plotfield and DEplot to
plot the vector fields and their flow lines. For each model, students
are expected to use the computer to plot the relevant vector fields
and flow lines, to analyze the critical points of the system and then
to interpret the flow lines and critical point analysis in terms of
the physical system being modelled.
The first set of models consists of two physical models, the damped pendulum and the bead on the wire. This assignment builds on the simple pendulum model developed the text. In the first part, the students are asked to analyze the motion of a linearly damped pendulum for varying initial conditions and to analyze how the motion depends on the magnitude of the damping coefficient. In the second part, the students are asked to analyze the motion of a bead on a rotating wire subject to the force of gravity. In particular, they are asked to consider the effect of increasing the initial angular velocity of the rotating wire on the bead both for a frictionless bead and for a bead that is also subject to a linear resistive force. This model is of interest because the number of critical points depends on the angular velocity of the wire.
The second set consists of two population models, a predator-prey model and a competition model. This assignment builds on the predator-prey model developed in Section 2.5 of the text. In the first part, the students are introduced to the classical predator-prey model of Volterra as it was applied originally to study the populations of selachians and fish in the Mediterranean Sea during World War I. They are asked to consider the effect of adding an external factor, fishing, at various times during the population cycle. In the second part, the students are introduced to a model for two populations which compete for a single resource. They are asked to analyze this model and comment on the ecological principle of competitive exclusion.
The third set consists of two models for the spread of an epidemic, an SIR model and an SI model with reinfection. This builds on the SIR model developed in Section 2.3 of the text. In the first part, the students are asked to analyze the basic SIR model and determine under what initial conditions an infection will evolve into an epidemic. In the second part, the infection is not assumed to confer immunity from reinfection, so that there are only two populations, the susceptibles and the infecteds, and it is no longer assumed that the total population is constant. The students are asked to analyze this model and compare the spread of infections governed by this model to infections governed by the SIR model.
startsection section10mm-.5 Chapter 3: Differentiation of Real-Valued Functions
This chapter introduces real-valued functions of two and three
variables and develops a number of concepts related to the
differentiability of these functions. In Section 3.1 of the text we
begin by developing an intuitive understanding of these functions.
This section focuses on the different ways to represent functions of
two or three variables and provides a number of examples of functions
that occur in the sciences. In Section 3.2 we introduce the concept
of the directional derivative and in Section 3.3 we develop partial
derivatives. In Section 3.4 we discuss limits and continuity for
functions of two variables, which are necessary in order to establish
the notion of differentiability for functions of several variables.
In Section 3.5 we introduce differentiability through the concept of
local linearity and use the language of the gradient vector to restate
the ideas of directional derivatives and the chain rule. In the next
chapter we will use the gradient vector field to establish a geometric
classification of the critical points of a function of two variables.
The discussions include in-class discussions to interpret the contour
plot of a function and to motivate and use the concept of the
directional derivative, as well as computer laboratories to enhance
students graphical intuition and to develop a graphical understanding
of local linearity in two variables. Discussion 14 and Discussion 17
are in-class discussions and do not require access to the computer.
Discussion 15 and Discussion 18 are in-class computer laboratories.
Discussion 16 is an exploratory exercise using the computer which can
be assigned as an out-of -class collaborative discussion.
The following is a daily schedule for Chapter 3 incorporating discussions, laboratories, lectures and reading assignments.
- Day 1
- A lecture on Section 3.1: Representation and Graphical Analysis of Functions, introducing real-valued functions of several variables, emphasizing the different representations. Read Section 3.1 for homework.
- Day 2
- Discussion 14: Contour Plots in class.
- Day 3
- Discussion 15: Plotting Functions of Two Variables in Maple in lab. Assign Discussion 16: Analyzing Functions Using Maple to be done outside of class.
- Day 4
- Discussion 17: Rates of Change in class. Read Section 3.2: Directional Derivatives for homework.
- Day 5
- Lecture summarizing Discussion 17 and Section 3.2. Read Section 3.3: Partial Derivatives for homework.
- Day 6
- Lecture on Section 3.3. Read Section 3.4: Limits and Continuity for homework.
- Day 7
- Lecture on Section 3.4.
- Day 8
- Discussion 18: Local Linearity of Functions of Two Variables in lab. Read Section 3.5: Differentiable Functions for homework.
- Day 9
- Lecture summarizing Discussion 18 and Section 3.5.
- Day 10
- Lecture on the gradient vector.
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.