Math 562 Fall 2007

The final exam will be held on Wednesday, December 12, from 8am to 10am, in UNIV 119.

The exam will be proctored by Prof. Monica Torres, and will cover all the material up to and including the generalized Stokes’ theorem.

Graded homeworks are available for collection outside my office, MATH 430.

The course notes are available online.

We introduced the generalized Stokes’ theorem for differential forms, and show how it recovers the classical Stokes’ theorem and the divergence theorem.

This corresponds to VI.5 in the textbook.

We also briefly discussed the Hodge star operator, which together with the sharp and flat operators that identify one-forms with vector fields, allows us to express the usual vector calculus operators like div, grad, and curl, in terms of the exterior derivative.

We introduced the notion of a manifold with boundary. It is analogous to the definition of a manifold, except that it is modeled on the Euclidean half-space, as opposed to Euclidean space. One can show that a choice of an orientation on a manifold with boundary induces an orientation on the boundary.This is achieved by a choice of an orientation in the form of an atlas of coordinate charts whose transition maps are orientation preserving. Then, one can show that the natural restriction of the coordinate chart on a manifold induces an atlas on the boundary, and the transition maps between these restricted coordinate charts is still orientation preserving.

This correspond to VI.4 in the textbook.

The following homework was assigned, and is due on Tuesday, December 4, 2007

p 207, Ex 5,6

p 212, Ex 1,3

p 219, Ex 1,5

p 251, Ex 4

We discussed the extension of integration on R^n to manifolds, and showed that the notion of integrability of a function is independent of the choice of the n-form that encodes the orientation of the manifold. We then consider the integral of a function on a manifold by using a partition of unity, and show that the result is independent of the choice of a partition of unity. We also considered integration on Riemannian manifolds.

This corresponds to VI.2 in the textbook. 

We proved that the exterior derivative commutes with pullback, and showed how to express the Frobenius theorem in terms of differential forms. We also reviewed properties of the Riemann integral in preparation for extending the integration theory to manifolds.

This corresponds to V.8 and VI.1 in the textbook.

The following homework is due on Tuesday, November 27, 2007.

p 181, Ex 8
p 186, Ex 3, 5
p 192, Ex 1, 9
p 207, Ex 5

I will be giving a seminar during the Applied Mathematics pizza lunchtime seminar on Friday, November 16, 2007, from 11:30am to 12:15pm in REC 122. It will be geared towards first and second year graduate students. The title and abstract follows:

Lie Group and Homogeneous Variational Integrators and their Applications to Geometric Optimal Control Theory

Melvin Leok

The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory. These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation.

We will discuss the application of geometric structure-preserving numerical schemes to the optimal control of mechanical systems. In particular, we consider Lie group variational integrators, which are based on a discretization of Hamilton’s principle that preserves the Lie group structure of the configuration space. In contrast to traditional Lie group integrators, issues of equivariance and order- of-accuracy are independent of the choice of retraction in the variational formulation. The importance of simultaneously preserving the symplectic and Lie group properties is also demonstrated.

In addition, we will introduce a numerically robust shooting based optimization algorithm that relies on the conservation properties of geometric integrators to accurately compute sensitivity derivatives, thereby yielding an optimization algorithm for the control of mechanical systems that is exceptionally efficient. The role of geometric phases in these control algorithms will also be addressed.

Recent extensions to homogeneous spaces yield intrinsic methods for Hamiltonian flows on the sphere, and have potential applications to the simulation of geometric exact rods, structures and mechanisms.

We will place recent work in the context of progress towards a coherent theory of computational geometric mechanics and computational geometric control theory, which is concerned with developing a self-consistent discrete theory of differential geometry, mechanics, and control.

The research has been supported in part by NSF grants DMS-0726263, DMS-0714223 and DMS-0504747.

We proved that the wedge product is associative, and showed that the wedge product of distinct 1-forms (in increasing order) which are dual to the canonical basis yield a basis for the exterior algebra.

We introduced the exterior derivative and show that it is the unique R-linear map that satisfies the Leibniz rule, d^2=0, and agrees with the derivative on 0-forms (functions).

This corresponds to V.6 and V.8 in the textbook.

We introduced the symmetrizing and alternating maps on r-covariant tensors, and show that they are projections, and commute with pullback. Introduced the tensor product, and showed that the tensor product of 1-covariant tensors form a basis for r-covariant tensors.

This corresponds to V.5 and V.6 in the textbook.

We introduced the notion of a partition of unity that is subordinate to a covering, and show that every covering of a differentiable manifold has a subordinate partition of unity. We use this to show that every differentiable manifold has a Riemannian structure. Also, we show that a compact differentiable manifold can be embedded into Euclidean space. We noted the existence of sharper embedding theorems, including the Whitney embedding theorem, and the Nash isometric embedding theorem.

We also started discussing (r,s) tensors as multilinear maps on V^r X (V*)^s, and showed that this a vector space of dimension n^(r+s). Examples of tensors we encountered previously include covectors which are (1,0) tensors, bilinear forms which are (2,0) tensors, and vectors which are (0,1) tensors.

This corresponds to V.4 and V.5 in the textbook.

The following homework was assigned:

p 126, Ex 1
p 127, Ex 9
p 134, Ex 3
p 152, Ex 3
p 159, Ex 2

It is due on Tuesday, November 13, 2007

I have finished grading the exams, and returned them in class on Thursday. If you have not picked up your exam, they are available for collection in the slot in front of my office MATH 430. The distribution was somewhat bimodal, with a concentration around the 90’s and another around the 60’s, as well as a few outliers.

I am happy to discuss your performance in the exam during my office hours on Friday, November 2, from 1:30pm to 3:30pm, during my regularly scheduled office hours, or by appointment.

It might be worthwhile to review the exam solutions to get a sense of what I was expecting:

MATH 562 exam

MATH 562 exam solutions