0708-1300/Class notes for Tuesday, November 6

0708-1300/Navigation Panel   Add your name / see who's in! # Week of... Links Fall Semester 1 Sep 10 About, Tue, Thu 2 Sep 17 Tue, HW1, Thu 3 Sep 24 Tue, Photo, Thu 4 Oct 1 Questionnaire, Tue, HW2, Thu 5 Oct 8 Thanksgiving, Tue, Thu 6 Oct 15 Tue, HW3, Thu 7 Oct 22 Tue, Thu 8 Oct 29 Tue, HW4, Thu, Hilbert sphere 9 Nov 5 Tue,Thu, TE1 10 Nov 12 Tue, Thu 11 Nov 19 Tue, Thu, HW5 12 Nov 26 Tue, Thu 13 Dec 3 Tue, Thu, HW6 Spring Semester 14 Jan 7 Tue, Thu, HW7 15 Jan 14 Tue, Thu 16 Jan 21 Tue, Thu, HW8 17 Jan 28 Tue, Thu 18 Feb 4 Tue 19 Feb 11 TE2, Tue, HW9, Thu, Feb 17: last chance to drop class R Feb 18 Reading week 20 Feb 25 Tue, Thu, HW10 21 Mar 3 Tue, Thu 22 Mar 10 Tue, Thu, HW11 23 Mar 17 Tue, Thu 24 Mar 24 Tue, HW12, Thu 25 Mar 31 Referendum,Tue, Thu 26 Apr 7 Tue, Thu R Apr 14 Office hours R Apr 21 Office hours F Apr 28 Office hours, Final (Fri, May 2) Register of Good Deeds Errata to Bredon's Book

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 $X_{1},\ldots ,X_{k}\in T_{p}M\!$ by feeding these vectors into some function $\omega :(T_{p}M)^{k}\to \mathbb {R} \!$. We would like $\omega \!$ to satisfy a few properties:

1. $\omega \!$ should be linear in each argument: for example, if we double the length of one of the sides, the volume should double.
2. If two of the vectors fed to $\omega \!$ are parallel, the volume assigned by $\omega \!$ 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 $V\!$ be a real vector space, let $p\in \mathbb {N} \!$ and let $L(V^{p};\mathbb {R} )$ denote the collection maps from $V^{p}\!$ to $\mathbb {R}$ that are linear in each argument separately. We set

$A^{p}(V)=\left\{\omega \in L(V^{p};\mathbb {R} ):\omega (\ldots ,v,\ldots ,v,\ldots )=0\ \forall v\in V\right\}$

Proposition

Suppose that $\omega \in A^{p}(V)\!$ and $v_{1},\ldots ,v_{p}\in V\!$. The following statements hold:

1. $A^{p}(V)\!$ has a natural vector space structure
2. $A^{0}(V)\!$ is $\mathbb {R}$
3. $A^{1}(V)=V^{*}\!$ is the dual space of $V\!$
4. $\omega (v_{1},\ldots ,v_{j},\ldots ,v_{k},\ldots ,v_{p})=-\omega (v_{1},\ldots ,v_{k},\ldots ,v_{j},\ldots ,v_{p})\!$ for every $j<k\in \{1,\ldots ,p\}\!$
5. If $\sigma \in S_{p}\!$ is a permutation, then $\omega (v_{\sigma (1)},\ldots ,v_{\sigma (p)})=(-1)^{\sigma }\omega (v_{1},\ldots ,v_{p})$

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 $A^{0}(V)\!$ 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 $A^{1}(V)\!$ and $V^{*}\!$ coincide.

As for the fourth, note that $0=\omega (v_{1},\ldots ,v_{j}+v_{k},\ldots ,v_{j}+v_{k},\ldots ,v_{p})$ so that using linearity we obtain

$0=\omega (v_{1},\ldots ,v_{j},\ldots ,v_{j},\ldots ,v_{p})+\omega (v_{1},\ldots ,v_{k},\ldots ,v_{j},\ldots ,v_{p})+\omega (v_{1},\ldots ,v_{j},\ldots ,v_{k},\ldots ,v_{p})+\omega (v_{1},\ldots ,v_{k},\ldots ,v_{k},\ldots ,v_{p})$

and hence $\omega (v_{1},\ldots ,v_{k},\ldots ,v_{j},\ldots ,v_{p})+\omega (v_{1},\ldots ,v_{j},\ldots ,v_{k},\ldots ,v_{p})=0$.

The fifth statement then follows from repeated application of the fourth.

Remarks

Our computation in the previous proof shows that we could equally well have defined $A^{p}(V)\!$ to consist of all those multilinear maps from $V^{k}\!$ to $\mathbb {R} \!$ 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 $A^{p}(V)\!$ with $A^{q}(V)\!$. This multiplication is called the wedge product and is defined as follows.

Definition

For each $p,q\in \mathbb {N} \!$ the wedge product is the map $\wedge :A^{p}(V)\times A^{q}(V)\to A^{p+q}(V),(\omega ,\lambda )\mapsto \omega \wedge \lambda$ defined by

$(\omega \wedge \lambda )(v_{1},\ldots ,v_{p+q})=\sum _{\sigma \in S_{p+q}}{\frac {(-1)^{\sigma }}{p!q!}}\omega (v_{\sigma (1)},\ldots ,v_{\sigma (p)})\lambda (v_{\sigma (p+1)},\ldots ,v_{\sigma (p+q)})$

for every $v_{1},\ldots ,v_{p+q}\in V$.