Syllabus: MTH 324, Spring 2016

Vector Calculus (via differential forms)

This elective course builds on student’s knowledge of multivariable calculus and introduces calculus on manifolds. The goal is to acquire facility with calculations using differential forms and integrals over manifolds in any dimension. The focus is not on proof writing, but rather on calculational techniques.

The official prerequisites are MTH 254 and MTH 261. Please do not take this course if

• you struggled with calculus, or
• you have no patience for abstraction.

In the first few lectures, the students will make the transition, in notation and concepts, from earlier multivariable calculus courses like MTH 254 to those continually used in this course. Then, viewing curves, surfaces, and solids as instances of general manifolds, we generalize the integrals over curves and surfaces to integrals of \(k\)-forms over \(k\)-manifolds in \(n\)-dimensions. Applications, discussed sparsely, include interpretation of work done against a force, flux in fluid flows, and electromagnetics in the language of forms.

1. Integration in \(n\)-dimensions: review of determinants, Jacobian, \(n\)-dimensional change of variable formula, volume of balls, simplices, and other useful \(n\)-dimensional objects, \(k\)-parallelograms in n-dimension.
2. Abstract vector spaces: change of basis matrices, orientation of finite-dimensional vector spaces, computing dual bases, multilinear forms, vector spaces of derivations and differentials, classical and modern notions of covariance and contravariance, tensors with and without coordinates.
3. Differential \(k\)-forms: multilinearity and antisymmetry, shuffles and permutations, exterior product of forms, elementary forms, geometric interpretation of forms, forms representing work, mass and flux.
4. Manifolds: local, implicit, explicit and parametric representations, inverse and implicit function theorems, tangent spaces, orientation of manifolds, orienting curves by tangents, orienting surfaces by normal vector, orientation preserving parameterizations.
5. Integration of forms over oriented manifolds: vector fields and form fields, paving a manifold using infinitesimal \(k\)-parallelograms, using parameterizations to calculate integrals, manifolds with boundary, inheriting orientation on boundary.
6. Exterior derivative: grad, div, curl, the generalized Stokes theorem, the particular cases that reduce to integral theorems of calculus.

Lectures will be followed by problem solving sessions. Students are expected to present solutions to homework problems in the problem solving sessions.

Textbook: Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by John Hubbard and Barbara Burke Hubbard, 5th Edition, September 2015.

Note that the learning material will not come a single text, hence it is important to follow the class activities closely. Students missing classes will not be able to catch up using the textbook alone.

• 10% of the grades are determined by class participation in the problem solving sessions.
• 90% of the grades are determined by 5 quizzes in class during these dates:
  1. April 7
  2. April 26
  3. May 12
  4. May 26
  5. June 2

There is no make-up for missed quizzes. However, the lowest scoring quiz will be dropped. Pieter Vandenberge (piet2@pdx.edu) will serve as grader and all questions on grading should be addressed to him first.