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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 by 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 of 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)
and if
, we say that the degree of
is
and write
. ![{\displaystyle \Box \!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfa56256ab229c63c5b940fc3a960aedae487ce)
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. ![{\displaystyle \Box \!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfa56256ab229c63c5b940fc3a960aedae487ce)
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
, where
.
The idea behind this definition is to feed vectors to
in as many ways as possible. We could equally well have set
![{\displaystyle (\omega \wedge \lambda )(v_{1},\ldots ,v_{p+q})={\frac {1}{p!q!}}\sum _{\sigma \in S_{p+q}}(-1)^{\sigma }\omega (v_{\sigma (1)},\ldots ,v_{\sigma (p)})\lambda (v_{\sigma (p+1)},\ldots ,v_{\sigma (p+q)})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0690ba31f5a526d1c9b5d4637dc72d0ca3c3bed3)
.
The factor of
compensates for the overcounting that we do by summing over all permutations, since their are
ways of rearranging the
vectors fed to
if we don't care about order, but only one way if we do care. The same argument accounts for the
.
Of course, as we have defined it, it is not immediately clear that
. However, multilinearity is obvious and it is fairly clear that the
takes care of the skew-symmetry.
In fact,
has a number of nice properties:
Proposition
The following statements hold:
-
is a bilinear map.
-
is associative.
-
is supercommutative:
.
Proof
Bilinearity is clear. Associativity and supercommutativity follow from some combinatorial arguments.
It turns out that we can use the wedge product to find bases for
:
Proposition
If
is a basis for
then
is a basis for ![{\displaystyle A^{p}(V)\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1184d7a6a1aba2ee8c1e05e5b0d2d26963df397f)
Proof
Let
be the dual basis to
, so that
. Let
. For
with
and
, let
, and let
. Then
if
and
otherwise.
We claim that if
, then
. But
, so equality holds for ordered sequences of basis vectors. Equality then holds for any sequence of vectors by skew-symmetry and linearity. We claim further that the
are linearly independent. But if
, then
by applying
to
. Hence the
form a basis.![{\displaystyle \Box \!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfa56256ab229c63c5b940fc3a960aedae487ce)
Corollary
, where
. ![{\displaystyle \Box \!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfa56256ab229c63c5b940fc3a960aedae487ce)
We may now define differential forms. The idea is to smoothly assign to each point
in a manifold
an element of
.
Definition
Let
be a smooth manifold of dimension
. For
, a differential
-form on
(or simply a p-form) is an assignment to each
an element
that is smooth in the sense that if
are smooth vector fields on
then the map
is
.
The collection of
-forms on
will be denoted by
.![{\displaystyle \Box \!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfa56256ab229c63c5b940fc3a960aedae487ce)
If
are such that
form a basis for
for each
with
open, then
can be written (for
) as
where the maps
are smooth. In fact, we could have taken this property as our definition of smoothness on
.