Saint Louis University

Department of Mathematics and Mathematical Computer Science

Applets and Maple Worksheets for Calculus III

(For Maple 10)

At Saint Louis University, we are using graphing calculators as the primary technology incorporated into teaching the first two semesters of our main calculus sequence. The easy arguments for graphing calculators are that they are relatively inexpensive (about the cost of a standard calculus textbook), moderately easy to learn to use, and are in the students' hands so that the students have the same set of tools available in class, for homework, and during quizzes and tests. Hopefully they will understand that the tools are tools for doing mathematics rather than tools for working in mathematics classes. Quite simply, we felt that graphing calculators made the best fit for the needs of the calculus students we have.

Nevertheless, as the students move through the calculus sequence there are topics where a computer is much more effective than a graphing calculator. This is particularly true in multi-variable calculus with the need to visualize in 3 dimensions. Our third semester calculus is being taught in a computer classroom where, among other tools, the students have access to the computer algebra system Maple and to Java applets that we have developed.

Thoughts on Maple

One of the drawbacks of a program as powerful as Maple is that the difficulties of learning Maple and learning to use a computer need to be consciously factored in when planning the course. One pedagogical strategy is to make using Maple a routine part of the course and consistently teach use of the program along with the mathematics. This strategy works best if Maple is to be used heavily in a sequence of courses.

A second strategy (and the one I have used) is to introduce Maple through carefully designed worksheets. Depending on the material I was covering, the worksheets were designed to be used as one of the following:

• An in class lecture aid with the instructor running the worksheet with a projection system.
• A handout to be printed up, copied, and handed out to the students.
• A lab assignment that the class will start together as a substitute for a lecture.
• A supplemental homework assignment that the students are expected to find and do on their own.

Worksheets to be done by the student are set up so that the first time through a student can get through the worksheet by hitting enter repeatedly. These worksheets include a significant amount of exploratory text. The exercises tend to ask to student to repeat the examples from the worksheets with minor modification. I use the template model because I want them to use the power of the Maple to look at problems where I could not expect them to produce the code, but I can expect them to copy and modify a code template, focusing on the results of the problems.

More recent versions of Maple also have a document that can be used for nicer visual presentations.

Thoughts on Applets:

A parallel strategy to the use of computer algebra systems is the introduction of java applets into the course.  The applets I have constructed are designed to be exploratory tools that students can use to look at a single issue in the course.  By the nature of their design they are not as flexible as Maple, but they also are easier to explain to the students.  Most of the applets for Calculus III are local implementations of an applet written at Brown University for the calculus class of Tom Banchoff.  (More precisely the links are to web pages that contain parameter tags that call that applet in a variety of configurations useful for projects and demonstrations.)  The applet is used with permission.  The applet was designed for use with multivariable calculus.  A help page is available for these applets.

The full set of applets developed at SLU is available at the applet page.

Instructional Material:

• Preliminary Material
  
• Graphing functions of several variables
  
• Vectors
  
• Derivatives
  
• Optimization
  
• Integration
  
• Vector Fields, parameterized Curves and Surfaces
  
• Line Integrals
  
• Flux Integrals
  

With that long winded introduction, here are some worksheets I produced for topics in Calculus III. We are using the "Harvard Calculus" book for this course, so the worksheets are organized in line with that book. Most of the material is pretty standard across multivariable calculus texts.

You can download a zipped archive of the Calculus 3 Maple 10 worksheets.

• Preliminary Material
  • Just Enough Maple for worksheets - A worksheet designed to introduce the students to just enough Maple syntax for them to be able to run worksheets.
  • Writing Maple 10 documents - A worksheet to walk the students through some of the new features of the document interface in Maple 10.
    
• Chapter 12 - Graphing functions of several variables
  • Plot With Maple - An introduction to plotting with Maple, reviewing 2D plotting and introducing plotting in 3D.
  • The 3D Grapher Applet page collects 3 applets for graphing in 3-space.  One can either use the Cartesian Grapher to graph z as a function of x and y, or use the Cylindrical Grapher to graph r as a function of theta and z, or use the Spherical Grapher to graph r as a function of theta and phi.
    
  • The Cross Section applet looks at how the graph of a 2 variable function is built up from the cross graphs of the cross sections, which are each graphs of single variable functions.
  • The Level Curve applet builds up the graph from its level sets.  It connects the contours of a three dimensional surface with the contour graph obtained by looking down the z-axis.
    
  • Understanding Limits - is a worksheet on the formal definition of limit for functions of two variables. To simplify graphing, square neighborhood in the domain are used in the definition.
• Chapter 13 - Vectors
  • The Adding Vectors Applet lets you visualize two vectors with their sum and difference and the vectors measured both in polar and rectangular coordinates. The Adding Vectors in 3D Applet is a Banchoff Applet that lets you move the visualization to 3 dimensions.
  • Cross and Dot Products - a fast handout explaining the syntax used for dot and cross product in Maple. The intent of this worksheet is simply to show the students how to use Maple to check their work with vector computations.
  • The Cross Product applet gives a visual approach to cross products of vectors in 3-space with the coordinates of the two vectors controlled by sliders.
    
  • The Projections, Dot and Cross Products Applet lets you specify two vectors by dragging their endpoints.  It then gives bit a visualization as well as numerical data for the projection of one vector on another, their dot product of the two vectors, and the cross product of the two vectors. with their sum and difference and the vectors measured both in polar and rectangular coordinates.
    
  • Visualizing Vectors - is a demonstration worksheet to show the students how to visualize vectors, vector arithmetic, and cross products with Maple. The worksheet does not include exercises.
• Chapter 14 - Derivatives
  • Partial Derivatives - walks the students through the syntax for finding partial derivatives with Maple.
  • The Partial Derivatives applet finds partial derivatives as tangent lines to the curves in slices where x or y is held constant.
    
  • Easy Tangent Planes - is a Maple worksheet that starts with an easy construction of tangent planes. We construct tangent planes to a surface by finding the lines tangent to the paths that fix x and y in turn.
    
  • The  Tangent Planes applet, connects the tangent lines of the x and y cross sections at a point to the tangent plane to the surface at that point.
    
  • Local Linearity - is a Maple worksheet that looks at visualizing the book's definition, that a function is differentiable at a point if the graph near the point is locally approximated by the tangent plane. The definition is used to understand the corresponding delta-epsilon definition.
  • The Multivariable Linear Approximation applet is a for exploring the region where the tangent plane can be used as a good approximation of a function.
    
  • Visual Gradients - is a Maple worksheet that visualizes directional derivatives and connects them to the gradient for differentiable functions.
  • Multivariable Chain rule - is a Maple worksheet that walks the students through the chain rule in several variables, starting with a visualization of the chain rule in a single variable.
    
  • Taylor Series - steps the students through the construction of Taylor polynomials for functions of 2 variables.
  • Animated Taylor Series - Shows an animation of the Taylor polynomial wrapping down to the surface. (This was broken off the other worksheet on this section due to the memory demands of animated 3D-graphics.)
  • Checking Differentiability- Steps the student through the process of checking differentiability of a function at a point.
  • Checking Differentiability, an example -Since the student have such trouble with this section, this is a worked example that looks at checking differentiability.
  • The Polar Functions Applet looks at a collection of functions that are useful in seeing what can go right or wrong when thinking about differentiability of functions in two variables.
    
• Chapter 15 - Optimization
  • The Gradient-Contour Applet uses the gradient field and contours to find local extrema of a function of two variables.
    
  • Rotations - is a worksheet to connect the use of the discriminant with rotations of axes and the elimination of the cross term in polynomials of degree 2.
  • The Constrained Extrema Applet addresses the problem when restricted to a parameterized constraint function.
    
  • Lagrange Multiplier Problem - an extra credit homework worksheet on section 14.3. It shows how to use maple to solve systems of equations.
  • Solving with a Gradient Search - Shows how to mechanize a gradient search to find a solution to a system of equations.
• Chapter 16 - Integration
  • Integration Checker - is a fast note that demonstrates for the students the Maple commands needed to do integration. It seems useful in this chapter where many of the problems reduce to "and finish by evaluating the two or three integrals."
  • Plotting in Other Coordinate Systems - Is another "refresher in Maple commands." It looks at how to plot in coordinate systems other than Cartesian. It also shows how to combine objects described in different coordinate systems on a single plot.
  • Multiple Riemann sums - The worksheet looks at the Riemann sum definition of double integrals. It follows the usual pattern of the course by reviewing the definitions in the one variable case, then generalizing.
  • Double Integrals (Cartesian) - The worksheet looks at visualizing limits of integration in Cartesian coordinates in 2 dimensions and in changing the order of integration.
  • Triple Integrals (Cartesian) - This is similar to the double integral worksheet, but with triple integrals.
  • Extended Triple Integral Example - This is an extended example for setting up a triple integral.
  • Extended Triple Order of Integration Example - This gives an extended example of changing the order of integration for a triple integral.
    
  • Double integrals in Polar Coordinates - This Worksheet was done by the students in class. It looks at setting up integrals in polar coordinates and switching between rectangular and polar coordinates.
• Chapter 17 - Vector Fields, parameterized Curves and Surfaces
  • Parametric Curves - is a Maple worksheet that looks at parameterizing curves in R2 and R3,
  • The Parameterized Curve Applet is set up to look at a parameterized curve in x-y as a space curve in x-y-t space.  Rotating the axes lets you see the parameterized curve as well as the x(t) and y(t) curves in the x-t and y-t planes respectively.
    
  • Parametric Surfaces - This Maple worksheet looks at parameterization of surfaces.
  • The Parameterized Surface Applet lets you see the graph of a parameterized surface.  As we did with the cross section applet above, we can also see a wire frame constructed from the graphs of lines in the domain.
    
  • Planetary motion - This worksheet steps through parameterizing planetary motion under gravity. This is a demonstration worksheet. It solves for planetary motion with a flow line solution of a vector field in 4 dimensions.
  • Vector Fields and Gradient Fields - is a worksheet that has the students look at plotting vector fields and gradient fields in 2 and 3 dimensions.
  • The Vector Fields Applet plots a vector field  in either 2 or 3 variables.
    
  • Plotting Flow Lines - Looks at plotting flow lines for vector fields in 2 and 3 dimensions. It also shows how Euler's method works as a numerical method.
  • The Integral Curves Applet  allows the user to plot integral curves to a vector field.
    
• Chapter 18 - Line Integrals
  • Line Integrals - This Maple worksheet steps the students through the procedure of setting up and evaluating a line integral along a parameterized curve. It is intended as a homework checker.
  • The Line Integral Applet explores a visualization of the line integral of a vector field over a parameterized curve.
    
• Chapter 19 - Flux Integrals
  • Flux Integrals - This Maple worksheet steps the students through the procedure of setting up and evaluating a flux integral through a parameterized surface. It is intended as a homework checker.
  • The Flux Integral Applet explores a visualization of the flux integral of a vector field over a parameterized surface.