Matthias Kawski Dept. of Mathematics and Statistics Arizona State University
MAPLE resources Back to technology page

This work has been partially supported by the National Science Foundation through several grants that include EEC 92-21460, DUE 97-52453, and DMS 00-72369.

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Getting MAPLE


MAPLE Primes: Help, users group, archives...
MAPLE User group - archive


MAPLE runs in virtually every public access computer lab at ASU.
Some of the best choices are the math dept's undergrad computer labs in ECA 221/225.
Look for very affordable student versions in the university 
bookstore/computer store in the Computing Commons.

0. Tutorials 0. Tutorials Contents

interactive.mws
r8, 2005

tutorials.mws	Contents.
mk01.mws r5, 1998	File management, navigating a worksheet (no math).
mk02.mws r5, 1998	Worksheet formatting
mk03.mws r5, 1998	Exact arithmetic, algebra, symbolic calculations
mk04.mws r5, 1998	Basic data structures, pure functions
mk05.mws r5, 1998	Plotting
mk06.mws r5, 1998	Reading and writing data
mk07.mws r5, 1998	Calculus and the student package
mk08.mws r5, 1998	Basic programming
getstart.ms r3, 1995 ?	Basic programming
Code fragment to demonstrate interactive input from keyboard.

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1. Calculus 1.a. From Brief Calculus, MAT 210 Contents

directory listing	Contents.
cubic51.mws r5, 1999	Description coming sometime ....
improper.mws r5, 1999	Description coming sometime ....
logistic.mws r8 2003 (r5, 1999)	Uncommented simple commands (for demo in an an otherwise CAS-free class) for calculating the inflection point of a genral logistic curve y=a/(1+b*exp(-c*x)) via straightforward differential calculus (as opposed to arguing via symmetry, rewrite as tanh).
logistica.mws r5, 1999	Description coming sometime ....
sec_t1_4.mws r5, 1999	Description coming sometime ....

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1.b. From Calculus I and II, MAT 270 and 271
Contents

directory listing	Contents.
03-10-270-diff.mws 03-10-270-diff.html r8, 2003	Sample syntax for entering simple formulas into MAPLE and using MAPLE to check one's paper&pencil derivatives. Plenty of common pitfalls, and different appearances... elementary, but important. Worksheet 03-10-270-diff.pdf 03-10-270-diff.ps.
usederiv.mws r5, 1999	Using the 1st and 2nd derivative to obtain info about monotonicity, local extrema, convexity/concavity, inflection points.
ecg.mws r5, 1999	Description coming sometime ....
freefall.mws r5, 1999	Sample solution for time it takes for a free falling body to hit the ground. E.g. work w/ R-sums and error estimates.
trapbox.mws r5, 1999	Procedure for drawing "trapezoidal boxes"; examples.
frommoon.mws r5, 1999	How much of the Earth do you see from the shuttle?
shuttle.mws r5, 1999	How much of the Earth do you see from the shuttle?
literally.mws r5, 1999	Description coming sometime ....
taylorcos.mws r5, 1999	Taylor approximations: Intro, some syntax, animations.
convtaylor.mws r5, 1999	Taylor approximations: Explorations of convergence.
sinearc.mws r5, 1999	Taylor approximations at work for arc-length of a parameterized family of curves. Contrast with numerical simulations. Very explorarory in character -- open-ended!

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1.c. From Calculus II, MAT 271
Contents

directory listing	Contents.
CDholes.mws r5, 1999	Description coming sometime ....
Si_converg.mws r5, 1999	Description coming sometime ....
cycloid.mws r5, 1999	Description coming sometime ....
harmonics.mws r5, 1999	Description coming sometime ....
sigmas.mws r5, 1999	Description coming sometime ....

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2. Ordinary Differential Equations 2.a. Ordinary Differential Equations Contents

directory listing	Contents.
us-pop.mws r8, 2005 (r3, 1995)	Graphical and numerical exploration of logistic model and time-varying linear model for US-population. Motivated by model from Lomen/Lovelock book early 1990s. More on this in my MATLAB pages.
slopefields.mws fieldplots.mws r8, 2005	Templates to draw lots of slope fields for use in class/tests.
picard.mws r8, 2005	Illustration of Picard iteration. Fairly raw worksheet, step by step. With animations. See also more advanced worksheets in sections 6.b and 7.e below.
euler3.mws euler3.html r8, 2005 (r5, 1999)	Picture (and code for it) to suggest that Euler's method produces underestimates if solution curve is convex (concave up). (uncommented). ( euler3.html: html/gif version).
lotka-rk.mws r8, 2005	Introductory discussion of numerical solutions of DEs in the test-case of the Lotka Volterra system. Comparison of Euler, Heun=RK2, and Runge Kutta 4th order methods. Since MAPLE since to revert automatically to default step sizes once it determines that the chosen steps are too big, this worksheet also includes "hand-written" code to implement guaranteed fixed size steps, suitable for comparisons. In this system RK2 performs even much better than RK4. ???
pendulum.mws pendulum0.mws pendulum.html r8, 2005 (r5, 1999)	Phase-portrait of simple pendulum. Compare Euler's method and advanced algorithms. Focus on whether numericalsolutions are nearly periodic, both near equilibrium, and when going over top. Animations of a simple pendulum and real-time animation of solution curves. pendulum.mws w/ live animation 750kB, pendulum0.mws no output,

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2.b. Ordinary Differential Equations
Contents

directory listing	Contents.
dsolve.mws (r5, 1999)	As the DEtools package and with it the dsolve command have recently become much more sophisticated, the help pages, too, have become much harder to read. Here are some no-frills samples for the first time user. HTML-version.
secondorderlinear.mws secondorderlinear.html r8, 2005	Introductory code for second order linear constant coefficient homogeneous ODEs. Both quick code for beginners, and trying to make MAPLE work solution step by step. Background discussion of complex-exp versus real-trig soltuions in underdamped case. Animations to see effect on solutions of varying damping coefficient, incl. root-locus plot animations.
sysDE.mws r8, 2005	Introduction to solving systems of linear const coefficient ODEs -- special feature are highly nested worksheet -- using any level of high-level commands that do much in one step, or allowing the reader to zoom in into high level of detail of what is being done.
ex3dim.mws r8, 2005	Back-up for a paper-pencil calculation -- with many many double checks, and lots of playing with (advanced) MAPLE code..... Nice grafix of slightly unstable spiral inside a stable (attracting) plane.
varpar-exa.mws r8, 2005	Uncommented worked text book exercise y''-2y'+y=exp(t)/(1+t^2) Variation of parameters. Step by step, and using compact formula.
LT-antideriv.mws r8, 2007	Uncommented quick commands for inverse transforms of transforms of antiderivatives, moment integrals, and when init cond are all zero.
stepLT.mws r8, 2005	Uncommented worked text book exercises Laplace trafo and Heaviside function.
undet-coeff.mws undet-coeff.html r8, 2005	Introductory code developing the method of undetermined coefficients for nhomogeneous second order linear constant coefficient ODEs through a systematic, hands-on experimenatl approach that emphasizes "differential operator" thinking and again and again shows the importance of linearity. Treats all major cases of combinations of polynomial, sinusoidal, and exponetial right hand sides, inclusing terms that are in the kernel of the differential operator.
oscill.mws (r5, 1999)	A primer to symbolic solutions of 2nd order (SYMBOLIC!) const coeff linear DE's with forcing. Includes brief exploration of nonlinear DE y''+sin(y)=0 and preview of how to utilize Fourier approximations for general periodic (here: triangle wave) forcing terms. (Kawski, April 1999). HTML-version.
forced.mws r8, 2005 (r5, 1999)	Explorations of forced 2nd order (linear, const coeff) diff equns. Sinusoidal forcing, resonance, numeric soln and Fourier approx in case of nonsinusoidal periodic forcing. (Kawski, April 1999, runs well after minor changes in r8, 2005) HTML-version.
convolution.mws convolution.html r8, 2007	Many animations relating to convolutions and impulses in the context of 2nd order linear time-invaraint forced DEs. The main focus is on equating the effect of forcing with summing lots of unforced DEs each starting at a different time, but w/ new init conditions that correspond to an impulse at that time. Main purpose: Use for dynamic visualization as backdrop for classroom presentation/discussion.
coldpills.mws coldpills.html r8, 2007	First order linear DE with periodic impulsive forcing -- interpreted as taking cold-medicine. Exploring how MAPLE's laplace/invlaplace handle (or do not handle) infinite series, then pragmatic work around. Final plots show accumulation blanced by metabolization, and briefly look at results of doubling the dose or changing the frequency.
series.mws r8, 2005	First attempts of series solutions: y''+y=0 and Legendre eqn. Plots of successive approximations.
firstBessel.mws r8, 2005	Raw unfinished exploration: Bessel DE as just another linear 2nd order DE. MAPLE "knows" the solutions. First attempts of series solutions. Plots of successive approximations.

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2.c Ordinary Differential Equations
Contents

directory listing	Contents.
complex.mws	General soln of const coeff linear 2nd order DE w/ symbolic paramters. Evaluate soln in case of real or complex roots, and plot.
de-1-2ex.mws r8, 2005 (r3, 1995)	An integrating factor example -- grafically: DEplot, dfieldplot
de-6-1lt.ms r8, 2005 (r3, 1995)	Comparison of using Laplace transform on one second order linear const coeff DE, or on one time-invariant system of two first order DEs. Includes some nice plots.
de-hw7.mws r8, 2005 (r3, 1995)	Many worked exercises with Laplace transforms, incl impulsively forced DEs. Original use as back-up comparison w/ paper&pencil work, to practice MAPLE syntax, and to see some plots.
de-hw8.mws r8, 2005 (r3, 1995)	Many worked exercises with Laplace transforms, inclystems of two 1st order DEs. Original use as back-up comparison w/ paper&pencil work, to practice MAPLE syntax, and to see some plots.
de-hw9.mws r8, 2005 (r3, 1995)	Many worked exercises with matrix exponentials, incl numerous phase protraits. Original use as back-up comparison w/ paper&pencil work, to practice MAPLE syntax, and to see some plots.
de-lin36.mws r8, 2005 (r3, 1995)	A quick procedure that computes, separates, and plots/overlays the contributions of initial conditions and of forcing, nice pix of transients plus steady state yields whole solution.
de-t2sol.mws r8, 2005 (r3, 1995)	Sample calculations in small steps and plots. Sinusiodally driven const coeff linear second order IVPs
de-t3alt.mws r8, 2005 (r3, 1995)	One example of undetermined coefficients "worked by hand" -- lots of Fourier terms, uses concatenation operator to define variables a||k.
de-t3sol.mws r8, 2005 (r3, 1995)	Sample solutions to 3rd test. Focus on input-output view of second order linear constant coefficient DEs. includes first pic of "low pass" filter.
fc-final.mws r8, 2005 (r3, 1995)	Sample solutions of a final exam given in the integrated program of the Foundation Coalition. These problems come from the EEE circuits class. In 1995, EEE was not all familiar with the notion of computer algebra systems (as opposed to numerical programs).
de-final.mws r8, 2005 (r3, 1995)	Sample solutions to final exam. Includes comparison between pendulum and its linearization (nice pics comparing damped responses to various init velocities ) and linearity of flow of linear second order DE.
fundmtrx.mws r8, 2005 (r3, 1995)	Introduction to fundamental matrix solution and matrix exponential. Very old -- 1995.
hw-361a.mws r8, 2005 (r3, 1995)	Explorations of fitting together pieces of solutions of the inhom DE y''+y=1/cos(t). Many plots of various solutions on subintervals, discussion of the singularities. Apparently MAPLE r3 had more trouble w/ branch-cuts of the logarithm....

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2.d. Ordinary Differential Equations
Contents

directory listing	Contents.
Lambert.mws Lambert.html r8, 2005	Introduction to the two real branches of the Lambert function. Graphical images that clearly explain domains and ranges. Application to predator prey system -- both branches are needed.
dance.mws r5 ?, about 1997 ?	Animating time-varying / parameter dependent vector fields? A very simple 3 line trial.
mathieu.mws r5 ?, about 1997 ?	Time-varying vector fields visualized as a bundle of planar vector fields stacked on top of each other. Overlay the appropriately colored solution curve(s). Application: Even if for each fixed time, the "frozen" system is stable, the time-varying system may still be unstable. (trouble with old, release 3 DE-commands -- needs clean-up).

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3. Linear Algebra 3. Linear Algebra a For most of my linear algebra classes I have been using MATLAB, compare my commented MATLAB index. The few worksheets below are either instructional aides, or come e.g. from advanced classes that provide a simnple linear algebra review to provide examples for an intro to MAPLE.

directory listing
fruit.mws r5, about 1999.	Used as an intro to matrix multiplication. Generate individual assignments so that each student gets her/his own mix of four fruit (and number of servings). Together with a nutrient chart, the 1st task is to calculate the total percentage of the RDA covered by the salad. The 2nd exercise is about expressing (100%,100%,100%) as a (positive?) linear combination of the fruit....
det.mws det.html r8, 2004 (r6, 2001)	Seeing the formula for the determinant arise in denominators of the formulae for solution of linear systems (or inverse of matrices). Useful for quick demo after students have worked the 2 x 2 system by hand.
jordan.mws r5, about 1999.	Some aides to generate matrices that guarantee that the (manual) row reduction to rref will involve e.g. only small integers (used to cook up problems for mastery tests).
realnearJordan.mws realnearJordan.html r8, 2006	Should there be ONE or TWO ones above each 2 x 2 block in nontrivial real almost Jordan forms in the presence of repeated nonreal eigenvalues? Are the respective different normal forms similar? Orthogonally similar?
phx.mws r5, about 1999.	Generate nice 3D-grafix for introduction to orthogonal transformations. Basically show both a frame aligned with polar axis of Earth and aligned with 0, 90 degree longitude, and a second frame based in Phoenix (easy to change) that is aligned with (Phoenix') up, East, North....
rsgrid.mws rsgrid.doc r5, 2000	Generate two overlaid grids to help work out coordinate changes as e.g. in 1-dim wave equn changing from u_xx=u_tt to u_rs=0. Main use, transform individual points, then level curves from one coordinate set to other.

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4. Multivariable and Vector Calculus 4.a. Functions of two or more variables Contents

directory listing	Contents.
cross_sections.mws r8 2003	Plotting graphs of z=f(x,y), horizontal cross-sections (contours), vertical cross-sections parellel to coordinate planes, in radial directions, and along vertical cylinders. Includes animations. All done once for pure functions, and once for expressions. Sample code -- user should clip preferred samples for fast execution of many examples.... Supersedes very old worksheets (releas 4, approx 1996): plots3d.mws and volcano.mws
maketable.mws r4 ??, 1996	Only release 5 of MAPLE has a built-in spread-sheet function. Until then EXCEL is more appropriate for making tables of function values. Nonetheless, MAPLE an do it, too; it is only more cumbersome.
day1.mws r4 ??, 1996	Description sometime...
interest.mws r4 ??, 1996	Naive way (guess and try w/ MAPLE) to derive the formula for paying of loans/mortgages (i.e. solve difference equation).
heateqn.mws heateq1.mws heateq2.mws r4 ??, 1996	Learning how to read 3d-graphs. This graph is revisited later when working with partial differential equations.
heateqn0.mws r4 ??, 1996	This is "heateqn-LITE", an uncommented version of the previous worksheet for those afraid of functions, afraid of MAPLE, and who just want to know how to do it, but don't care about the story.

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4.b. Vectors in the plans and 3-space
Contents: Most of these worksheets date back to 1996 or 1997, generated in MAPLE V releases 3, 4 and 5. Much has changed since -- in particular, the since release 6 (?) the package linalg has been superseeded by the package LinearAlgebra -- linalg still works, and the samples below still use it in release 8 -- but we recommend anyone to try to do the same using LinearAlgebra.

directory listing	Contents.
airspeed.mws r8 2003 (r4, 1996)	A sample MAPLE-calculation (Problem CCH 12.2/11) involving an air-plane, climb-rate, wind-velocity and ground-velocity.
planes.mws r8 2003 (r5 1998)	The usual problems with triangles and intersections of planes.
vectors97f.mws r8 2003 (r4 1997)	Getting started with vectors and parameterized curves in MAPLE.r Some comments about lists versus vectors.
T1980211.mws r8 2003 (r5 1998)	Sample solutions to MAT 272, Calculus III, Test 1, 2/11/98. Several items related to triangles and equns of planes.
satellite.mws GPSangles.mws r8 2003	Satellite (space shuttle) -- distance from Phoenix and angle at which it appears above the horizon. (The first worksheet summarizes earlier calculations in class, not yet commented, the second is all just as a start -- only useful part is an animation -- but it uses too much memory... and freezes MAPLE.... both worksheets need much work, almost too raw for consumption at this time.)

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4.c. Parameterized curves
Contents

directory listing	Contents.
16-1-4.mws r5 ??, 199x	From tables of values, and the side-views x=g(t), y=h(t) to the view of the parameterized curve (x(t),y(t)). Problem CCH 16.1.4
viviani.mws r8 2003	Simple picture (w/ code) of cylinder intersecting sphrere yielding Viviani's curve.
curves.mws r5 ??, 199x	Calculate al the usual objects associated with parameterized curves. Includes composition with a parameterized surface. (This is an old worksheet written for release 3, and needs to be up-dated.)
acc_2d_curv.mws vel-acc.mws r8, 2003 (r4, 1995?)	Animations of velocity and acceleration vectors on Lissajous figures, colored by the magnitude of the parallel acceleration component (speeding up = green, braking = red). Completely reworked in 2003. Dramatic constrast of animations of curves parameterized as usual Lissajous and by arc-length. In particular, constant speed animations, and nice osculating circles.
rollercoaster.mws r5, 2000	An attempt to create a fun rollercoaster -- 3D-analog of project 1 -- as a piecewise smooth parameterized curve. Not as easy as it looks..... Should plan ahead more!
repara.mws r5 ??, 199x	Reparameterizations of curves (1998). An elaborate worksheet addressing everything from shifts and scaling (essentially precalculus) to reparameterizations by arclength. It addresses the algebraic procedures in detail, and it also provides fantastic 2- and 3-dimensional graphics. (Also very useful for the first team project).
project1a.mws r5 ??, 199x	Help for convenient MAPLE syntax for the first team project.
T1980211.mws r5 ??, 199x	Sample solutions for Test 1, 98/02/11. Several items related to parameterized curves.
T1980216.mws r5 ??, 199x	Sample solutions for Test 1, makeup, 98/02/16. Several items related to parameterized curves.
arclngth.ms r3 ??, 199x

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4.d. Partial derivatives, optimization
Contents

directory listing	Contents.
mws-a.html r5 ??, 199x	Vertical slices of 3D-graph, animated vertical cross-sections.
partials.mws r5 ??, 199x	Similar to volcano.mws, but older (spring 1996), and more oriented towards partial derivatives.
chainrule.mws chnrule.mws r5 ??, 199x	Part 1: Specializing in the composition of (x(t),y(t)) with z(x,y). (See also the worksheets of the section on parameterized curves). Part 2: General case: Matrix multiplication (Jacobian matrices).
cubics.mws r8 2003 (r4 1995) maxima.mws r8 2003 (r3 1995)	Two examples of cubic polynomial in two variables. Emphasizes graphics to find the critical points as intersections of two conic sections (zero-sets of partials). ( cubics.mws was rewritten in 3/98 with some 2nd derivative tests added.) ( maxima.mws was updated directly from the 1995 worksheet, with select other changes.) Need to sometime combine these two...
2nd_deriv_test.mws r8 2003 (r4 1997)	Mostly graphical (and some algebraic) explorations related to the discriminant: Saddle versus parabolid (and parabolic cylinder). Includes animation as coefficient of mixed term x*y changes its value while coefficients of pure quadratic terms x^2 and y^2 remain fixed.
optim2.mws r8 2003 (r4 1997)	Unconstrained and constrained optimization w/ silly cubic polynomial example (as in old-fashioned textbooks), but consrained to a polygonal area!!! Emphasis on actually creating meaningful pictures which help organize and making a strategy. Along each edge one soln via paramterization and single-var-calculus, and one soln via Lagrange multipliers -- compare both, and the latter is very hard to automate due to difficulty to numerically finding the "right" critical point along finite edges that are parts of unbounded curves.
lstsq98.mws r8 2003 (r5 1998)	Sample calculations for a least squares fit (no matrices or linear algebra, straightforward calculus). Fitting quiz performance (averages of 8 consecutive quizzes) with linear function (March 98).
steepdescent.mws r5 ??, 199x	Use (naive, yet powerful) numerical techniques from differential equations to calculate the location of minima and maxima.
lagrange.mws r8 2003 (r4 1995)	Sample calculation and background material txo explain how Lagrange multipliers work. Includes grafix to relate to single-variable calculus approach. NEW: The dual problem, and ABNORMAL critical points.
15_5_30.mws r5 ??, 199x	Sample solution for CCH 15.5/30. Use Lagrange multipliers for the constrained optimization problem: Find that point on the surface xy+yz+xz=12 that is closest to the origin.
optim.mws r5 ??, 199x	A complete (though academic example) that includes it all: Find the extrema of a cubic of two variables over a polygonal region in the plane. Use 2d, 3d plots, animations, algebraic techniques and make it all fit together.

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4.e. Iterated integrals
Contents

directory listing	Contents.
overview page (old) asu.zip asulib r7 (r4 1996)	The package with intdraw/intdraw3d. Originally written for Maple V release 3, this package has held up remarkably well, and apparently works also in MAPLE 7. Usually the eaistes is to download the file asu.zip and install it according to the comments in readme.txt. If this fails, simply download all files from the directory asulib into a local directory with the same name and proceed from there.
s01t3.mws r6 2001	Sample solutions -- finding limits for iterated integrals: A skew pyramid six times. Novelty: mini-procedure diplays limits in title. (Saved with output - 6 grafix.)
iter-int.xls	EXCEL worksheet w/ simple plots and table of values: Basically to motivate Fubini's theorem: Sum (slice) in either order, sum of areas times thickness yields volume.
iter_int.mws r6, 2001	An introduction to setting up and evaluating iterated integrals. The worked example: average distance of points on triangle from another point. Lots of grafix. Worked vertical slices, guided exercise: horizontal slices. Advanced exploration: Repeat in polar coordinates. Throughout emphasis on structure of problem, and visual guidance. (Requires the package asu.zip (Completely rewritten in 2001, based on a 1997 worksheet w/ same name.)
ex15_2_4.mws r5 ??, 1997?	Sample solution to CCH exercise 15.2/4, using the package asu.zip to aid visualizing (also, rewrite double int as triple integral!).
dbl_int.mws r5 ??, 1997?	Double integrals illustrated. A precursor (spring 1996) to the worksheet on_edge.mws. (Requires the package asu.zip
on_edge.mws r5 ??, 1997?	A sample calculation for a region in the plane, asking for the specific size of the region that will cause the center of mass to fall on the boundary of the region. (Requires the package asu.zip
outlines.mws r8 2003 (r4 1997)	Provides outlines of regions in the plane. The student is asked to "fill in" the regions by finding "limits of integration". Also provides templates for evaluating all the "usual" integrals for area, center of mass etc. over these regions using found limits of integration. (Requires the package asu.zip
outlines3d.mws r8 2003 (r4 1997)	Similar to outlines.mws, but now in 3D. Main examples: straight and skew pyramids, ice-cream cones with flat top, shallow top, or big ball, wedge cut out of tree, a torus, and two intersecting pipes. Includes automated generation of outline (edges) when given the corners of a skew pyramid. Completely rewritten February 2003. Completed all four classes of skew pyramids Mar 2003. (Requires the package asu.zip
coordchg.mws r5 ??, 1997?	Coordinate changes: Work in progress. To illustrate where the Jacobian comes from as magnification factor.

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4.f. Parameterized surfaces
Contents

directory listing	Contents.
easyvase.mws r6 2001 (1997)	A simple intro to surfaces of revolution.
newvase.mws r8 2003 (1997)	Animations that demonstrate how to think of parameterized surfaces as a parameterized family of parameterized curves, using a "vase" as example. Now allowing for revolution of general parameterized curves, not only revolution of graphs of functions. Many new examples created with minimal command; beautiful illumination....
oldvase.mws r6 2001 (1997)	(formerly: makevase.mws) Old precursor for the newer worksheets easyvase.mws and newvase.mws.
polar.mws r6, 2001 (1995)	Graphs of equations in polar coordinates, first visualized as ordinary graphs in a rectangular (r,theta)-plane and then transformed by a transformation of the plane (composition with coordinate change).
curvsurf.mws r6, 2001 (1997)	Composing parameterized curves in the plane and parameterized surfaces in 3-space (examples to help w/ pure-function notation, uncommented worksheet).
lagrange_identity.mws r8, 2003	uncommented sample calculations for dA via cross products and via Lagrange identity yielding dA=sqrt(EG-F^2) for general parameterized surface, sphere and torus.

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4.g. Calculus of vector fields in the plane
Contents

directory listing	Contents.
studplots.mws r6 2001	Some 50 pictures of vector fields in plane (and a short macro for plotting). To be used as problems-set: "Given picture, decide whether linear etc., 2. find formula, 3. decide whether zero/pos/neg divergence/curl, 4. decide whether gradient.
introvf0.mws r5 ??, 1997?	First introduction to vector fields. Generate pictures (plot syntax), and discuss what the picture might represent. Includes divcolor and rotcolor.
introvf.mws r5 ??, 1997?	Comprehensive intro to MAPLE and vector fields in a class that only once a week had computer access. This worksheet combines visual games (like mystfield.mwes), but also notes on symbolic calculations (grad, curl, div and jacobian from linalg package). (Old, spring 1996).
vfplots.mws HTML r6 2001	Uncommented. Commands to generate large collection of sample plots. Used to practice formula from picture, decide whether gradient, whether divergence free etc. HTML-version with pictures only, prints on 2 pages.
mystfield.mws r5 ??, 1997?	Practice finding formulas for linear vector fields, and visually determining whether field might be a gradient field. Worksheet randomly generates plots of linear fields with small integer coefficients. User checks by plotting own field. Also includes templates for contour diagram of associated potential function.
lineintro.mws r5, 11/98	Riemann sums for line integrals: Development, calculation, and visualization
line_exa.mws r5 ??, 1997?	A simple example with circles in a nonlinear field. Preparation for more sophisticated worksheets like line_int.mws
line_int.mws r5 ??, 1997?	From evaluating line integrals by hand (via parameterizations), towards automation. Application to line integrals over closed contours, discover scaling by area of enclosed region for linear fields, and also for nonlinear fields after shrinking contours.
halfauto.mws r5 ??, 1997?	Evaluating line integrals of the same field over many different contours: different locations, shapes and sizes. Not yet fully automated -- i.e. with still tangible parameterizations. Typically use for demos, and to cap/summarize class discussion from student explorations.
autoline.mws r5 ??, 1997?	Procedures to automatically evaluate line integrals (both circulation and flux) for polygonal curves in the plane. Targeted for discovering Green's theorem for linear fields.
magline.mws r5, 2000	Uncommented sample commands for line-integrals (especially triangular/polygonal paths) in the magnetic field.... Simple images. Use when developing winding number, simple connectivity.
autoinclass.mws r5 ??, 1997?	Yet another example of procedures for automating the evaluation of line integrals. Geared towards the discovery of winding numbers: triangles and the the field 1/z.
loops_linear.mws r5 ??, 1997?	Templates for evaluating line integrals around various closed contours, with emphasis on linear fields (i.e. symbolic evaluation), and discovery of integrals scaled by area of enclosed region , otherwise independent of shape, size and location of the contour. (Section for formal discussion on linear vector fields is still unwritten).
conserv.mws r5 ??, 1997?	Visualizing the dichotomy "conservative - controllable" by lifting loops in the base space, and seeing whether they live in a "potential" surface.
zoom1vf.mws r5 ??, 1997?	Zooming for the "linear part" of a vector field at a point. Complete with lots of calculus-I review (spring 1997).
0310-362-t1sol.mws r8, 2003	Selected sample solutions for test exam MAT 362 in fall 2003.
f00final.mws r5, 2000	Selected sample solutions for final exam MAT 362 in fall 2000. Additional excursions. Primarily written for grading purposes, still useful to glance effective syntax.
flux_vis.mws. flux_vis0.mws. r6, 2001	Dramatic animations of the flux across a surface (this worksheet served as test-bed for JAVA program) -- colored depedning on whether in or out flowing. Zooming examples -- useful as backdrop for defn of Riemann integral for flux across a curve. Also using color-coding to illustrate flux across a surface -- dramatic images! The larger worksheet flux_vis0.mws (650 kB) contains some live animations and plots as the systems of DEs for large number of init cond's can take some time to solve.

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4.h. Calculus of vector Fields in 3-space Contents

directory listing	Contents.
fluxint.mws. r6 2001 (1998)	Create a procedure for automating evaluation of flux integrals over parameterized surfaces (very small worksheet).
fluxcalc.mws r6 2001 (1997)	.... formally updated to release 6, but content has not yet been revised to 2001 standards ... Sample calculations for flux integrals over parameterized surfaces, with section on divergence theorem. Emphasis on fixed field integrated over various surfaces (shapes), that in the limit as they shrink to zero yield flux integrals scaled by volume.
fluxcal2.mws r6 2001 (1997)	Sample calculations for a typical pair: vector field and surface: finite cylinder and electric field about a point-charge, first naively, then using the divergence theorem. Complements handwritten calculations (directly from this index).
newstokes.mws r6, 2001	Under preparation. To replace stokes.mws Typical use of Green's theorem and Stokes' theorem for fields with singularities (magnetic field etc.), with many animations, and various explorations how to visualize the definition of the curl in 3D. (Old, spring 1996, and comprehensive, but somewhat clumsy). ( newstokes0.mws with output, 1.2 MB -- still under construction).
stokes.mws r4, about 1996	Completely outdated -- kept for sentimental reasons. See newstokes.mws for rewritten version. Typical use of Green's theorem and Stokes' theorem for fields with singularities (magnetic field etc.), with many animations, and various explorations how to visualize the definition of the curl in 3D. (Old, spring 1996, and comprehensive, but somewhat clumsy).
rotate.mws r4, about 1996	Visualize the curl by an animation of small volume elements inside a flow field. Emphasis on the harmonic oscillator and the magnetic field.
divcurl.mws divcurl.html r8, 2003	Background images that should help derive formulas for the curl and the divergence in polar, cylindrical, and spherical coordinates using a geometric approach. Provided are pairs of opposing faces (for the divergence) and pairs of opposing edges for the curl of plar/cylindrical a,d spherical "boxes" with overlaid unit vector fields in the coordinate directions. Practically no comments are given -- assumption is that this simply replaces hand-drawn still-sketches (these can be resized and rotated!), but the derivation of the formulas is still left as in-class or homework exercise.
stokes_exa5.mws r8, 2003	A simple example from Marsden's book: With Stokes' thereom only elemntary arithmetic is needed -- the worksheet explores and compares several different ways to directly evaluate the corresponding surface integrals. (Field is rotation about axis z=y=x, surfaces are part of unit sphere above plane x+y+z=1 and part of plane x+y+z=1 unit sphere.) Includes direct parameterizations via Cartesian and spherical coord's, and via adapted cylindrical and spherical coordinates. Useful grafix even if not interested in direct evaluations. (With full output: dodeca.mws.)
dodeca.html dodeca0.mws (r4, 1996) r8, 2009	The 3d-version of rotate.mws. However, the vector field is no longer visible, and there is still one gap in the integration on the rotation group. Nonetheless, a beautiful animation. (Large worksheet, animation is running in saved version.) dodeca0.mws: Same as dodeca.mws, but with all output removed. (Much smaller, but the animation takes a long time to re-create).

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4.i. Further explorations
Contents

directory listing	Contents.
divrotcol.mws r5 ?, about 1997 ?	First explorations towards coloring a vector field by divergence/rotation, and towards the animations in rotate.mws.
schwarz.mws r5 ?, about 1997 ?	Surface area cannot be defined via limits of triangulations. Calculations and visualizing the famous accordion-like, standard counter-example.
zoom11.mws r5 ?, about 1997 ?	Resources for zooming in all different ways. Contains: 1.1 Limits of sequences, 2.1 slopes and lines, 2.6-7 telescoping sums and 2nd fundamental theorem, 6.5 Schwarz' surface, 11.2 skew symmetric, even and odd; 11.3 curl in coordinates, 12.6 homotopies in then magnetic field.
zoom2nd.mws r5 ?, about 1997 ?	Zooming of second kind at critical points
zoom_loops.mws : Pictures for fundamental theorem (calc I, telescoping sums), and for individual contours in prep of Green's theorem.
lenses.mws r5 ?, about 1997 ?	Preliminary work for the JAVAscope. Views of the various components of the linear part of a vector field at a point. Exploration how much zooming will yield apparently linear field.
get_skew.mws r5 ?, about 1997 ?	A fancy way of coding the projection onto the direct sum of multiples of the identity and the skew symmetric matrices (conjugation with rotation by 90 degrees).
pipe.mws r5 ?, about 1997 ?	Attempts working with cut-off functions that simulate the vector field flowing in a pipe, and that vanish outside.

MAT 272:

Optim2.mws
 basics.xls
 day1.mws
 dbl-int.ms
 fallcat.mws
 iter-int.xls
 project1
 project2
 project3
 tests

5. Basic mathematical structures 5.a. Recursions and Induction Explicit implementations of Counting functions, and of recursive/inductive solutions (with great grafix!) of basic tiling problems

directory listing	Contents.
trian.mws r5, 2000	Tiling with "triangular triominos" -- very nice images!
multiplictable.mws multiplictable.html r8, 2004	Quick generation of multiplication tables in modular arithmetic for explorations for which values the linear equation b*x=a (mod n) has a unique or multiple or no solutions.
Ltiles.mws r5, 2000	Tiling with "L-shaped triominos" -- very nice images!
coin.mws coinweighing.mws r8, 2006	work in progress -- given 12 coins one of them false (either light or heavy) devise an alogorithm to find the bad one. testing power of "random" algorithms.
bitstrings.mws r8, 2006	work in progress -- HEADS and TAILS game: winner is whose sequence of three occurs first.
permute.mws r8, 2003 (r5, 2000)	explicit coding of counting functions and their inverses (???)
infin_many_primes.mws infin_many_primes.html r8, 2005	infinitely many primes -- background calculations for common mistake.

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5.b. Advanced Calculus / Intro to Analysis
Contents

directory listing	Contents.
interval_of_limitpts.mws interval_of_limitpts.html r8, 2004	A simple formula for a sequence whose set of limit points is the entire interval [0,1]. Includes a table and a plot of the graph of the initial segment.
sqrt2.mws r8, 2004	Background sample data and algebra simplifications for AGH inequality and sequences of rationals converging to sqrt(2) and exp(1) [used in prep of sup-axiom].

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5.c. Abstract Algebra
Contents

directory listing	Contents.
Fquot.mws Fquot.html r8, (2005)	Make list of polynomials in Fn[X] of small degree, calculate their remainders mod a given polynomial P(x), and partititon the set into congruence classes. Simple calculations, useful for transparencies showing larger set (e.g. after having done many by-hand calculations first).
Fext-tables.mws Fext-tables.html r8, (2005)	Addition and multiplication tables for the congruence classes of irreducible quadratic and cubic polynomials over Z2[X] and Z3[X]. For comparison, also tables for the rings Z4, Z8, Z9, Z2 x Z2, Z2 x Z2 x Z2, Z3 x Z3. As first steps towards splitting fields, also tables of function values of quadratic and cubic irreducible polynomials evaluated on the fields of congruence classes.
poly_Zp.mws poly_Zp.html r8, (2005)	Tables of polynomials of small degree in Zp[x] for p small primes, their factors and roots with multiplicities. Includes some slick and fun MAPLE code.
irred-red-modp.mws r8, (2005)	Uncommented calculations illustrating that x4+3x2+1 is irreducible over Q, but factors nontrivially in every Zp[x].
ringisos.mws ringisos.html r8, (2005)	Addition and multipklication tables for small rings, comparing Zm x Zn and Zmn. Uncommenetd.
mat444.mws r4, 1997	A very brief intro to MAPLE for an abstract algebra class. Just to create awareness that the usual polynomial manipulations related to calculating Galois groups need not all be done by hand. A very raw, exploratory worksheet ...
octahedral.mw octahedral.html r9.5, (2005)	Trying out the "group" package (but not the generators/relations parts), applied to the problem of finding the subgroup lattice of the octahedral group.
cubesym.mw cubesym.html r9.5, (2005)	A programming exercise (lots of nested map(...)) to explore going back and forth between representations of the rotational-symmetry group of the cube (which is of order 24) in terms of matrices and permutations of faces (S6), of vertices (S8), and of edges (S12). Includes animations of orbits of colored cubes under selected subgroups, ready to take any cyclic (ordered) subgroup as input for an animation.
game20-4.mw r9.5, (2005)	A partial answer to the questions raised here about a simple childrens' toy that involves the permutations (1 4)(2 3) and (1 2 .... 19 20).
fieldext.mws fieldext.html r8, (2005)	Some very simple, sample calculations about a cubic extension of the field F2 (compare Fraleigh exercise 27/25). Uncommented.
actonLH.mws actonLH.html r8, (2005)	Explicit tables of function values for the homomorphism of a group G into the groups of permutations of the set of left cosets of a subgroup H, permutations of the set {1,2,...m} where m=[G:H]. Very suggestive tables for the cases of G=S3 and G=Z3, and H being a subgroup of index m=3.
primetables.mws primetables.html r8, (2005)	In the spirit of Eratosthenes' sieve, select from the first N (e.g. 200) natural numbers those that are prime, prime power, prod of 2 or 3 primes etc. in order to zero in into those group-orders where there may be a simple group possible (i.e. exclude all those which are immediately taken care of using the Sylow theorems in a standard way).

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5.d. Geometry
Contents

directory listing	Contents.
trisectors.mws trisectors.html r7, 2001	An algebraic demonstration that three of the intersection points of the trisectors of any triangle form an equilateral triangle. The purpose was to show that a formal proof is a rather trivial undertaking, straightforward in an computer algebra system -- however, the main mathematical enterprise is finding this result, and for this the ideal tool appears to be CABRI.

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6. Differential Equations 6.a. Linear Differential equations See also sections 2. Elementary Differential equations, or 3. Linear algebra. Contents

directory listing	Contents.
C2sincos.mws C2sincos.html r8, 2006	Use matrix language (similarity transformations) to rewrite complex matrix exponential of real 2 x 2 matrix in real terms.
jordanexa.mws jordanexa.html r8, 2006	Homework exercise involving nontrivial Jordan form of a 4-dimensional system worked in many different ways. Tutorial. With some fun MAPLE-coding experiments.
realnearJordan.mws realnearJordan.html r8, 2006	Should there be ONE or TWO ones above each 2 x 2 block in nontrivial real almost Jordan forms in the presence of repeated nonreal eigenvalues? Are the respective different normal forms similar? Orthogonally similar?
excursion.mws excursion.html r8, 2006	Solution curves of asy-stable (and dtable) systems may make quite big excursions ---a 3 simple examples: nontrivial Jordan block, or ellipses (elliptical spiral) whose (almost) diagonalizing matrix has barely linearly independent columns, or ebenm though columns are orthogonal, they are of very different magnitudes. Use for slides when discussing ε and δ in definition of Lyapunov stability.
expAoft.mws expAoft.html r8, 2009	A simple example to illustrate that naive exponentials of integrals of time varying matrices do not make fundamental matrix solutions.

6.b. Existence and uniqueness

directory listing

Contents.

picard.mws picard.html r8, 2006

Advanced implmentations of Picard iteration (successive approximations of solutions of initial value problems) for both scalar and systems cases. Worked examples include y'=y, y''+y=0, Bessel eqn and the nonlinear DE y'=exp(-y), whose iterates are complicated expressions in terms of elliptic integrals, yet the limit is simply log(t). Updated version allows nonconstant y0=φ0 in scalar case. Includes various animations in 2d and 3d. Useful for experimenation, demos, and to raise convergence questions. See also more basic and more worksheets in sections 2.a above and 7.e below.

6.c. Stability

directory listing	Contents.
harvest.mws harvest0.mws r8, 2007	Worked exercise (Hirsch-Smale-Devaney 1.6) on logistic DE w/ constant harvesting; bifurcation-point of view (animations). Worksheet contrasts quick numeric work w/ laborious manual calculations for closed form solution formulas that yields little immediate insight.
lyap1.mws lyap1.html r8, 2006	Mainly pictures for Lyapunov function for damped and undamped simple (nonlinear) pendulum: Phaseportrait, energy surface, lifts of trajectories to energy surface, and lifts of trajectories to surface of dV/dt(x,y) ! Mainly as back-up for class-discussion, esp. focus on V(x,y) vs. V(x(t),y(t)). Elaborate pictures that allow to see both the local picture and the multiple basis of attractions. Code uses sneaky nested map and zip commands in several places.
not-rad-unbounded.mws"> not-rad-unbounded.html"> r13, 2009	Pictorial counteraxample: Going to infinity along every ray from the origin does NOT imply radially unbounded (i.e. does not imply proper).
lyap-pf.mws lyap-pf.doc r8, 2006	Just a quick picture for the proof of Lyapunov's LaSalle's theorema (about decrease of V to a finite positive limit)
lyap-not-proper.mws lyap-not-proper.html lyap-not-proper.doc r8, 2006	An illustrated story that constructs a Laypunov function V and a vector field f such that V is trictly positive definite, the derivative Vdot of V along f is strictly negative, and f is NOT asystable. Includes some examples of how to use gradient and Hamiltonian fields. Main use: Just discusss what can go wrong if V is not proper (not radially unbounded). One picture (to be prouid of) tells the story.
Mathieu-aver.mws Mathieu-aver.html r11, 2008	Completey raw examples/ w/ illustrations of averaging theory: van der Pol and Mathieu.
pb.mws pb.html r8, 2006	Some sample calculations to justify the application of the Poincare Bendixson theorem. The key issue is to rigorously establish (prove) that some compact annulus is forward invariant. Three examples from Khalil, and one more (picture only) from Hirsch-Smale.

6.d. Advanced geometric topics

directory listing	Contents.
08-02-19-poincareNF.mws r8, 2008	Exploration of resonances obstructing fromal coordinate change to bring system into Poincaré normal form.
poincareNF-example1.mws r13, 2009	Uncommented naive example (homework) bring system into quadratic normal form.

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7. Partial Differential Equations 7.a. Fourier series for PDEs See also sections 7.c, 7.d, and 7.e for Fourier calculations embedded into worksheets primarily addressing PDEs. Contents

directory listing	Contents.
harmonics.mws harmonics.html r?, ????	Sums of sinusoids (musical tones): Overlaid 220Hz and 880Hz and either of their arithmentic means (550Hz) or geometric mean 440Hz (missing is harmonic mean 352Hz).
DEsample.mws r5, 2000	Quick illustrations of second order linear DEs with different kinds of forcing. To motivate Fourier expansions. intro-level MAPLE.
introfourier.mws r5, 2000	Intro to Fourier expansions for MAPLE novices, first with only elementary commands, then repeated with for loops, seq, sum, map. Also included basic syntax for plots/animations. First part useful; as template, second intended mainly for demo's. Also included some fun applic's/animations of forced 2nd order DEs via Fourier analysis, incl. unexpected results of higher harmonics near resonance. Main issue: The use of functions a:=n->int(...) gives occasional trouble with only "generically correct" antiderivatives. Thus in Spring 2001 we prefer to work with tables a[k] ...
fouriersyntax.mws r5, 2000	NEW! Detailed discussion, with examples and sample code (suitable as template!!!), of the relative advantages and problems of ak, a[k], a(k), a:=k->int(...) a:=unapply(...,k). NEW!
four1.mws r6, 2001	Sample commands from class in January 2001, just for temporary record, to be deleted soon.
jan23.mws r6, 2000	Sample commands from class in January 2001, just for temporary record, to be deleted soon. Main advantage over Fall-2000 worksheets is the use of tables a[k], b[k] as opposed to functions a(k),b(k).
abssin.mws r5, 2000	Simple examples of Fourier approximations. Contrasting Fourier series for periodic signals, half-range expansion and variations thereof for finite signals, and Fourier integrals, Sine integrals for (semi)-infinite signals with finite energy. Basic example used throughout is abs(sin(t)).
rectified.mws r5, 2000	Sample calculations of Fourier coefficients w/ grafix fro rectified sinusoid. Main issue is that the "generically correct" formula for the integral a(k) is not correct for special cases (here k=1). This encourages the safer (but wasteful approach of "for k ... do a[k]:= int(...); od;)
halfrange.mws r5, 2000	Contrast different half-range expressions for finite signals. Compelling grafix of first reflect, then extend periodically with impressive animations of convergence. Also impressive: speed of convergence versus continuity/smoothness of extension.
fourier2d.mws r5, 2000	Fourier expansion in 2d for a square pyramid. Main issue is that use of functions a(m,n) as opposed to tables a[m,n] leads to zero series as "generically" the integrals are zero.... Uncommented "working sheet".
rsgrid.mws rsgrid.doc r5, 2000	Generate two overlaid grids to help work out coordinate changes as e.g. in 1-dim wave equn changing from u_xx=u_tt to u_rs=0. Main use, transform individual points, then level curves from one coordinate set to other.
test1.mws r5, 2000	Fourier expansion of exponential charging/discharging capacitance, both using real and complex forms. (Sample solution to test problem w/ many comments and excursions).
f00final.mws r5, 2000	Selected sample solutions for final exam MAT 362 in fall 2000. Additional excursions. Primarily written for grading purposes, still useful to glance effective syntax.
s01test1.mws r5, 2001	Sample solution for test 1, spring 2001 in MAT 362 in spring 2001. No comments. Just quick calculations (useful as template for intermediate users).

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7.b. Fourier integrals for PDEs
See also sections 7.c, 7.d, and 7.e for Fourier calculations embedded into worksheets primarily addressing PDEs.
Contents

directory listing
FserInt.mws r6, 2001	Illustration of how one might intuitively step from Fourier series to Fourier integrals by considering families of periodiuc functions whose period increases to infinity. Very nice animations: Family of functions, amplitude spectra against normalized frequencies, Fourier approximations with fixed and with increasing number of terms as period increases.
Fint.mws r5, 2000	A simple exploration / demonstration of MAPLE's capabilities to work with simple Fourier integrals -- the issue is when to work numerically with the improper integrals
FTintro.mws r6, 2001	An intro to Fourier transforms -- focus on algebraic properties such as linearity, transforms of derivatives, applic to PDE. Large initial section reviews analogue properties of Laplace transforms as taught in first ODE course.
convolve.mws r6, 2001	Some simple animations regarding convolutions.

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7.c. Partial Differential Equations 1 dimensional wave and heat
Contents

directory listing
poppyra.mws r6, Jan 2001	Aging, or the 1st order wave equation. Explorations with characteristics. Emphasis: Working with variables, some visualization...
wave1.mws r8, 2003 (r5, 2000)	One dimensional wave equation: Animations of how two traveling waves add up to a standing wave. Examples include triangular, sinusoidal, sums of sinusoidal initial deflections. (Colors and visual aids to see periodic extensions).
dAlembert2hats.mws r6, 2001	Wave eqn on infinite interval. Piecewise linear initial data (2 hats some distance apart). Sample solution for a final exam problem. Nice animations of d'Alembert's solution, and experimentation with solving PDE via Fourier transforms by hand -- i.e. using only int(...), w/o reference to with(inttrans) and with(PDEtools) -- quite informative steps.
thirdstring.mws r8, 2003	One dimensional wave equation - vibrating string. Sample problem (hold own string at 2/3 from end w/ thumb and pluck it at 1/3 from end). Solution via separation of variables and via d'Alembert's method. Includes nice animations and error-estimate using Parseval's identity.
0010-362-T2.mws r8, 2003 (r5, 2000)	Vibrating string worked example: Initial deflection: thumb down in center and pluck string at one quarter. Sol by sep of var's -- syntax not optimal, somewhat exploratory but OK. Several animations. Simple calcualtions related to vibrating rectangular membrane, including nodal lines.
vibstring.mws vibstring.html r8, 2003 (r6, 2001)	One dimensional wave equation - vibrating string. Linearity of (PDE w/ its BC) -- i.e. split problem with many inhomogeneities into superposition of several problems that are inhomogeneous in only one place. Emphasize operator notation combining PDE and BC. Animated 2d-plot versus 3d-surface (basic syntax). [[Solutions of subproblems are only posited and checked, not derived.]] -- Main flaw: No attention to usual technique of transferring inhomogeneities such as moving endpoints to PDE.
eigvalprblm.mws eigvalprblmB.mws HTML (HUGE)!!! r6, 2001	Pictorial (animations) and algebraic (incl. linear algebra) presentation of (ODE-)eigenvalue problems as they arise in sep.of.var's approach to solving PDEs. Incl. Dirichlet, Neumann, and some mixed BC, as well as freely supported and clamped endpoints for a beam. Section on Bessel's equation added on March 21.. eigvalprblmB.mws is much larger (900kB), containing live output of Bessel-section which takes long to recalculate otherwise. HTML is a HUGE HTML/GIF version of on older file lacking the Bessel section.
beam.mws r6, 2001	Eigenvalues of the beam equation (no load, clamped and freely supported endpoints. Emphasis of structure of linear eigenvalue problem. Graphical solution. Includes only minimal conclusions. (In some ways similar to eigvalprblm.mws, but much more focused, much smaller, and lacking grafix).
Multi-var calc. heateqn.mws	For several good views of the 1-dimensional heateqn, background, solution etc. see the section on plotting (of/and) multivariable fnctions. There the focus is not on solving the PDE, but on making sense of it, of the 3D-graph, its cross-sections and working with functions of two or more variables.
heat2hats.mws r6, 2001	Heat eqn on finite interval, Dirichlete boundary conditions. Piecewise linear initial data (2 hats, or upside down W). Sample solution for a final exam problem. Complete separation of variables, periodic odd extension, Fourier-sine series, several plots. Extras: Error estimates using Parseval's identity, and "how HOT does the midpoint get?".
infiniterod.mws r6, 2001	Diffusion equation on the (infinite) real line: "Enjoying the fruit" after the hard derivation of the kernel of the 1D-heat equation on 1/2 plane. Many animations and cross-sections.
infrodexa.mws r6, 2001	Diffusion equation on the (infinite) real line: Example w/ IC u(x,0)=1/(1+x^2), solution in integral form. To serve as background for discussion of structure of the various terms involved. Includes plots and animations.

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7.d. Partial Differential Equations, Laplace and 2-dim heat and wave, rectangular domain
Contents

directory listing
laplace-annulus.mws r8, 2003	Simple example of Laplace's eqn on annulus. First parts of sep of var assumed to have been completed by hand. Includes mainly the linear systems arising from Dirichlet BC, and a graphical check (animation of successive approximate solutions).
Ltransfo.mws	Laplace transform, variation of parameters, Green's kernels. Revisit soln techniques for ODEs, comparison, and discuss appropriate notation. Pushing the limits: From specific examples to general formulas. Some idiosyncratic notation.
drum.mws : Laplace eq, sep of var (not yet complete)	Analyze in detail steps of separation of variables, focus on eigenvalue problem Release 5 PDEtools package, implement sep of var, incl. eigenvalue problem, by hand.
greens.mws r8, 2003 (r4, 1998)	Intro to Green's functions : Develop idea of fundamental singularity via discrete approximations Focus on ODE BVP. Advanced book-keeping, some tricky sums and lists, many visuals. Explore the limits of symbolic integration of Green's functions.
wave2.mws r6, 2001	Template for visualizing (animations!) vibrating rectangular and circular membranes -- i.e. given a solution u(x,y,t) formula, a ready-made procedure generates carefully crafted animations. For circular domains simply specify the desired linear combination as linear combinations of Fourier-Bessel-series (generally NOT radially symmetric). [[This worksheet provides the animations only, for the development of the solution on a disk, see the worksheet wave2disk.mws.]]
resonance.xls	EXCEL-spreadsheet for quick view of resonances m^2+n^2=i^2+j^2.

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7.e. Partial Differential Equations, more advanced cases
Contents

directory listing
wave2disk.mws r6, 2001	(Formerly part of wave2.mws ). Laplacian into polar coordinates. Making MAPLE carry out separation of variablkes step-by-step using precise commands. Exploration of Bessel-functions. Animations of solutions (generally not radially symmetric)!
BesselFun.mws r6, 2001	Explorations of Bessel's DE: vector fields, varying phaseportrait, comparison w/ harmonic oscillator and trig functions, numerical solutions, Picard iterates w/ impressive animations of convergence,... Main objective is to convince student that it is fair to DEFINE Bessel functions as the existing unique solutions of initial value problems. This worksheet compellingly proves that these are very computable, very tangible objects -- even before (analytic series) formulas are developed... ( BesselFun1.mws is larger, about 900 kB, saved with live output, but older version). Animation of convergence of Picard iterates (and derivatives) for n=1, f(1)=0, f'(0)=1. (155 kB, animation too fast)

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8. Complex Analysis 8.a. Complex Analysis a Contents several others ... and ... Working with colormaps to visualize complex functions. Many explorations, that later helped MATLAB and JAVA implementations.

directory listing
461logo.mws r8 2003 (r5 1999)	Creating a color-coded image of several sheets of the Riemann surface of z=sqrt(z^2-1). Few comments, but an intriguiung final grafix which served as the logo for an introductory complex analysis class.
complex.mws r8 2003 (r5 1999)	First intro to MAPLE in complex analysis class. 1. simple calculus example: find harmonic conjugate use CAS for diff and int). 2. first example of plotting curves and their images under a complex mapping.
contours.mws r8 2002 (1999)	Simple procedures that plot (piecewise defined) contours in complex plane and evaluet contour integrals. Input may use "complex" variables, procedures disassemble the expressions using Re and Im -- this seems to work well for simple examples (as included), but may crash if contours of function include symbolic parameters (w/o assumptions).
converge.mws converge.html r8 2006 r8 2002 (1999)	Great images and animations to illustrate convergence of sequences and series of both real and complex scalras (e.g. epsilon tubes and disks), of pointwise and uniform convergence of functions, and even animations of effects of termwise integration and differentiation of series of functions -- default animations integrate and differentiate Fourier series of rectangular wave.
sinh.mws r8 2002 (1999)	Some first efforts to plot functions of a complex variable. Mostly disappointing, but still pretty. (E.g. height for real part or magnitude, and color/shading, e.g. ZHUE for the imaginary part or the argument. Precursor for the more refined colormaps used later.
colorwheel.mws r8 2002 (1999)	Early explorations of how to define a color map. Mainly for archival purposes, why we did not go along w/ "piecewise linear maps" etc. Several examples.
essential.mws r8 2002 (1999)	First tries to use colormappings to eventually zoom in into essential singularities. Much nicer images were later obtained in MATLAB and in JAVA II. See also the power-point presentations from several recent conferences (Odense 2000, Crete 2002).
homotopy.mws r8 2002 (1999)	Sample commands to generate animations of a homotopy. Sample example "homotopes" a triangle (piecewise defined!) into an ellipse. Some discussion about "possibly not all intermediate curves being simple".
laurent.mws r8 2002 (1999)	A very "real" worksheet. Main focus on alternatives to Taylor expansions using negative powers -- lots of animations of respective successive approximations for REAL functions. The corresponding 3d-animations for complex counterparts are disappointing --leading to the later development of colormaps (see MATLAB, JAVA II, and power-point presentations from conferences (Odense 2000, Crete 2002).
lineint.mws r8 2002 (1999)	Uncommented. Sample syntax for evaluating line integrals almost by hand. Comparison w/ real plane. Linear forms w/ rectangular curves.
lopez.mws r5 1999	description coming soon
circle.mws r8 2006 (1999)	Pictorial and algebraic reflection of various circles across the circle by the special Moebius transformation f(z)->1/z. The algebraic proof that the image is again a circle is hard since the map does not preserve uniform parameterizations -- very nicely shown grafically (think of electric charges on a loop and a mirrored loop).
mapping.mws mapping.html r8 2002 (1999)	Main tool for visualizing complex functions thru their effects on finite rectangular or polar grids (centered anywhere). Simple input, and carefully crafted color-codes (tracking each edge and inside grid) allow one to play with many examples, and focus on the geometry, not the code. Special routines for even faster work w/ Moebius transformatrions. Eventually, want this to run in JAVA (note: the commercial program "f(z)" from Lascaux graphics has the desired mouse-interface, but lacks sophisticated coloring which we deem essential. Send e-mail if interested in working together on this.
oneover1plusz4.mws r8 2006	Just a few very quick calculations for the integral of 1/(1+x^4) yusing real calculus and using residues.
taylorfourier.mws r8 2006	Just a few very quick pix relating Fourier-series of square and triangular wave to Im and Re part of corresponding complex Taylor series.
pisqrover6.mws r8 2006 (r5 1999)	Residue theorem application. Summing the series sum(1/k^2,k=1..infty): using MAPLE to mimic by-hands-calculation of residue at thirs order pole at zero.
poisson.mws r8 2003 (r5 1999)	Several ill-fated attempts to use MAPLE for solving a boundary value problem via Poissson / Cauchy integral formula and plot the solutixuons.
residue.mws r8 2002 (1999)	Sample commands to perform residue calculations in MAPLE, incl. convert, parfrac, limit, residue, ...
t2soln.mws r8 2003 (r5 1999)	Sample solutions (w/ extended pictures), trying to do almost all in MAPLE for test 2 in introductory class (mainly contour integrals, some Cauchy integral formula etc.)

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9. Differential Geometry 9.a. Differential Geometry a Contents: Elementary geometry of curves in the plane and 3-space.

directory listing	Contents.
acc_2d_curv.mws vel-acc.mws r8, 2003 (r4, 1995?)	Animations of velocity and acceleration vectors on Lissajous figures, colored by the magnitude of the parallel acceleration component (speeding up = green, braking = red). Completely reworked in 2003. Dramatic constrast of animations of curves parameterized as usual Lissajous and by arc-length. In particular, constant speed animations, and nice osculating circles.
ellipse.mws, r8 2007 (r4, 1998)	An intro (meant for a presentation w/ guided discussion) to basic capabilitiues and limitations of CAS. Suitable to be shown to CAS-novices (but knowing calculus). Example: Arclength and curvature of an ellipse. Invariance of plane-curvature under rotations.
arclength.mws r8, 2003 (r5 1998)	Calculating arclength of parameterized curves. Invariance under the action of the Euclidean group. Limitations of the int() command. Implementing "group action" via explicit parameterization. Proof by direct calculation.
curvature.mws r8 2007 (r5 1998)	Curvature of plane curves: Arbitrarily parameterized curves and curves parameterized by arc-length. Osculating circles. Straightforward calculations. Visualization using color nontrivial for curves!). Animations of circles give problems due to scaling.
rotcurve.mws r8 2003 (r5 1998)	Rotate the graph of a parabola. Observe how second derivative and curvature depend and are independent of rotation. No abstract chain-rule, but down-to earth application in special example.
torusknot.mws r8 2007 (r4 1998)	"One paragraph" worksheet providing "test-curve" for e.g. Frenet-frame animation project. Composition of paramterized curve and parameterized surface. Show only torusknot, or overlaid surface.
frenet.mws r8 2003 (r4 1998)	Integrating the Frenet equations -- still using e.g. linalg, no LinearAlgebra, but runs o.k. with procedures that allow for fun experimentation: Give kappa(s),tau(s) and obtain a space-curve and animated Frenet frame.....
serret.mws r8, 2007 (r4, 1997) intcurve.mws frenetold.mws	Integrating the Serret formulas: Curvature (and torsion) completely determines the curve. Time-dependent constant-length loops. Curvature evolving according to a PDE (diffusion or 1-dim wave equation In serret.mws now constant center of mass and also vibrating loops. Closed form symbolic and numeric integration. Visualization and animations.
moebiuspix.mws r8 2003	Just some fun pix of a Moebiusstrip and together w/ another (orientable) surface w/ same boundary.

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9.b. Differential Geometry b
Contents: Basic work on differiantiable manifolds: Vector fields, flows, coordinates,
maps between manifolds and their (co)-tangent mapes, integrability (Frobenius), ...

directory listing	Contents.
corner.mws From 494 AdvMTech	Standard example of smooth (C^infty) curve that has a corner (distinguish the curve - a function - and its image).
sterografic.mws, sterografic.html r8, 2007	Illustrations and formulas for stereographic projections, both for using equatorial plane and for using polar tangent planes. Includes inverses and transition maps (but not yet tangent maps and geodescis ans their images.)
hopf.mws, hopf.html r4, 1998	Explorations of the Hopf map: S3->S2. Local coord's via stereographic projection. rank?
coords.mws From 494 AdvMTech	Orthonormal normal frame of normal vector, and pair of tangent vectors for a surface in 3D. Purely symbolic calculations involving derivatives of unspecified functions.
coord0.mws From 494 AdvMTech	Coordinate transformations of functions and equations. Normal forms for conic sections. Implementing coordinate changes using equations. Assignments can't possibly work!
coord2.mws From 494 AdvMTech	Coordinate transformations of systems of differential equations. Working with derivatives of coordinate transformations, implemented as sets of equations. From specific examples to general formulas.
coord1.mws From 494 AdvMTech	Coordinate transformations of vector fields: differential equations, column and row vectors, 1st order partial differential operators differential forms: Tangent maps, pullbacks and push-forwards. Working with Jacobians. The need to be explicit about changes from (x,y) to (x(t),y(t)) and vice versa. Implementing vector fields as differential operators, and transforming these.... From substitutions in concrete examples with specific formulae to the general case.
coord4.mws From 494 AdvMTech	Differential forms, exterior algebra. More details on the D operator. The advanced implementation in the tensor package.
AltSym.mws AltSym.html r8, 2007	Sample calculations / demos of the Alt and Sym maps for tensors. Barehanded implementation. Use for demo of the images, especially their dimensions.
flow.mws r8, 2003	Contrast vector fields as column vectors and as (pure) functions Many images, animations, flows, nonintegrable distributions, loops that do not close and holonomy, Lie brackets algebraically and dynamically, tangent maps, tangent bundle maps push forwards,... Somewhat experimental -- and still unfinished: Did not manage to completely implement the pushforward as a single line procedure of type nested-arrows (input is pure function that maps point to pure function which maps pure function to pure function) -- and output is of the same kind -- the tricky part is to implement the inverse of the flow and compose it with the input vector field......
coords1st2nd.mws, coords1st2nd.html r4, 1998	Demonstrations of changes to coordinates of the first kind, and to coordinates of the second kind, respectively.

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9.c. Differential Geometry c
Contents: Riemannian geometry, geodesics, curvature, parallel transport, connections.

directory listing	Contents.
viewRmetric.mws r8, 2003 (r4, 1998)	Calculate the Riemannian metric for imbedded surfaces in R3 and visualize 2-D Riemannian manifolds.
gij.mws r8, 2003 (r4, 1998)	Record of simple in-class example: quick calculation of the matrix gij for sphere and torus as parameterized surfaces.
doublepend.mws 8 2003 (r4, 1998)	Example from mechancis demonstrating origin of the metric as inertia tensor. Includes Christoffel symbols.
riemann1.mws, riemann1.html doublepend.mws 8 2003 (r4, 1998)	Example from mechancis demonstrating origin of the metric as inertia tensor. Includes Christoffel symbols.
riemann1.mws, riemann1.html r8, 2003 (r4, 1998)	Demonstration of calculations of Riemannian metric, by hand w/ the tensor package.
metric.mws From 494 AdvMTech	A very basic development of Riemannian metrics based on a concrete example, the graph of a function z=f(x,y): Local coordinates, the metric G=g^ij, Christoffel symbols Gamma_ij^k, geodesic equation, visualization. Handling more complex expressions that lead to large symbolic output: Calculating G for a concrete example as a matrix, calculating its derivatives to form Gamma_ij^k, constructing the geodesic equations, solving them numerically and plotting the solutions.
euler.mws r6, 2000	Sectional curvature for a surface. Euler's theorem for curvatures of intersection curves of smooth surface with normal planes. Somewhat more complex calculations, but working with specific surfaces. Includes animations.
gaussmap.mws r8, 2003 (r4, 1998)	Implementing the Gauss-map, and visualization using side-by-side plots (color keeps track of location). Color alone as attempt to visualize the Weingarten map.
gauss.mws r8 2003 (r4, 1998)	Calculate the Gauss curvature of parameterized surfaces in 3-space. Visualize the curvature using color-coding.
egregium.mws r8 2003 (r5, 1998 ?)	A semi-manual, semi-CAS calculation that PROVES that the Gauss curvature depends only on the metric. w/ huge effort to make MAPLE output appear somewhat similar to usual type-set text...
parallel.mws r4, 1998	Parallel transport on 2-dim R-manifolds. Set up and solve DE. Visualize as still-images or as animation, in coord plane or on imbedded surface (graph or parameterized).
connexpolar.mws r6, 2000	Christoffel symbols for polar /spherical coordinates, picture. MS.doc picture and comments
connex1.mws r6, 2000	Explorations on the graph of a function z=f(x,y): The relation of the Christoffel symbols to the derivatives of the metric and first steps towards a covariant derivative of vector fields. (Includes careful discussion of ambiguities of imbedded surfaces and "projecting out normal directions".)
geocurv.mws r4, 1998	Geodesics on graphs of functions z=f(x,y): Compare geodesic eqn, curves whose acceleration is normal to the graph, and solving a constrained minimization problem via Lagrange multipliers.
geodesics.mw r10, 2006 geodesics.mws r8, 2006 geodesics.html r8, 2006 (r4, 1998)	geodesic flow and geodesic spheres. Great images combining geodesics with color-coding by Gaussian curvature. The worksheet has been saved with all output as some of it takes time (basic execution w/ all standard examples 450 sec CPU-time, on a 450MHz Pentium) to recalculate -- and results in a 4MBplus file. A small file without output is geodesics0.mws. The .html file is based on the r8 version

MAT494

 aa_new_one.mws
 dirac.mws
 ellipseL.mws
 lap.mws
 mk08.mws
 mk09.mws
 mk10.mws
 multifcn.mws
 pde_soln.mws
 side-rels.mws
 tables.mws
 z_new_one.mws

10. Geometric Control Theory 10.a. Geometric Control Theory Contents

directory listing	Contents.
Vessiot package	Vessiot package by Ian Anderson of the Formal Geometry and Math. Physics group at Utah State

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