COMPUTER RESOURCES FOR CALCULUS III

This page contains a selection of resources for Calculus III. They are currently organized by type of sofware, and are being reorganized by topic in the syllabus, as has been done for Calculus I.

Page Contents

  • Maple explorations: A collection of Maple worksheets designed for use with Calculus III.
  • Java Explorations: A collection of Maple worksheets designed for use with Calculus III.

    • Maple Explorations for Calculus III
    The following items were developed for Georgia Tech's Math 1509.

  • Three Dimensional Graphics. Maple can be used to help visualize the shapes of surfaces, as well as for the computation of algorithms. Some tools are suggested in these two worksheets. The first one illustrates the commands with straightforward examples, while the second one develops some graphics tools in a specific context.
  • Projections Onto Lines and Planes. Given a line (or a plane) and a point A not on the line (or plane), how do you find the point on the line (or plane) that is closest to A? The dot product provides a good computational way to do this. The techniques of this worksheet generalize to more complicated situations in many dimensions. In fact, the methods are independent of the dimension.
  • The Volume of a Parallelepiped. The area of a parallelogram and the volume of a parallelepiped can be found using dot-products and cross-products. The worksheet makes clear the ease of finding these geometric quantities without resorting to angle measure. It's all in the arithmetic!
  • Tangents, Normals, and Curvature. Perhaps as much as any other place, the computations for the tangent and normal to a space curve are best accomplished with a computer. The calculus to obtain these geometric notions for even simple functions can be formidable. Because these ideas arise so prominently in the task of resolving both motion and the forces causing motion, it is well that the ideas be understood at the beginnings of a study of multidimensional calculus. We will give computational tools and illustrate the ideas with examples.
  • A Water Whirl. If a cylinder is partially filled with water and whirled about the major axis, the surface of the water changes shapes depending on the speed of the revolution. This worksheet resolves the forces acting on the water into components and determines the shape of the spinning surface.
  • Parabolic and Spherical Mirrors. A classroom project on this subject by Xu-Yan Chen. No software is required to read this link, although you may wish to look at the Maple worksheet on the water whirl, which is an active document with related calculations.
  • Tangent Planes and Normal Lines. This worksheet will provide an explanation for how a tangent plane and a normal line for a surface S can be constructed in case f has continuous partial derivatives in x and y. Graphical illustrations are provided. Also, a pathological example is given to illustrate when the procedure can fail.
  • Surface Area and Pediatric Pharmacology. Medical technologists often decide dosage levels for medication based on the size of the intended recipient. In particular, if the drug is to be administered intravenously, the surface area of the individual is important. This worksheet, provides a correlation between height, weight, and surface area for humans as determined by the commonly used West Nomogram. Additionally, partial derivatives (and the chain-rule) are used to find the rate of change of the surface area for a rapidly growing adolescent.
  • Newton's Method and Newton's Law of Cooling. Real data are given for a warming body. A model is provided for the rate of warming. The task of finding the coefficients for the model requires the solution of a nonlinear equation. For this latter, we use the nonlinear, multidimensional Newton's method for finding a zero of a function.
  • Maximization, with Constraints. A common tool for finding the maximum of a function with constraints is to use the method of Lagrange multipliers. In this worksheet we not only give algebraic tools for solving the equations associated with the method, but also illustrate the geometry that serves as the motivation for the method.
  • Cobweb diagrams It often happens in science, mathematics, and engineering that is on interest to find a fixed point of a function. In this worksheet we investigate how the process of seeking such fixed points through iteration can be visualized geometrically. The iteration process is called the cobweb construction.

    • APPLETS AND PROJECTS FOR CALCULUS III

    Contents

  • APPLETS: The applets in the fast-downloading jar file format
  • APPLETS:The applets in the unarchived format required by Navigator 3
  • EXPERIMENTS AND PROJECTS:A collection of projects and experiments to be done with these applets

    • The applets in jar files
    Use these links if you are using Netscape Navigator 4, or another browser that understands the "archive" tag, and your downloads will be faster.
  • Applet on level sets -- contour lines -- for functions of two variables.

    • The applets in unarchived form
    Use these links if you are using Netscape Navigator 3, or another browser that does not understand the "archive" tag, and the applets will be downloaded the old-fashioned way: one class at a time.
  • Applet on level sets -- contour lines -- for functions of two variables.

    • Projects and experiments for use with these applets
    The following links are to pdf files outlining mathematical experiments designed to be done with these applets. Since several of the applets are on closely related subjects, some of these experiments involve the use of two or more of the applets, while others just use one.

    Section Contents

  • Experiments with contour curves and gradients

  • Experiments with contour curves and gradients

     This project is designed to lead to a deeper understanding of the gradient, and its relation to contour curves and critical points.

    Project resources

  • The questions to be explored and answered.
    • Eric A. Carlen (send message)