As time allows we may also do additional material from 11.6,11.7, 13.8, and 13.10 and from
the notes ``Motion in the plane and in space", ``Average Velocity and Velocity"
Chapter 11 introduces functions of two variables, and studies them from the algebraic, geometric and numerical points of view. The easiest functions to understand are the linear functions. The next simplest are quadratic functions, and you will study these in the first computer lab.
In the first three sections of Chapter 13 you will use the notion of the partial derivative to find the best local linear approximation for a "good" function. This means that functions which are hard to understand can be approximated by linear functions, which are easy to understand. Newton's method for solving systems of non-linear equations is a good example of how useful this technique is. The second computer lab will help you get a feel for how well the linear approximation of f at (a,b) approximates f.
At this point, we need an introduction to vectors to continue with the calculus, so we go back and do Chapter 12. The notes will show you how to use vectors to describe the motion of a point in the plane.
In the rest of Chapter 13, you will study the idea of the gradient of a function. The gradient of a function at a point is a vector which points in the direction in which the function is increasing the fastest, and the magnitude of the gradient is the rate of increase of the function in that direction. The gradient has many important applications. For example, heat flows in the direction opposite to the gradient of the temperature, and water tends to run down a hill opposite to the direction of the gradient of the height. Lab 3 will help you to explore the behavior of the gradient close to critical points of the function.
In Chapter 14 you will learn how to use calculus to find the maximum and minimum values of a function of 2 or more variables, and get some more practice in modeling problems from Engineering, Physics and Economics. The fourth computer lab will give you a lot of experience in recognizing critical points from the pattern of level curves of a function, and a chance to explore the relation between the number of critical points and the domain of a function.
Using the notes, you will continue to explore the connection between the gradient, fluid mechanics, differential equations and conservative forces.
In Chapter 15 you will study integration in 2 and 3 variables; this will be used to find the volume and mass of a solid. Many of you will use the double and triple integral to find the center of mass of a plate or solid in your engineering courses.
As you can see the computer labs are an important part of the course. Most of them use the MAPLE software package. The notes contain enough material on MAPLE to get you started on the first lab. We recommend that you do the labs in the Excel lab which is the computer facility run by the Math department. The Excel Lab is located in 553 Lake Hall. We have experienced mentors working there who know both the lab materials and the software who are ready to help you. If you have any questions about MAPLE or the Labs, feel free to send e-mail to gaff@neu.edu.
Created: December 23, 1996. Last modified: January 5, 2000.