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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
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.
We will now shift our attention to the theory of integration on smooth manifolds. The first thing that we need to construct is a means of measuring volumes on manifolds. To accomplish this goal, we begin by imagining that we want to measure the volume of the "infinitiesimal" parallelepiped [1] defined a set of vectors
by feeding these vectors into some function
. We would like
to satisfy a few properties:
-
should be linear in each argument: for example, if we double the length of one of the sides, the volume should double.
- If two of the vectors fed to
are parallel, the volume assigned by
should be zero because the parallelepiped collapses to something with lower dimenion in this case.
Inspired by these requirements, we make the following definition:
Definition
Let
be a real vector space, let
and let
denote the collection maps from
to
that are linear in each argument separately. We set
![{\displaystyle A^{p}(V)=\left\{\omega \in L(V^{p};\mathbb {R} ):\omega (\ldots ,v,\ldots ,v,\ldots )=0\ \forall v\in V\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb651ca22dd5537717db88afc079d4ef1d0b24e)
Proposition
Suppose that
and
. The following statements hold:
-
has a natural vector space structure
-
is
-
is the dual space of
-
for every
- If
is a permutation, then
Proof
The first statement is easy to show and is left as an exercise. The second statement is more of a convenient definition. Note that
consists of all maps that take no vectors and return a real number since the other properties are vacuous when the domain is empty. We can thus interpret an element in this space simply as a real number. The third statement is clear as the defintions of
and
coincide.
As for the fourth, note that
so that using linearity we obtain
![{\displaystyle 0=\omega (v_{1},\ldots ,v_{j},\ldots ,v_{j},\ldots ,v_{p})+\omega (v_{1},\ldots ,v_{k},\ldots ,v_{j},\ldots ,v_{p})+\omega (v_{1},\ldots ,v_{j},\ldots ,v_{k},\ldots ,v_{p})+\omega (v_{1},\ldots ,v_{k},\ldots ,v_{k},\ldots ,v_{p})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16e892cfe4d6a8eae4fa773f698ac50c1feb3a41)
and hence
.
The fifth statement then follows from repeated application of the fourth.
Our computation in the previous proof shows that we could equally well have defined
to consist of all those multilinear maps from
to
that change sign when two arguments are interchanged.
One of the nicest things about these spaces is that we can define a sort of multiplication of elements of
with
. This multiplication is called the wedge product and is defined as follows.
Definition
For each
the wedge product is the map
defined by
for every
.