Difference between revisions of "07081300/Class notes for Tuesday, October 2"
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==Class Notes==  ==Class Notes==  
<span style="color: red;">The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.</span>  <span style="color: red;">The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.</span> 
Revision as of 12:42, 2 October 2007

Class Notes
The notes below are by the students and for the students. Hopefully they are useful, but they come with no guarantee of any kind.
General class comments
1) The class photo is up, please add yourself
2) A questionnaire was passed out in class
3) Homework one is due on thursday
First Hour
Today's Theme: Locally a function looks like its differential
Pushforward/Pullback
Let be a smooth map.
We consider various objects, defined with respect to X or Y, and see in which direction it makes sense to consider corresponding objects on the other space. In general will denote the push forward, and will denote the pullback.
1) points pushforward
2) Paths , ie a bunch of points, pushforward,
3) Sets pullback via Note that if one tried to pushforward sets A in X, the set operations compliment and intersection would not commute appropriately with the map
4) A measures pushforward via
5)In some sense, we consider functions, "dual" to points and thus should go in the opposite direction of points, namely
6) Tangent vectors, defined in the sense of equivalence classes of paths, [] pushforward as we would expect since each path pushes forward.
CHECK: This definition is well defined, that is, independent of the representative choice of
7) We can consider operators on functions to be in a sense dual to the functions and hence should go in the opposite direction. Hence, tangent vectors, defined in the sense of derivations, pushforward via
CHECK: This definition satisfies linearity and Liebnitz property.
Theorem 1
The two definitions for the pushforward of a tangent vector coincide.
Proof:
Given a we can construct as above. However from both and we can also construct and because we have previously shown our two definitions for the tangent vector are equivalent. We can then pushforward to get . The theorem is reduced to the claim that:
for functions
Now,
Q.E.D
Functorality
let
Consider some "object" s defined with respect to X and some "object u" defined with respect to Z. Something has the property of functorality if
and
Claim: All the classes we considered previously have the functorality property; in particular, the pushforward of tangent vectors does.
Let us consider on given a
We can arrange for charts on a subset of M into (with coordinates denoted )and on a subset of N into (with coordinates denoted )such that and
Define
Now, for a we can write
So,
Now, we want to write
and so,
where the i is at the kth location.
So, , i.e., is the differential of at p
We can check the functorality, , then
This is just the chain rule.