| MATH 710: | Special Topics in Applied Mathematics |
| Continuum Mechanics |
Spring 2021 Course information
| Class Time/Place: | MoWe 2:30pm–3:45pm, online |
| Email: | rostamian@umbc.edu |
| Online office hours: | TBA |
Course Description
Continuum mechanics is the study of the relationship between forces and deformations in continuous media. The subject encompasses fluid mechanics, linear and nonlinear elasticity, viscoelasticity, gas dynamics, and many other types of material models.
This presentation of continuum mechanics is aimed at graduate and advanced undergraduate students of mathematics, but it may also be of interest to mathematically inclined students of engineering and physics.
Course contents
A First Course in Continuum Mechanics by Oscar Gonzalez and Andrew M. Stuart is the textbook. I will follow it pretty closely, and occasionally add supplementary materials. Here is a brief summary of the topics:
- Review of the prerequisite mathematics: linear algebra and operator theory
- Tensor algebra, eigenvalues, and the principal invariants
- Kinematics, deformations, the transport theorem
- Balance of mass, momentum and angular momentum
- Stress and strain; Cauchy's theorem on the existence of the stress tensor
- Constitutive equations
- Frame-invariance (objectivity)
- Symmetry, isotropy
- Representation theorems for isotropic tensor functions
- Non-Newtonian fluids; derivation and examples
- The Navier–Stokes equations: derivation and examples
- Nonlinear elasticity: derivation and examples
- Linear elasticity: derivation and examples
This spans the textbook's first 7 chapters. There won't be time for the remaining two chapters on thermal effects.
Prerequisites
- a working knowledge of basic concepts of physics, such as pressure, force, mass, acceleration, and Newton's laws.
- a solid knowledge of undergraduate-level linear algebra including eigenvalues and eigenvectors
- facility with multi-variable calculus
- Math 301 (for undergraduate students enrolling in this course)
Online instruction
This is a fully synchronous remote class which means you will be expected to log in and participate in class sessions at established dates and times for all sessions. The class meets on Blackboard Collaborate on Mondays and Wednesdays between 2:30 and 3:45pm.
Homework and course evaluation
There are no exams in this course. Your work will be evaluated solely based on your performance on homework assignments. I will assign homework problems as we go along. Homework problems assigned during any Monday–Friday week are due on the Wednesday of the next week. I will email the homework assignments to you, and you will email your solutions back to me. Please write your solutions in LaTeX (but don't worry about meticulous formatting if that's too tasking) and include explanations and logical arguments to make it understandable.
Each homework problem will have a designated point value, indicative of the amount of work needed to solve it. You will be credited with a problem's full or partial point value depending on how close it comes to being a correct and complete solution. At the end of the semester I will add up your accumulated point values and compare it against the maximum possible. Your course grade will be A, B, C or D (possibly with ±) if you score greater or equal than 85%, 75%, 65%, 55%, respectively.
Notable Quotes
This paper gives wrong solutions to trivial problems. The basic error, however, is not new.
Clifford A. Truesdell in Mathematical Reviews #12,561a
[Halmos and I] share a philosophy about linear algebra: we think basis-free, we write basis-free, but when the chips are down we close the office door and compute with matrices like fury.
Irving Kaplansky in Paul Halmos: Celebrating 50 Years of Mathematics
Miscellaneous notes
Registrar's Office Dates and Deadlines
The Official UMBC Honors Code
By enrolling in this course, each student assumes the responsibilities of an active participant in UMBC's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong. Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal.
Student Disability Services (SDS)
Services for students with disabilities are provided for all students qualified under the Americans with Disabilities Act of 1990, the ADAA of 2009, and Section 504 of the Rehabilitation Act who request and are eligible for accommodations. The Office of Student Disability Services is the UMBC department designated to coordinate accommodations that would allow for students to have equal access and inclusion in their courses.