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Week of...
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Links
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Fall Semester
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1
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Sep 10
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About, Tue, Thu
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2
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Sep 17
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Tue, HW1, Thu
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3
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Sep 24
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Tue, Photo, Thu
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4
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Oct 1
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Questionnaire, Tue, HW2, Thu
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5
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Oct 8
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Thanksgiving, Tue, Thu
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6
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Oct 15
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Tue, HW3, Thu
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7
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Oct 22
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Tue, Thu
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8
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Oct 29
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Tue, HW4, Thu, Hilbert sphere
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9
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Nov 5
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Tue,Thu, TE1
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10
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Nov 12
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Tue, Thu
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11
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Nov 19
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Tue, Thu, HW5
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12
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Nov 26
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Tue, Thu
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13
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Dec 3
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Tue, Thu, HW6
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Spring Semester
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14
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Jan 7
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Tue, Thu, HW7
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15
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Jan 14
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Tue, Thu
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16
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Jan 21
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Tue, Thu, HW8
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17
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Jan 28
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Tue, Thu
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18
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Feb 4
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Tue
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19
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Feb 11
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TE2, Tue, HW9, Thu, Feb 17: last chance to drop class
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R
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Feb 18
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Reading week
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20
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Feb 25
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Tue, Thu, HW10
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21
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Mar 3
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Tue, Thu
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22
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Mar 10
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Tue, Thu, HW11
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23
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Mar 17
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Tue, Thu
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24
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Mar 24
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Tue, HW12, Thu
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25
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Mar 31
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Referendum,Tue, Thu
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26
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Apr 7
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Tue, Thu
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R
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Apr 14
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Office hours
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R
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Apr 21
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Office hours
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F
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Apr 28
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Office hours, Final (Fri, May 2)
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Register of Good Deeds
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Errata to Bredon's Book
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Announcements go here
Typed 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.
First Hour
Recall we are ultimately attempting to understand and prove Stokes theorem. Currently we are investigating the meaning of
.
Recall we had that
Now we want to compute
on the parallelepiped formed from k+1 tangent vectors. For instance let us suppose k= 2 then are interested in the parallelepiped formed from the three basis vectors
.
Feeding in the parallelepiped we get ![{\displaystyle \sum _{I}df_{I}(v_{1})dx^{I}(v_{2},v_{3})-\sum _{I}df_{I}(v_{2})dx^{I}(v_{1},v_{3})+\sum _{I}df_{I}(v_{3})dx^{I}(v_{1},v_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63fa2bcfdb8dd95881e3f96a73c318e4ac65e832)
Now,
, or, loosely
So this corresponds to the difference between
calculated on each of the two faces parallel to the
plane.
Hence, our
on the parallelepiped is just the sum of parallelograms making the boundary of the parallelepiped counted with some signs.
We can see the loose idea of how the proof of stokes theorem is going to work: dividing the manifold up into little parallelepiped like this,
will just be the faces of the parallelepipeds and when summing over the whole manifold all of faces will cancel except those on the boundary thus just leaving the integral of
along the boundary.
We note that this is similar to the proof of the fundamental theorem of calculus, where we take an integral and compute the value of f' at many little subintervals. But the value of f' is just the difference of f' at the boundary of each sub interval so when we add everything up everything cancels except the values of the function at the endpoint of the big interval.
Claim
d exists if
and is unique.
Define
Proof
We need to check that this satisfies the properties:
1)
and so satisfies
Note
We now adopt Einstein Summation Convention which means that if in a term there is an index that is repeated, once as a subscript and once as a superscript, it is meant as implicit that we are summing over this index. This just cleans up the notation so we don't have to have sums everywhere.
2)
because the mixed partial is symmetric under exchange of indices but the wedge product is antisymmetric under exchange of indices. That is, each term cancels with the one where j and j' are exchanged.
3) let
and
then
so
Via assignment 3 this is unique.
Q.E.D.
Now, we can extend this definition on manifolds by using coordinate charts.
Claim
Properties 1-3 imply that on any M, d is local. That is, if
then
Proof: Exercise.
Definition
For
the
Then
has compact support if the supp
is compact.
Define
:= the compactly support
Definition
We define
by
and thus
then
I.e.,
Second Hour
Coming soon to a wikipedia near you.