Discussion 6: Introductions to Optics

This discussion is an extended application of the material on vectors in Section 1.2 and Section 1.3 to several problems in optics. The goal is to have students use vectors and vector algebra to explore Fermat's principle for reflection and refraction and then to investigate some of the implications of these principles. The discussion is self-contained, all of the necessary material on optics is given in the introduction to the discussion.

The first exercise develops Fermat's principle for reflection, and the second develops Fermat's principle of refraction. In the remaining three exercises students apply these principles to determine the differences between our visual perception of reflected and refracted images of objects and the objects themselves. This discussion can be assigned as an out-of -class collaborative exercise.

startsection section10mm-.5 Chapter 2: Parametric Curves and Vector Fields 

In this chapter, we begin the study of calculus by considering functions of one variable with images in the plane or in space. In addition to presenting these functions, parametrizations and vector fields, and developing the calculus of these functions, this chapter investigates a variety of physical, biological and mathematical systems that can be modelled using these functions and analyzed using the mathematics developed here. In Section 2.1, we begin by introducing parametrizations, functions defined on subsets of ${\bf R}$ with images in ${\bf R}^2$ or ${\bf R}^{3}$, as a way to give a mathematical description of motion in the plane or in space. We then use these functions to model a variety of other physical, biological and mathematical systems which can be described by the interaction of two or three quantities that change over time. In Section 2.2, we use the idea of velocity to motivate the definition of the derivative of a parametrization and then use techniques from one-variable calculus to calculate the derivative and develop its properties. Section 2.3 utilizes the mathematics introduced in Section 2.1 and Section 2.2 to develop detailed models of different biological and physical systems. In particular, we explore a model of an epidemic, a model of a mechanical pump, and a model of the absorption of lead in the human body. In Section 2.4, we use an intuitive understanding of fluid flow to introduce vector fields. We also introduce the concepts of a flow line of a vector field and of a critical point of a vector field, which connects the study of vector fields to parametric curves. These concepts will be used again in Chapter 4 to analyze the gradient vector field of a function of two variables and to give a geometric classification of the critical points of the function. Finally, in Section 2.5 we use the mathematics developed in Section 2.4 to develop a model of a system of interacting populations, a physiological system and a mechanical system. In particular, we develop a model of a predator-prey system, a model for human glucose regulation and a model for the motion of a simple pendulum.

The discussions in this chapter include in-class discussions in which students develop the basic definitions and properties of parametrizations and vector fields, computer laboratories in which the students use the computer to develop a graphical understanding of these functions, and extended assignments in which students explore in depth several mathematical models of physical systems. These models are extensions of the models introduced in Section 2.3 and Section 2.5 of the text.

Discussion 7, Discussion 9, and Discussion 11 are in-class discussions and do not require access to the computer. Discussion 8 and Discussion 12 are in-class computer laboratories which use the computer algebra system Maple. Students should be able to complete most of the work on these discussions in one class period, although they will probably need time outside of class to complete the discussion and a write-up of their results. Discussion 10 is an exploration of a model of the human cardiac cycle which uses parametric curves. Discussion 13 consists of three extended modelling assignments. Each of these assignments consists of two models which build upon models given in the text. These assignments require more extensive computer exploration as well as a more detailed write-up. Each set of models can be assigned as an out-of-class collaborative exercise which would require approximately one week to complete.

The following is a daily schedule for Chapter 2 incorporating discussions, laboratories, lectures and reading assignments.

Day 1

Discussion 7: Parametric Curves in class. Read Section 2.1: Parametric Representations of Curves for homework.

Day 2

A lecture on Section 2.1.

Day 3

Discussion 8: Plotting Parametric Curves Using Maple in the computer laboratory.

Day 4

Lecture on Section 2.2: The Derivative of a Parametrization. Read Section 2.2 for homework.

Day 5

Discussion 9: The Derivative of Parametrization in class. Read Section 2.3: Modelling with Parametric Curves for homework.

Day 6

A lecture on the models from Section 2.3, including the pump model as a prerequisite for Discussion 10: Applications of Parametric Curves.

Day 7

Discussion 10 in class.

Day 8

Brief introduction to vector fields including the definition of a vector field and the definition of flow lines, followed by Discussion 11: Vector Fields in class.

Day 9

Discussion 12: Critical Points of Vector Fields in the lab. Read Section 2.4: Vector Fields for homework.

Day 10

Lecture summarizing Section 2.4 and Discussions 11 and 12. Read Section 2.5: Modelling with Vector Fields for homework.

Day 11

Lecture on Section 2.5. Assign the models in Discussion 13: Vector Fields as Models as extended projects or exercise sets to be done outside of class.

A brief description of each discussion including a statement of the goals of the discussion, a synopsis of the discussion, suggested reading assignments and an indication of which questions or models will be used again in subsequent discussions is given below.