**Math 3593H Honors Mathematics II Spring Semester 2006
**

Differential Forms, 2nd ed., Prentice Hall, 2002.

http://www.math.umn.edu/~webb

This course is a continuation of Math 3592 taught in the Fall semester, and we will study material from Section 2.9 and Chapters 3, 4, 5 and 6 of the book by Hubbard and Hubbard. Part of what we study will have to do with integrals and part with derivatives. At the end of the Fall Semester we used derivatives to find maxima and minima of functions in Chapter 3 and we will start by continuing with this, doing the method of Lagrange multipliers. We will study the implicit and inverse function theorems from Section 2.9 and go on to say what a manifold is at the start of Chapter 3. We will introduce multiple integrals on regions of space in Chapter 4, use them to compute volumes of manifolds in Chapter 5. In Chapter 6 we put all this together and study integrals of differential forms. The main theorem in Chapter 6 is the theorem of Stokes, which has as particular cases theorems of Gauss and Green, and may be regarded as the extension to arbitrary dimensions of the fundamental theorem of calculus. Familiarity with multiple integrals and the theorems which relate to them is essential working with the differential equations of Physics.

In terms of mathematical sophistication this course falls between the other undergraduate courses on multivariable calculus (for example, the IT Honors course) and the graduate level course 'Manifolds and Topology'. We will work with differential forms (not done in other undergraduate courses) but be less complete than the graduate treatment.

It may appear that we have a large part of the book to cover this semester, although in fact the number of pages in Chapters 3 - 6 is not that much more than the number of pages in Chapters 0 - 2. Nevertheless we will need to get through the pages faster than we did in the first semester. I propose to do this in part by omitting certain sections. For instance, in Chapter 4 I anticipate that we will only study sections 0, 1, 5, 8, 9, and 10.

There will be three full-period mid-term exams, to be held on

Assignments will usually be handed out on Wednesday. Some of the problems are to be handed in on Thursday of the following week at the beginning of your recitation period (8 days after it is assigned). Late homework will receive a very reduced grade (no credit for problems already solved in class). If it is handed in after the assignment has been graded, there will be no credit given.

There will be a short quiz at the beginning of most of the Thursday recitation periods covering homework due that day. The first quiz will be on

Students are expected to attend all lectures and recitations. Attendance may be checked and included in the grade line.

In a number of cases in the homework problems and the questions in the exams you will not get full credit if you simply write down the correct answer. To get full credit you will need to write an explanation of how you got your answer. Where explanations need to be given, these should be written out in sentences i.e. with verbs, capital letters at the beginning, periods at the end, etc. and not in an abbreviated form.

I encourage you to form study groups. However everything to be handed in must be written up in your own words. If two students hand in identical assignments, they will both receive no credit.

Computers may not be used on quizzes and exams. Calculators will be allowed on some quizzes and exams.

These will only be given in exceptional circumstances. A student must have satisfactorily completed all but a small portion of the work in the course, have a compelling reason for the incomplete, and must make prior arrangements with the

The University requires the following be on all syllabi.

University Grading Standards

A achievement that is outstanding relative to the level necessary to meet course requirements.

B achievement that is significantly above the level necessary to meet course requirements.

C achievement that meets the course requirements in every respect.

D achievement that is worthy of credit even though it fails to meet fully the course requirements

S The minimal standard for S is to be no lower than C-. The instructor or department must

inform the class of this minimal standard at the beginning of the course.

F (or N) Represents failure (or no credit) and signifies that the work was either (1) completed but

at a level of achievement that is not worthy of credit or (2) was not completed and there was no

agreement between the instructor and the student that the student would be awarded an I.

I (Incomplete) Assigned at the discretion of the instructor when, due to extraordinary

circumstances, e.g. hospitalization, a student is prevented from completing the work of the

course on time. Requires a written agreement between instructor and student.

Academic Dishonesty. Academic dishonesty in any portion of the academic work for a course shall

be grounds for awarding a grade of F or N for the entire course.

Credits and Workload Expectations. For undergraduate courses, one credit is defined as equivalent

to an average of three hours of learning effort per week (over a full semester) necessary for an

average student to achieve an average grade in the course. For example, a student taking a

three credit course that meets for three hours a week should expect to spend an additional six

hours a week on course work outside the classroom.

Date of this version of the schedule: 1/15/2006