Multivariable Calculus in the Lab

A collection of Maple V R3 Worksheets

Used in Math 222 at Cornell, Fall 1994

Copyright Notice

Some Maple Release 7 Versions (and Related) Worksheets are now available here.

This collection of worksheets was written for Maple V R3 on the Macs. It was a first time using these materials, so there are rough edges to be improved upon.

The Mac interface to Maple is a very nice one, although significantly different from the Unix xmaple and Windows interfaces. In particular it supports the pasting of color pictures into worksheets. Also, these worksheets use multi-line input at times, and hence rely on the Mac option that only the enter key sends input to Maple.

The worksheets can be accessed by any of:

Table of Contents
Worksheet Summaries
Maple Command Index

Download Entire Collection (as Math_222.sea.hqx) in Mac Format

Using Maple:
Week 1:
- Surfaces 1
Week 2:
- Limits
Week 3:
Week 4:
Week 5:
- Flow Results
- (Draw Flows Package)
- Zero Div and Zero Curl Movie
- Zero Div and Nonzero Curl Movie
- Nonzero Div and Zero Curl Movie
- Nonzero Div and Curl Movie
Week 6:
Week 7:
Week 8:
Week 9:
Week 10:
- Parameterized Surfaces 2
Week 11:
Week 12:
- Green's Theorem for a Triangle

Worksheet Summaries

Using Maple

3D Plotting:
An introduction to 3 dimensional plotting with Maple. Serves as both a reference for many possible options as well as a collection of suggestions on some especially useful ones.
2D Plotting:
A less extensive introduction to two dimensional plotting.
Mac Interface Intro:
A brief substitute for the 50 page Mac Interface manual. Covers vital topics like the ENTER key and scratchpads.
Basic Maple:
Some basic computational Maple. A substitute for the first chapter of First Leaves.
Maple Data Structures:
A one page listing with examples of Maple data structures. To help people recognize which items are primitives in Maple.
Week 1

Surfaces 1
An introduction to three dimensional graphs. Lots of examples of contours. Includes some cases where computer pictures are misleading. Hyperboloids of one and two sheets with an opportunity to look at sections and what they tell us about graphs.
Week 2

Limits
This worksheet uses a variety of graphical and numeric methods to study multi-dimensional limits. It encourages people to continue thinking about the graphical elements of functions of several variables. Much attention is placed on examples where limits fail to exist. Looking at the graph of a function over a small domain is perhaps the most naive strategy introduced. Techniques such as restriction to families of lines and curves (algebraically, by 2D graphs or by animations) are also used. Subtleties involving loss of precision due to catastrophic cancellation also arise.
Week 3

Linear Approximation:
Some examples of tangent lines and planes graphically. Efficient generation in Maple of the linear approximation of a function at a point. Use of the transform plots package to geometrically compare the mapping properties of a function with those of its linear approximation.
Non Differentiable Surfaces:
This worksheet looks geometrically at three different kinds of singularities. A high point is the sketch of an argument at the end about why the cube root of (x^3-3xy^2) fails to be differentiable at the origin. A good opportunity to talk about the idea of what it means for an approximation to be linear.
Gradplot Examples:
Examples of using gradplot to plot the gradient vector field associated to a function. Relation to level sets and the graph of the function discussed.
Transform Plots Package:
A collection of locally written Maple procedures to graphically map both general two dimensional plot structures and rectangles specifically by a function from R^2 to R^2.
Using Transform Plots
Instructions for using the transform plots package. Efficient generation in Maple of the linear approximation at a point of a transformation from R^2 to R^2. Comparison of the mapping properties of a transformation with those of its derivative.
Week 4

Curves:
This worksheet draws some curves and Frenet frames. It shows how readily differentiable curves can develop self-intersections and corners. The Frenet frame package is illustrated and used to draw some Frenet frames for a helix.
Vector Identities:
A real "kitchen sink" worksheet. Covers both major Maple usage issues and mathematical content ones. Most worksheeets prior to this one were geometric in focus. This one pointed out in passing the power of the algebraic paarts of the system and attempted to use them non-trivially. In the usage category here are partial derivatives, vector differential operators, and the important map function. Three vector identities are proved; one for scalar triple product and two for the divergence and curl of cross products. The transformation to spherical coordinates is then studied and the natural orthonormal moving frame produced. Finally these are used to compute a formula for gradient in spherical coordinates.
Numerical Solution:
Illustrates the use of the Maple command fsolve for numerically solving algebraic and transcendental equations or systems of equations. Mostly a "how to do it in Maple" worksheet. Covers one variable, multi-variable, roots in specified rectangles, and complex solutions.
Frenet Frame Display Package:
A fairly simple minded set of routines allowing the simultaneous display of Frenet frames at several points along a parameterized curve.
Week 5

Flow Results:
The major mathematical focus here is to illuminate the relationship of curl and divergence to geometric properties of the flow of a vector field. The worksheet also shows how to compute numerical solutions to systems of differential equations, use the draw flows package, and piece the results together into a quicktime movie. Linear vector fields in the plane are most of the examples here. One nonlinear vector field is also contrasted with its linearization.
Fieldplot and Numerical ODE's
While specialized programs such as MacMath and DsTool do a better job at integrating systems of differential equations, there are advantages to the integrated environment of Maple. This worksheet shows how to numerically integrate an initial value problem, and graph the solutions (two coordinates at a time) to a system of differential equations. It also shows how to use Maple to draw gradient fields and slope fields of planar vector fields.
Draw Flows Package:
This package aids in drawing the flow of a rectangle for a planar vector field.
Nonzero Div and Curl Movie:
A generic case (5x + 15y, -10x + 5y) where both expansion and rotation are visible.
Nonzero Div and Zero Curl:
The vector field (5x + 10y, 10x + 5y). A shear transformation together with an expansion is visible.
Zero Div and Nonzero Curl:
The vector field (10y, -10x). Rotation without change of size is readily visible.
Zero Div and Zero Curl Movie:
The vector field (5x + 10y, 10x - 5y). Hear the rectangle undergoes a sheear transformation but remains unchanged in size.
Some Vector Fields to Look at in MacMath:
A list of vector fields for the students to quickly explore with MacMath. The students did this for about 5 or 10 minutes during the lecture. (MacMath is of course much better at generating phase plane pictures than Maple.)
Week 6

Least squares:
Uses least squares to fit data in the file "Plane Data 0 " to a plane z = a x + b y + c. Readily adaptable as a template for other least squares problems. Also covers the syntax point of reading data in from an external file.
Taylor:
Explores multivariate Taylor series. Compares graphically various Taylor polynomials to the original function. Investigates how errors depend upon the size of the region on which the approximation is being used. Uses estimates on the size of partial derivatives to bound errors in Taylor polynomial approximation. (These latter are obtained from Maple graphs of the partial derivatives.)
Quadrics and Eigenvalues:
This worksheet explores the relationship between symmetric matrices and quadratic forms. It shows how the eigenvalues of such a matrix relate to the geometric character of the graph of the quadratic form. It also discusses in the context of an example how the eigenvectors of the symmetric matrix determine a rotation of coordinates making the quadratic form diagonal.
Quadrics Worksheet:
A worksheet intended for people to use as a reinforcement on the correlation between eigenvalue signs and the graphs of quadratic forms. Helping with max-min as well.
Week 7

Implicit Function Example:
This example studies the function x = h(y,z) defined implicitly by z = x^3 - x y. Singular behavior is viewed geometrically when the hypotheses of the implicit function theorem fail to apply.
Lagrange Multipliers:
The geometry of Lagrange multipliers is explored in the context of the optimization problem for y e^x on an ellipse. Solutions are also obtained numerically using fsolve.
Lagrange Numerical Example:
Extrema for a quadratic form are sought numerically along the intersection of an ellipsoid with a hyperboloid. Intended to show people how Maple can support a generic Lagrange multiplier problem numerically.
Singular Inverses:
This worksheet studies geometric behavior near singular points for a mapping of the plane to itself. Relationships with the problem of numerically solving for the inverse are discussed . The collpase of areas near singular points is brought out by the Transform Plots package. And the development of a cusp as the image of a smooth curve is analyzed in detail within the context of an example.
Week 8

Random Riemann Sums:
Shows how such can be computed for double integrals. Behavior with respect to mesh diameter looked at. Variation among sums correlated with the size of the gradient of the function.
old_plot_region_4:
The original routine for displaying regions of integration in the plane. Can be used as a check on conventional techniques. This version draws Maple curves to describe the region. To improve performance in exchange for lower quality, this is to be replaced by a polygon based version like plot_region_3D.
Riemann Sums Concise:
A concise set of commands for calculating a Riemann sum.
Setup Example:
A double integral for a volume is setup. Maple's numeric and graphical capabilities are used in fairly essential ways.
Using Plot Region:
This worksheet shows how to use the function old_plot_region_4 for displaying an elementary region of integration in the plane.
Week 9

Circular Cylinder and Planes:
An introductory worksheet using the 3D Region Plot Package and other Maple tools to work up the description as an elementary region of a volume bounded by a circular cylinder and several planes.
Parabolic Cylinder and Planes:
A worksheet using the 3D Region Plot Package and other Maple tools to work up the description as an elementary region of a volume bounded by a parabolic cylinder and several planes.
3D Region Plot Package:
A package to show how elementary 3D regions of integration appear. In the interests of better performance, a polyhedral approximation is displayed. Default is a fairly crude but quick picture.One can speciify higher resolution if one wants to improve the picture. Once one has generated the graphics structure for the entire volume, one can quickly inspect the "pieces" joining together to assemble it. The package is based on a natural map from the unit cube to an elementary region.
Using plot3d_region:
A simple example illustrating the use of the 3D Region Plot Package.
Week 10

Parameterized Surfaces 2:
A variety of parameterization and reparameterization examples are presented. Many are quadric surfaces. Singularities of parameterizations are discussed. The worksheet also presents the natural frame (T_u, T_v, N) associated with a parameterization.
Line Integrals
Line integrals of planar vector fields along piecewise polygonal curves are explored. There are routines both for numerical computation and display of paths and vector fields. The example of the gradient of the polar coordinate angle function is discussed, as is its connection with rotation numbers of curves.
Week 11

Using num_int_2d:
Use of the numerical integration package is illustrated and applied to the surface integral of a vector field.
2D Numerical Integration Package:
This package calculates double integrals numerically by using the trapezoidal rule in each direction.
Orientation 2:
The Moebius strip is generated as a parameterized surface and its non-orientability explored. A variation with two twists is also generated.
Week 12

Green For Triangle:
This worksheet does a variety of things related to Green's theorem for an infinitesimal triangle. Using Taylor series, it shows how to calculate the line integral of a vector field over a line segment to second order. Part of the interest here is noting how easily the method generalizes to produce higher order formulas. The result is also used to show that Green's theorem holds to second order for an infinitesimal triangle. Because of the naturality of approximately subdividing fairly arbritrary regions into infinitesimal triangles, an argument is briefly indicated by which one could use this result to give another proof of Green's theorem.

Handouts

Handout 1: Exploration of some quadric surfaces. Level curves.
Handout 2: Multivariable limits.
Handout 3: Linear approximation of mappings from a plane to itself, level curves, and gradient vector fields.
Handout 4: Velocity and acceleration of curves, geometry of reflection.
Handout 5: Phase plane sketches. Linear approximation. Relation to div and curl.
Handout 6: Least squares, approximation by Taylor series, error estimates., quadric surfaces worksheet.
Handout 7: Exploration of relation of local extrema of functions to flowlines of their gradient vector fields.Classification of equilibria.
Handout 8: Plots of three dimensional regions associated with triple integrals.
Projects for Math 222: A launching point for student projects involving computer use.

Topics and Maple Command Index

animation in 2D

Acknowledgement

Many thanks to Mathlab consultant Sung Bae Lee for helping to produce this World Wide Web version.