 


. . .  
x^{n}  = 

This makes sense, since the maps are not defined when ^{i} = 0 for all i, corresponding to the north pole.
Note Since is the distance from ^{i} to the origin, this map is hyperbolic reflection in the unit circle;

and squaring and adding gives
That is, project it to the circle, and invert the distance from the origin. This also gives the inverse relations, since we can write
In other words, we have the following transformation rules.
Change of Coordinate Transformations for Stereographic Projection
Let r^{2} = _{i }x^{i}x^{i}, and ^{2} = _{i }^{i}^{i}.

Note We can put all the coordinate functions x_{a}^{r}: U_{a}E_{1} together to get a single map
A more precise formulation of condition (c) in the definition of a manifold is then the following: each W_{a} is an open subset of E_{n}, each x_{a} is invertible, and each composite
W_{a }  x_{a}^{1}  E_{n} 
x_{b} 
W_{b} 
is a smooth function defined on an open subset.
We now want to discuss scalar and vector fields on manifolds, but how do we specify such things? First, a scalar field.
Definition 2.4 A smooth scalar field on a smooth manifold M is just a smooth realvalued map : ME_{1}. (In other words, it is a smooth function of the coordinates of M as a subset of E_{r}.) Thus, associates to each point m of M a unique scalar (m).
If U is a subset of M, then a smooth scalar field on U is smooth realvalued map : UE_{1}. If U M, we sometimes call such a scalar field local. 
If is a scalar field on M and x is a chart, then we can express as a smooth function of the associated parameters x^{1}, x^{2}, . . . , x^{n}. If the chart is , we shall write for the function of the other parameters ^{1}, ^{2}, . . . , ^{n}. Note that we must have = at each point of the manifold (see the transformation rule below).
Examples 2.5 (a) Let M = E_{n} (with its usual structure) and let be any smooth realvalued function in the usual sense. Then, using the identity chart, we have = .
(b) Let M = S^{2}, and define (y_{1}, y_{2}, y_{3}) = y_{3}.
Using stereographic projection, we find both and :
(x^{1}, x^{2})  =  y_{3}(x^{1}, x^{2})  = 

= 


(^{1}, ^{2})  =  y_{3}(^{1}, ^{2})  = 

= 

(c) Local Scalar Field The most obvious candidate for local fields are the coordinate functions themselves. If U is a coordinate neighborhood, and x = {x^{i}} is a chart on U, then the maps x^{i} are local scalar fields.
Sometimes, as in the above example, we may wish to specify a scalar field purely by specifying it in terms of its local parameters; that is, by specifying the various functions instead of the single function . The problem is, we can't just specify it any way we want, since it must give a value to each point in the manifold independently of local coordinates. That is, if a point p M has local coordinates (x^{j}) with one chart and (^{h}) with another, they must be related via the relationship
Transformation Rule for Scalar Fields

Example 2.6 Look at Example 2.5(b) above. If you substituted ^{i} as a function of the x^{j}, you would get (^{1}, ^{2}) = (x^{1}, x^{2}) (after some laborious albegra!).
1. Give the paraboloid z = x^{2} + y^{2} the structure of a smooth manifold.
2. Find a smooth atlas of E_{2} consisting of three charts.
3. (a) Extend the method in Exercise 1 to show that the graph of any smooth function f: E_{2}E_{1} can be given the structure of a smooth manifold.
(b) Generalize part (a) to the graph of a smooth function f: E_{n} E_{1}.
4. Two atlases of the manifold M give the same smooth structure if their union is again a smooth atlas of M.
(a) Show that the smooth atlases (E_{1}, f), and (E_{1}, g), where f(x) = x and g(x) = x^{3} are incompatible.
(b) Find a third smooth atlas of E_{1} that is incompatible with both the atlases in part (a).
5. Consider the ellipsoid L E_{3} specified by
 =  1, 
(a, b, c 0).
Define f: LS^{2} by f(x, y, z) = (x/a, y/b. z/c).
(a) Verify that f is invertible (by finding its inverse).
(b) Use the map f, together with a smooth atlas of S^{2}, to construct a smooth atlas of L.
6. Find the chart associated with the generalized spherical polar coordinates described in Example 2.3(c) by inverting the coordinates. How many additional charts are needed to get an atlas? Give an example.
7. Obtain the equations in Example 2.3(f).
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