Matthias Kawski
School of Mathematical & Statistical Sciences 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 9221460, DUE 9752453, and DMS 0072369. 
For uptodate info visit the manufacturer Waterloo MAPLE 
0. Tutorials
0. Tutorials

day0.mw
day0.html r15, 2011 (r13, 2010)  Getting started  basic features such as worksheet layout, casesensitivity, datatypes, packages. Written for calc III. 
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 
interactive.mw
interactive.html r15, 2011 r8, 2005)  Procedure that takes interactive input from keyboard (example) 
1. Calculus
1.a. From Brief Calculus, MAT 210

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 CASfree 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 .... 
1.b. From Calculus I and II, MAT 270 and 271
Contents
directory listing  Contents. 
0310270diff.mws
0310270diff.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 0310270diff.pdf 0310270diff.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/ Rsums 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 .... 
newton.mw
newton.html
r18, 2016  Newton's method and monotone rational sequences bracketing irrational square roots. 
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 arclength of a parameterized family of curves. Contrast with numerical simulations. Very explorarory in character  openended! 
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 .... 
2. Multivariable and Vector Calculus
2.a. Functions of two or more variables

directory listing  Contents. 
pendulumwave0.mw
pendulumwave.mw (11MB) pendulumwave.html animated gif only r15 2011  Animations of a pendulum wave (variable lengths). Easy to modify the lengths. 
cross_sections.mws
cross_sections.mw cross_sections.html (r8 2003) r18 2014 
Plotting graphs of z=f(x,y), horizontal crosssections (contours),
vertical crosssections 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 (release 4, approx 1996): plots3d.mws and volcano.mws 
maketable.mws
r4 ??, 1996  Only release 5 of MAPLE has a builtin spreadsheet 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 3dgraphs. This graph is revisited later when working with partial differential equations. 
heateqn0.mws
r4 ??, 1996  This is "heateqnLITE", 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. 
2.b. Vectors in the plans and 3space
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 MAPLEcalculation (Problem CCH 12.2/11) involving an airplane, climbrate, windvelocity and groundvelocity. 
dist2lines.mw
dist2lines.html r15 2011  Mainly pictures to illustrate the reasoning behind using vectors (cross product and projection) to find the distance between two lines, and the pair of points on different lines that minimize distance. Allows for random points  but good views are remarkably hard to generate. 
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.) 
2.c. Parameterized curves
Contents
directory listing  Contents. 
diff geom I  See also Diff Geom I diff geom I 
1614.mws
r5 ??, 199x  From tables of values, and the sideviews x=g(t), y=h(t) to the view of the parameterized curve (x(t),y(t)). Problem CCH 16.1.4 
introhelix.mw
introhelix.html r15, 2011  Intro to MAPLE. Plot parameterized curves. Calculate arclength integrals. 
paracurvesretrograde.mw
paracurvesretrograde.html r18, 2014 
Clipped commands for parameterizing a smileyface (project 1).
Animation of solar system, retrograde motion of mercury (vector addition of curves). 
mercury.mw
mercury.html r15, 2011  Uncommented worksheet  retrograde motion of mercury. nice animation of vector difference, translation to origin. To be combined with paracurvesretrograde.mw 
viviani.mws
viviani.html 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 updated.) 
acc_2d_curv.mw
acc_2d_curv.mws acc_2d_curv.html (r4, 1995?) (r13, 2010) velacc.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 arclength. In particular, constant speed animations, and nice osculating circles. 
rollercoaster.mws
r5, 2000  An attempt to create a fun rollercoaster  3Danalog of project 1  as a piecewise smooth parameterized curve. Not as easy as it looks..... Should plan ahead more! 
repara.mws
repara.mw repara.html r14, 2011 (r4, 1996)  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 3dimensional graphics. (Also very useful for the first team project). 
project1a.mw
(r4, 1993) r18, 2014  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  
2.d. Partial derivatives, optimization
Contents
directory listing  Contents. 
mwsa.html
r5 ??, 199x  Vertical slices of 3Dgraph, animated vertical crosssections. 
partials.mws
r5 ??, 199x  Similar to volcano.mws, but older (spring 1996), and more oriented towards partial derivatives. 
chainrule.mw
chainrule.mws chainrule.html r15, 2011 (r4, 1996) 
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 (zerosets 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... 
onlysources.mw
onlysources.html (2017)  Surface with infinitely many maxima, no saddles, and no minima. Polynomial surface w/ two maxima, no saddles , and no minima. Compare AMM (193), vol 100. no.3, Durfee, Kronefeld et.al., fig. 3.1. .pdf 
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. 
classifyCP.mw
classifyCP.html 2nd_deriv_test.mws r15 2011  Horizontal cross sections of graphs of quadratic functions. 
xyoverxsqrysqr.mw
xyoverxsqrysqr.html r16 2014  Pictures and animations of the graph and its crosssections of xy/(x^{2}+y^{2}). Partials exist everywhere but the function is not differentiable, not even continuous. (Written as review in Applied Complex Analysis course.) 
disc.mw
disc.html r18 2016 (r17 2014 ) 
Simple commands (suitable for simple polynomial examples, written for novice user)
to calculate partials, find critical points, and classify them.
fun examples with 3 saddles. animations: who is stronger X^2 and y^2 or x*y? Nonnovice commands to display all results in one nice table. Added on the side: Plotting commands  hard to find the right window. 
optim2.mws
r8 2003 (r4 1997)  Unconstrained and constrained optimization w/ silly cubic polynomial example (as in oldfashioned textbooks), but constrained to a polygonal area!!! Emphasis on actually creating meaningful pictures which help organize and making a strategy. Along each edge one soln via parameterization and singlevarcalculus, 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.mw
steepdescent.html r15, 2011 (r5, 1996)  Use (naive, yet powerful) numerical techniques from differential equations to calculate the location of minima and maxima. 
lagrange.mw
lagrange.html
r18 2016 (r4 1995) 
Sample calculation and background material
txo explain how Lagrange multipliers work.
Includes grafix to relate to singlevariable 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. 
2.e. Iterated integrals
Contents
directory listing  Contents. 
Visualizing the regions defined by the limits of iterated integrals.
Many options for slices, blocks, or columns. Accepts Cartesian, polar,
cylindrical amd spherical coordinates. Ideal for visual checking of
one's work, provides detailed visual feedback on mistakes  students
will "work until all answers are perfect" (grading is trivial).
Originally distributed in MAPLE V release 3 as the "package" asu.ms, for details see the accompanying journal article 3Dgraphics for iterated integrals (with S. Holland), MAPLETech, 4 no.1, (1997) pp. 9298.  
outlines.mw
outlines.html r15 2011 (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. 
outlines3d.mw
outlines3d.html r15 2011 (r4 1997)  Similar to outlines.mws, but now in 3D. Main examples: straight and skew pyramids, icecream 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. 
s01t3.mw
s01t3.html r15 2011 (r6 2001)  Sample solutions  finding limits for iterated integrals: A skew pyramid six times. Novelty: miniprocedure diplays limits in title. (Saved with output  6 grafix.) 
iterint.xls
iteratedintegral.xlsx"  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.mw
r15, 2011 (r4, 1996)  Double integrals illustrated. A precursor (spring 1996) to the worksheets on_edge.mw and outlines3d.html. 
on_edge.mw
on_edge.html r15, 2011 (r5, 1997)  A sample calculation for a region (quarter annulus) in the plane, asking for the specific dimensions of the region that will cause the center of mass to lie on the boundary of the region. 
coordchg.mw
r15, 2011 (r4, 1995)  Coordinate changes: Work in progress. To illustrate where the Jacobian comes from as magnification factor. Mapping curves and small rectangles from plane to plane. 
2.f. Parameterized surfaces
Contents
directory listing  Contents. 
easyvase.mws
r6 2001 (1997)  A simple intro to surfaces of revolution. 
newvase.mw
newvase.html r18 2016 (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 3space (examples to help w/ purefunction notation, uncommented worksheet). 
lagrange_identity.mws
r8, 2003  uncommented sample calculations for dA via cross products and via Lagrange identity yielding dA=sqrt(EGF^2) for general parameterized surface, sphere and torus. 
2.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 problemsset: "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 1 page 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. HTMLversion 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. 
lineintfence.mw
lineintfence.html r16 2014  3Dviews of lineintegrals $\int_C f(x,y) ds $ and $\int_C f(x,y) dx $, rotate to see Riemann sums as areas of a curved fence and of its projections. 
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 lineintegrals (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 calculusI review (spring 1997). 
0310362t1sol.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 testbed 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 colorcoding 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. 
2.h. Calculus of vector Fields in 3space
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 pointcharge, first naively, then using the divergence theorem. Complements handwritten calculations (directly from this index). 
newstokes.mw (8MB)
newstokes0.mw newstokes.html (r3, 1996) r6, 2011 
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. Many parts now superseeded by the JAVA Vector Field Analyzer II. With output: newstokes.mw (8 MB), without: newstokes0.mw (190 kB) 
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 handdrawn stillsketches (these can be resized and rotated!), but the derivation of the formulas is still left as inclass 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 3dversion 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 recreate). 
2.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 accordionlike, standard counterexample. 
zoom11.mws
r5 ?, about 1997 ?  Resources for zooming in all different ways. Contains: 1.1 Limits of sequences, 2.1 slopes and lines, 2.67 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 cutoff functions that simulate the vector field flowing in a pipe, and that vanish outside. 
Optim2.mws basics.xls day1.mws dblint.ms fallcat.mws iterint.xls project1 project2 project3 testsReturn to the top
3. Linear Algebra
3. Linear Algebra a

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.mw
det.html 2016 (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 3Dgrafix 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 1dim 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. 
4. Probability
4. Probability

directory listing  
NnoMatch.mw
NnoMatch.html
r16, 2015 
Simple experiments:
What is the probability that if N people are each randomly matched
with one of their N partners that none of them will be matched with
her/his own partner?
Simple experiment, random data generation for quick demo in a probability class. Basically create a random sequence, check whether any a(n)n=0, Then do this for larger values of n, and a few hundred trials ... see convergence to 1/e. 
5. Basic mathematical structures
5.a. Recursions and Induction

directory listing  Contents. 
twocyls.mw
twocyls.html r15, 2011  Two cylinders over regular polygons intersecrting at right angles. 
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 "Lshaped triominos"  very nice images! 
countQ.mw
countQ.html r18, 2014  Demonstrations of explicia formulas for counting off the product of countable sets and for its iverse (rationals are counatble). Also includes the fast formula using unique factorizations into primes. 
pidances.mw
pidances.html r15, 2011  Expanding as a binary. Inspired by ViHart's pidance 
coin.mws
coinweighing.mws r8, 2006  work in progress  given 12 coins one of them false (either light or heavy) devise an algorithm 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. 
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 supaxiom]. 
unifconv.mw (16MB)
unifconv0.mw unifconv.html r16, 2012  Animations of converging sequences of functions to motivate the need for different notions of convergence (pointwise/uniform). Loss of regularity (continuity/differntiability) and integral of limit not necessarily being equal to limit oif integrals. Worksheet w/ animations is huge unifconv.mw (16MB), ioutput removed: unifconv0.mw. 
see also:
converge.mw converge.html  Similar animatios originally written for complex analysis. But animations are for real examples. 
nothomotopy.mw (15MB)
nothomotopy0.mw nothomotopy.html r16, 2013  Graphs and animations of a purported homotopy that continuously (?) deforms a loop winding twice around a circle into one that winds once around the circle. Various preparatory graphs and animations of crosssections, and final animation of graphs(!) of deformed loops (i.e. "curves" in a torus). Mathematical issue: Continutiy in each variable does not imply continuity. Output removed: nothomotopy0.mw 
5.c. Abstract Algebra
Contents
directory listing  Contents. 
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 grouporders 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). 
RSAMC.mw
RSAMC.html
r16, (2014)  A first naive try to implement RSA encryption. Two primes w/ 55 digits (decimal) each work fine. 
Fquot.mws
Fquot.html
r8, (2005)  Make list of polynomials in F_{n}[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 byhand calculations first). 
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. 
irredredmodp.mws
r8, (2005)  Uncommented calculations illustrating that x^{4}+3x^{2}+1 is irreducible over Q, but factors nontrivially in every Z_{p}[x]. 
Fexttables.mws
Fexttables.html
r8, (2005)  Addition and multiplication tables for the congruence classes of irreducible quadratic and cubic polynomials over Z_{2}[X] and Z_{3}[X]. For comparison, also tables for the rings Z_{4}, Z_{8}, Z_{9}, Z_{2} x Z_{2}, Z_{2} x Z_{2} x Z_{2}, Z_{3} x Z_{3}. As first steps towards splitting fields, also tables of function values of quadratic and cubic irreducible polynomials evaluated on the fields of congruence classes. 
fieldext.mws
fieldext.html
r8, (2005)  Some very simple, sample calculations about a cubic extension of the field F_{2} (compare Fraleigh exercise 27/25). Uncommented. 
ringisos.mws
ringisos.html
r8, (2005)  Addition and multipklication tables for small rings, comparing Z_{m} x Z_{n} and Z_{mn}. 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 rotationalsymmetry group of the cube (which is of order 24) in terms of matrices and permutations of faces (S_{6}), of vertices (S_{8}), and of edges (S_{12}). Includes animations of orbits of colored cubes under selected subgroups, ready to take any cyclic (ordered) subgroup as input for an animation. 
game204.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). 
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=S_{3} and G=Z_{3}, and H being a subgroup of index m=3. 
5.d. Geometry
Contents
directory listing  Contents. 
hypell.mws
hypell.html hypell.pdf r15, 2011  pictures of orthogonal families of ellipses and hyperbolas with same foci. 
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. 
6. Differential Equations
Return to the top
6.a. Motivation. Modelling. Numerical

directory listing  Contents. 
uspop.mws
r8, 2005 (r3, 1995) 
Graphical and numerical exploration of logistic model and
timevarying linear model for USpopulation.
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.mw
r18, 2014 (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). 
lotkark.mws
r8, 2005 
Introductory discussion of numerical solutions of DEs in the testcase 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 "handwritten" 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) 
Phaseportrait 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 realtime animation of solution curves. pendulum.mws w/ live animation 750kB, pendulum0.mws no output, 
6.b. Elementary ODEs I
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 nofrills samples for the first time user. HTMLversion. 
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 complexexp versus realtrig soltuions in underdamped case. Animations to see effect on solutions of varying damping coefficient, incl. rootlocus 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 highlevel 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 
Backup for a paperpencil 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. 
varparexa.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. 
LTantideriv.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. 
undetcoeff.mws
undetcoeff.html r8, 2005  Introductory code developing the method of undetermined coefficients for nhomogeneous second order linear constant coefficient ODEs through a systematic, handson 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). HTMLversion. 
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) HTMLversion. 
convolution.mws
convolution.html r8, 2007  Many animations relating to convolutions and impulses in the context of 2nd order linear timeinvaraint 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 coldmedicine. 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. 
6.c. Elementary ODEs II
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. 
de12ex.mws
r8, 2005 (r3, 1995)  An integrating factor example  grafically: DEplot, dfieldplot 
de61lt.ms
r8, 2005 (r3, 1995) 
Comparison of using Laplace transform on one second order linear
const coeff DE, or on
one timeinvariant system of two first order DEs. Includes some nice plots. 
dehw7.mws
r8, 2005 (r3, 1995) 
Many worked exercises with Laplace transforms, incl impulsively forced DEs.
Original use
as backup comparison w/ paper&pencil work, to practice MAPLE syntax, and to see some plots. 
dehw8.mws
r8, 2005 (r3, 1995) 
Many worked exercises with Laplace transforms, inclystems of two 1st order DEs.
Original use
as backup comparison w/ paper&pencil work, to practice MAPLE syntax, and to see some plots. 
dehw9.mws
r8, 2005 (r3, 1995) 
Many worked exercises with matrix exponentials, incl numerous phase protraits.
Original use
as backup comparison w/ paper&pencil work, to practice MAPLE syntax, and to see some plots. 
delin36.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. 
det2sol.mws
r8, 2005 (r3, 1995) 
Sample calculations in small steps and plots.
Sinusiodally driven const coeff linear second order IVPs 
det3alt.mws
r8, 2005 (r3, 1995) 
One example of undetermined coefficients "worked by hand"  lots of Fourier terms, uses
concatenation operator to define variables ak. 
det3sol.mws
r8, 2005 (r3, 1995) 
Sample solutions to 3rd test. Focus on inputoutput view of second order
linear constant coefficient DEs.
includes first pic of "low pass" filter. 
fcfinal.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). 
definal.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. 
hw361a.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/ branchcuts of the logarithm.... 
6.x. unsorted
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 timevarying / parameter dependent vector fields? A very simple 3 line trial. 
mathieu.mws
r5 ?, about 1997 ? 
Timevarying 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 timevarying system may still be unstable. (trouble
with old, release 3 DEcommands  needs cleanup). 
6.d. Linear Differential equations
See also sections 6.b., and 6.c.
for more elemnatry approaches.
Also 3. Linear algebra.
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 4dimensional system worked in many different ways. Tutorial. With some fun MAPLEcoding 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 asystable (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. 
Monodromy)of_y0.mw
Monodromy)of_y0.html r18, 2014  Monodromy matrix depends on starting time [???], example. 
6.e. Existence and uniqueness
directory listing  Contents. 
picard.mw
picard.html [r8, 2006] [r18, 2017] 
Advanced implmentations of Picard iteration (successive approximations
of solutions of initial value problems) for both scalar and systems
cases. Worked examples include y'=y (incl. with nonconstant y0),
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=φ_{0} 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.f. Stability
directory listing  Contents. 
windingnumber.mw
windingnumber.html  Simple commands to generate plots that illustrate the concept of winding numbers (index of a vector field). First sample pictures suitable for handouts / overhead transparancies. Second part animations (incl. translated back to origin). 
logistictanh.mw
logistictanh.html r18, 2014  Trying to rewrtite the solution formula for the logistic equation using the initial data, and reorganizing exp and tanh in MAPLE. Comments and demonstreation of the mathematician's better model, and the price one pays when trying to make MAPLE perform the stand5rad abuse of notation y(tau)=y(tau(t))=Y(t).... 
harvest.mw
harvest0.mw harvest.html r18, 2014 (r8, 2007)  Worked exercise (HirschSmaleDevaney 1.6) on logistic DE w/ constant harvesting; bifurcationpoint of view (animations). Worksheet contrasts quick numeric work w/ laborious manual calculations for closed form solution formulas. 
supersol.mw
supersol.html r18, 2014  Quick example for sub/supersolution w/ focus on ibounds for finite escape time. 
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 backup for classdiscussion,
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. 
notradunbounded.mws
notradunbounded.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). 
lyappf.mws
lyappf.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) 
lyapnotproper.mws
lyapnotproper.html lyapnotproper.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. 
Mathieuaver.mws
Mathieuaver.html r11, 2008  Completey raw examples/ w/ illustrations of averaging theory: van der Pol and Mathieu. 
floquet.mw
floquet.html r11, 2008  Animations of a linearly unstable periodic orbit in 3d whose time frozen linearzation at every time is asystable. 
pb.mw
pb.html (r8, 2006) MAPLE 2017 
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 HirschSmale.
Many more examples added in 2014. 
6.g. Geometric/topological concepts
directory listing  Contents. 
straighten.mw
straighten.html r18, 2014  Example for algorithm to straighten out a vector field: first a constant affine change of local coordinates, followed by using the flow and coordinates on a transversal section. Even for the example of the harmonic oscillator (change to polar coordinates!) the usual algorithm has a few obstacles  but these can be circumvented with some foresight. 
080219poincareNF.mws
r8, 2008  Exploration of resonances obstructing fromal coordinate change to bring system into Poincaré normal form. 
poincareNFexample1.mws
r13, 2009  Uncommented naive example (homework) bring system into quadratic normal form. 
eulerchar.mw
.html
r2017, 2017 (2009)  Pictures of piecewise planar tori and double tori  for counting faces, edges,m vertices (Euler characteristic, genus). 
DEsingularities.mw
html
r16, 2016  First try  towards PoincareHopf: sink and saddle annihilate each other. 
poincarehopfvf.mw
.html
r2017, 2017  PoincareHopf visualization: Plot vector fields on the faces of polyhedra such that each face contains a source, each edge a saddle and each vertex is a sink. Interesting programming that disassemble the output of phaseportrait, manipulates it and reassembles 3Dplot structures. 
7. Partial Differential Equations
7.a. Fourier series for PDEs

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. introlevel 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 Fall2000 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, halfrange 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 halfrange 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 1dim 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). 
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. 
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 errorestimate using Parseval's identity. 
0010362T2.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 2dplot versus 3dsurface (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 Besselsection 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). 
Multivar calc.
heateqn.mws  For several good views of the 1dimensional 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 3Dgraph, its crosssections 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, Fouriersine 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 1Dheat equation on 1/2 plane. Many animations and crosssections. 
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. 
7.d. Partial Differential Equations, Laplace and 2dim heat and wave, rectangular domain
Contents
directory listing  
laplaceannulus.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.

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 bookkeeping, 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
readymade procedure generates carefully crafted animations.
For circular domains simply specify the desired linear combination as linear combinations of FourierBesselseries (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  EXCELspreadsheet for quick view of resonances m^2+n^2=i^2+j^2. 
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 variables stepbystep using precise commands. Exploration of Besselfunctions. 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) 
8. Complex Analysis
8.a. Complex Analysis a

directory listing  
461logo.mw
461logo.html r16 2014 (r5 1999)  Creating a colorcoded image of several sheets of the Riemann surface of z=sqrt(z^21). Few comments, but an intriguiung final grafix which served as the logo for an introductory complex analysis class. 
complex.mw
complex.html r16 2014 r8 2003 (r5 1999)  First intro to MAPLE in complex analysis class. 0. (~New, 2014): extended section for new MAPLE users. 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. 
exploreplots.mw
r16 2014 
Very simple procedures to create tables of function values,
plot the vector field, plot Re and Im parts as surfaces,
draw one surface colorcoded by the other part, first domain coloring.
For mappings of curves and regions, and good domain coloring see other wrksheets. this is a very basic introduction only. 
xyoverxsqrysqr.mw
xyoverxsqrysqr.html r16 2014  Pictures and animations of the graph and its crosssections of xyx^{2}+y^{2}). Partials exist everywhere but the function is not differentiable, not even continuous. (main entry in section multivar calculus / partials). 
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). 
conv.mw
conv.html r16 2014  Quick graphical demo of convergence of Taylor approximations for sin and for 1/(1+x^2). Suggest benefit of analyzing singularities of complexified function. 
converge.mw
converge.html r16 2014 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.mw
r17 2014 (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.mw
essential.html r15 2014 (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 powerpoint 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 3danimations for complex counterparts are disappointing leading to the later development of colormaps (see MATLAB, JAVA II, and powerpoint presentations from conferences (Odense 2000, Crete 2002). 
lineintfence.mw
lineintfence.html r16 2014  3Dviews of lineintegrals $\int_C f(x,y) ds $ and $\int_C f(x,y) dx $, rotatet to see Riemann sums as areas of a curved fence and of its projections. 
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.mw
circle.html r16 2014 (1999) 
(Completed reworked and much expanded in 2014.)
Inversion f(z)>1/z (across the circle) and Moebius transformations, pictorially and algebraically. Very nice images (think of electric charges on a loop and a mirrored loop) to demonstrate why (z_{0}+re^{it})^{1} describes a circle even though it is not easy to rewrite in the standard form. Algebraic exploration of the reparameterization. Moebius transformation form a group. Brief connection w/ linear algebra GL(2,R) 
mapping0.mw
(no images)
mapping.mw (10MB) mapping.html r16 2014 (1999) 
Main tool for visualizing complex functions thru their effects on finite
rectangular or polar grids (centered anywhere). Simple input, and carefully
crafted colorcodes (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 mouseinterface, but lacks sophisticated coloring which we deem essential. Send email 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 Fourierseries 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 byhandscalculation of residue at thirs order pole at zero. 
poisson.mws
r8 2003 (r5 1999)  Several illfated 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.) 
9. Differential Geometry
9.a. Differential Geometry a

directory listing  Contents. 
calc III  See also Calc III parameterized curves 
acc_2d_curv.mw
acc_2d_curv.mws acc_2d_curv.html (r4, 1995?) (r13, 2010) velacc.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 arclength. 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 capabilities and limitations of CAS. Suitable to be shown to CASnovices (but knowing calculus). Example: Arclength and curvature of an ellipse. Invariance of planecurvature under rotations. 
arclength.mws

Calculating arclength of parameterized curves. Invariance under the action of the Euclidean group.

curvature.mws

Curvature of plane curves: Arbitrarily parameterized curves and curves parameterized by arclength.
Osculating circles.

geocurv.mw
 Pictures and animations, and a few calculation as backdrop for discussion of geodesic curvature (covariant derivative, parallel transport). Stprysetup: driving "unstraight East" along I10 from PHX to ATL, or fly straight, or fly straight with initial heading East (to MCIA). Includes GOOGLEmaps images and quicklinks. 
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 chainrule, but downto earth application in special example. 
torusknot.mw
torusknot.mws r15 2011 (r4 1998) 
"One paragraph" worksheet providing "testcurve" for e.g. Frenetframe 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 spacecurve and animated Frenet frame..... 
serret.mw
serret.html serret.mws r15, 2011 (r4, 1997) intcurve.mws frenetold.mws 
Integrating the Serret formulas: Curvature (and torsion) completely determines the curve.
Timedependent constantlength loops. Curvature evolving according to a PDE (diffusion or 1dim 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

Just some fun pix of a Moebiusstrip and together w/ another (orientable) surface w/ same boundary. 
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

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.) 
adjointso3.mws,
adjointso3.html r13, 2010  A handson demosntration of the adjoint representation of R^{3} with the cross product considered as a Lie algebra. Eeverything is written out several times, emphasizing a connecting the abstract with the very familiar. rank? 
hopf.mws,
hopf.html r4, 1998  Explorations of the Hopf map: S3>S2. Local coord's via stereographic projection. rank? 
coords.mws

Orthonormal normal frame of normal vector, and pair of tangent vectors for a surface in 3D.

coord0.mws

Coordinate transformations of functions and equations. Normal forms for conic sections. Implementing coordinate changes using equations. Assignments can't possibly work! 
coord2.mws

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 
Coordinate transformations of vector fields: differential equations,
column and row vectors, 1st order partial differential operators
differential forms: Tangent maps, pullbacks and pushforwards.
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 nestedarrows (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. 
9.c. Differential Geometry c
Contents: Riemannian geometry, geodesics, curvature, parallel transport, connections.
directory listing  Contents. 
viewRmetric.mws
viewRmetric.html viewRie.doc viewRie.pdf r13, 2010 (r4, 1998)  Calculate the Riemannian metric for imbedded surfaces in R3 and visualize 2D Riemannian manifolds. 
gij.mws
r8, 2003 (r4, 1998)  Record of simple inclass 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
euler.html 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 Gaussmap, and visualization using sidebyside 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 3space.
Visualize the curvature using colorcoding. 
egregium.mws
r8 2003 (r5, 1998 ?)  A semimanual, semiCAS calculation that PROVES that the Gauss curvature depends only on the metric. w/ huge effort to make MAPLE output appear somewhat similar to usual typeset text... 
parallel.mws
r4, 1998  Parallel transport on 2dim Rmanifolds. Set up and solve DE. Visualize as stillimages 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 colorcoding 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 CPUtime, 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 siderels.mws tables.mws z_new_one.mwsThe old index from the class: MAT 494 MAPLE! :Advanced Math via Technology
Title and hyperlink 
Math focus 
MAPLE focus area 
mk00.mws : Commented and hyperlinked MAPLE index 
none 
Text and hyperlinks 
Part I: Introduction 
Calculus, 
Basic MAPLE skills 
mk01.mws : Saying hello 
Arithmetic, algebra. In exercise: Integration techniques, leading to discussion of ubiquitous need for keeping track of branch cuts of complex logarithm. 
Worksheet format 
purefcns.mws
Pure functions
purefcns0.mws 
Arrow notation and point of view for functions, operators, procedures, computer programs. Focus: No need for name of "x"variable 
Pure functions 
mk04.mws : (Calculus) optimization problems 
Calculus: Optimization problems. For which kinds of equations do there exist closed formulas for solutions? 
diff(), solve() 
mk07.mws : Root locus 
Rootlocus: Eigenvalues of parameterized curves in the space of 3x3 matrices ONLY UNORDERED set of evalues makes sense! 
Symbolic calculations of parameterdependent eigenvalues. Lots of visualization for "simultaneity". 
mk04.mws : Matrix exponentials 
Matrix exponentials: Diagonalization, flows of linear DEs 
Exact eigenvalue calculations in MAPLE. 
why_D.mws : Why D? 
The need to distinguish the x in top and bottom of (df(x)/dx) or (df/dx)(x). Use in ICs for IVPs in DE, coordinate changes, tangent maps and the like 
Pure functions as opposed to expressions, subs(), variables 
vidya_DE.mws : DE example 
2nd order linear SINGULAR DE, indicial equations, series solution 
Advanced work with dsolve(). 
midsoln.mws : Sample solutions 
Taylor expansions and Bernoulli numbers. Jacobian matrix. Evaluating the sineintegral function. GramSchmidt orthonormalization. Legendre differential equations. 
taylor(),convert(),linalg,jacobian(),fsolve(),dsolve(). Working with matrices and vectors, initial conditions for DEs, various plots, ..... 
10. Geometric Control Theory
10.a. Geometric Control Theory

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