College of Natural Sciences
Please direct all questions about the flag proposal process to the Center for the Skills & Experience Flags.
M 374M Mathematical Modeling in Science and Engineering
Department of Mathematics
Please give a brief description of the course. Include a description of the specific Quantitative Reasoning skills that students will learn and apply within this course.
The goal of the course is to develop tools for studying differential equation models that arise in applications, and to illustrate how the derivation and analysis of models can be used to gain insight and make predictions about physical systems. Emphasis is placed on examples and case studies, and a broad range of applications from the engineering and physical sciences are considered. As they progress through the course, students will learn the central questions and apply the main tools from four different areas of applied mathematics: dimensional analysis and scaling, dynamical systems, perturbation methods, and calculus of variations. For instance, in the area of dynamical systems, the central question is focused on characterizing the ultimate fate of a system that evolves in time, and on characterizing how this fate depends on the system parameters and initial state. Correspondingly, the main tools of dynamical systems are a set of tools that can be used to gain insight on this central question. For each of the four areas listed above, students will be expected to
- analyze a mathematical model of a physical system using a combination of graphical, analytical and computational tools
- draw conclusions from their analyses and discuss them within the context of the model assumptions
- characterize how the conclusions may depend on different parameters in the model.
Courses that carry the Quantitative Reasoning Flag must emphasize how QR skills can be applied in students’ everyday or professional lives. Please describe the kinds of applications the course uses to teach Quantitative Reasoning. Specific examples from assignments or exams are strongly encouraged.
Students will analyze a variety of different problems arising in the physical and engineering sciences. Highlights include: — study of optimal shape/isoperimetric problems; with activities using hanging cables and necklaces. — study of geodesic/shortest path problems; with activities using strings on cylinders, cones and spheres. — study of minimal surface problems; with activities using soap films. — study of a meniscus curve at a liquid-gas interface; illustration with water-air. — study of the orbit of Mercury using Newton’s equations with relativistic corrections; illustrate how corrections lead to precession of orbit. — study of a two-dimensional ballistic targeting problem; illustrate effect of gravity and air resistance on aiming angle and impact speed. — study of a three-dimensional spinning rigid body; with experiments showing stable and unstable behavior of a free-spinning block. — study of biochemical glycolysis model; illustration of equilibria, limit cycles and bifurcation in a two-dimensional non-linear system. — study of a two-person relationship dynamics model; illustration of phase portraits in a two-dimensional linear system. — study of a plant-herbivore ecosystem model; illustration of stability and bifurcation of equilibria in a one-dimensional system. — analysis of a chemical reactor model; illustration of dimensionless parameters and scaling. — analysis of a non-linear pendulum model; discovery of scaling law for period.
Courses that carry the Quantitative Reasoning Flag go beyond a superficial application of equations and strive for understanding of the underlying concepts. Please describe how you teach and assess conceptual understanding of Quantitative Reasoning. Specific examples from class, assignments, or exams are strongly encouraged.
The course has weekly homeworks, two midterm exams, and a final exam, all of which are focused on the analysis of models. Homeworks contain a mix of analytical and Matlab assignments, the goal of which is to further illustrate the core concepts of the course and expose students to applications beyond the case studies considered in the classroom. The highlight of each homework is a mini-project in which the student is expected to perform an in-depth analysis, draw and discuss conclusions within the context of model assumptions, and characterize how results may depend on certain parameters in the model. In some cases, when the subject of the mini-project pertains to a common physical system, the student can confirm their results and predictions by direct experiment.
To satisfy the Quantitative Reasoning Flag, at least half of the course grade must be based on the use of quantitative skills. Please describe the course grading scheme in such a way that clearly demonstrates at least half of the grade requires Quantitative Reasoning. Denote which components require Quantitative Reasoning and the total grade percentage these comprise.
The course grade is determined completely by the homeworks, midterm exams and final exam, and hence is based entirely on a student’s ability to analyze mathematical models using a combination of graphical, analytical and computational tools, and effectively communicate their results.