Date: 5-may-2018 Files by: bob jantzen, klaus volpert

[abandoned 2014 from VU Math website]

Many of the files in this subdirectory are MAPLE Release 4 and Release 5 .mws worksheets, some have been updated to more recent .mw files. All the worksheets are listed in this file, although the actual filenames are all lowercase. All .mws files need updating. Some of these are not so interesting, and others are extremely interesting: a mixed bag of toys. Eventually maybe I will clean up some of the less useful files. [nope.]

[Here is a short History of Curvature by Dan Margolit written when he was an undergraduate.]

**LambertW function in pharmaceutical problem**

- /lambert

This arises in the simplest drug absorption model when trying to express one of the two decay coefficients in terms of the peak time and the remaining decay coefficient.

**Miscellaneous stuff (filenames are actually all lower case):**

- animate.mw

Some examples of animations with plottools.

animate2.mw

An rotatable animated (rotating) 3d graphic.

animatedcardioidfamily.mw

Animated family of cardioids with a fixed tangent line slope tangent line found numerically included

animatedcardioid.gif - bin2norm.mw

Deriving the continuous normal distribution from the discrete binomial distribution in the limit of large n, by Les Wright. - calclpop.mw

Finding the best-fitting logistic equation to given data is not easy, because the parameters in the equation are non-linear. This worksheet discusses a method, that first makes a `good guess' at the parameters, and then uses graphical methods and calculus to find a local minimum of the error function near this guess. Regarding this problem see also several worksheets above: LGTRUDSM.MWS, LOGDOWN.MWS, LOGISTIC.MWS etc. - chaos-monks.mw

This is a comprehensive and well-written Maple notebook containing some tools for studying mathematical chaos. It was written by Ken Monks, who originally created this sheet for use in an undergraduate course on chaos and fractals which he developed and taught at the University of Scranton. [needs updating]

chaos5to10.mws

An amateur effort at coding iterations of functions, with animation of the iteration added with the Maple5 to Maple10 update. - philly/index.htm

Finding the geographic center of the city limits of Philadelphia using calculus. Updated in 2013 to find the circumference of the perimeter.

other updated files: philly/GeographicCenterWorksheet.mw, philly/NewHampshire.mw

- colors.mw

A discussion of the difference of two coloring systems that Maple uses: the RGB-spectrum and the HSV-spectrum.

New: Color2.mw - COMBINE.MWS

A differential equation dsolve example which shows the importance of the combine command. [fixed in later versions of Maple]

competingspecies.mw

Phase plane analysis of equilibrium points in the competing species nonlinear 2x2 system of ODEs - conics_from_rectangle.mws

[2009 update with trammel remarks: conics_from_rectangle.mw]

A cute geometric construction for the conic sections arising from pushing on a simple teacher test question. Apparently this rectangle construction has a hidden trammel in it [the "trammel of Archimedes"].

[related to this: ellipseinspace.mws, another curious question about an ellipse in space, how to identify it]

[a recent article on this topic:

**An Ancient Elliptic Locus**

By: John E. Wetzel

j-wetzel@illinois.edu

In the plane, two vertices*P*and*Q*of a triangular tile*PQR*move respectively on two intersecting lines*p*and*q*. What is the locus of the third vertex*R*? Frans van Schooten the younger showed in 1646 that the locus is an ellipse. We present two proofs of this result, one an elegant deduction from the special case in which*R*lies on the line*PQ*that was given by John Casey in 1885, the other a clever argument using motions that John Sullivan included in a 1996 paper that has never been published.

- cutsurf.mw

An incomplete worksheet showing the piece of a surface above a nonrectangular region of the plane. - cycloid.mw

A neat worksheet written by Gerhard Bitsch that visualizes the creation of a cycloid by a point on a moving cycle. - datainout.mw

How to read in or write out data from/to a file, which is then fit using least squares regression.

**dipstick.mw**

A very thorough discussion of the problem of calibrating a dipstick for a tank, that is of the shape of a surface of revolution. includes an interesting comment from MUG concerning the speed of numerical integration routines.- visualizefunctionfamily.mw display families of functions, Explore
- DSOLICFM.MWS

Patches the Maple V Release 5 differential equation solver to allow indexed function names to be used with initial conditions. - DSOLVNR5.MWS

Based on a discussion in MUG, this worksheet explains how to make MAPLE functions of the output of dsolve/numeric solutions which can themselves appear in other deq's or be integrated etc.

- earthoscillator.mw How long would it
take to fall through the Earth?

- EigenExplore.mw

Graphically explore eigenvectors and eigenvalues in the plane.

- ERRORBAR.MWS

A procedure that draws an 'error plot' for given statistical data. - FITODES2.MWS

Estimates of parameters in differential equations (by Preben Alsholm): Given an initial value problem diff(y(x),x) = F(y(x),a,b,c); , y(x[0]) = y[0];, where a, b, c, ... are unknown constants. Two lists are given: One containing the x-values and one containing the corresponding y-values. We are to determine the constants a, b, c, ... such that the solution curve y = f(a,b,c, ... , x) in a 'best possible' way passes through the given points.... -
**frenetellipse.htm**[see also the complete package]

Finding the points on an ellipse such that the line segments from a given point in the plane to the ellipse are perpendicular to the tangent direction, i.e., coincide with normals to the ellipse;

See also ellipsoid_normals.mws, .mw, Arthur Cayley investigated the geometry of the ellipsoid, Jacobi solved the geodesic problem: references, the analog of the evolute is called the centro-surface of the ellipsoid, this worksheet develops a plotting routine for these surfaces (there is one component of the surface for each of the two principal curvature centers of curvature)]

see also Evolute_Animation.mw.

- fourier.mw

A nice procedure, written by Christine Wilson, that calculates the Fourier Series for a given function, and animates the successive approximations. As an example there is a very nice visualization of the Gibbs phenomenon. - goldenratiotrick.mw

- gridline.mw

Provides a procedure that plots the graph of a function together with a user-adjustable underlying grid. **HersheyKiss.mw**

Fitting the Hershey chocolate Kiss profile with a Bezier (parametrized) curve, and finding its volume and center of gravity.-
linearprogrammingsequence.mw

An example of a sequence defined by a sequence of linear programs, solved using Maple's linear programming command. **logistru.mw [old, does not work]; new: logisticfit.mw (one has to be careful with large input numbers as explained here)**

Fitting the logistic curve to data using the statistics Fit command!

- mapleleaf.mw

Morphing Maple leaf. Area, centroid, circumference: original code courtesy of the book by Klimek and Klimek: Discovering Curves and Surfaces with Maple (1997, Springer), page 28. - newtoncotes.mw

Newton-Cotes coefficients generalize Simpson's rule (uses Vandermonde polynomial bases). - PLTARY3D.MWS

Displaying Arrays of Plots (still not entirely satisfactory in MAPLE V Release 4 or Release 5!): Either all 2D or all 3D plots. - POLYCOLO.MWS

Shows how to color a triangulated polygon (2D), and a polyhedra provided by the plottools or geom3d-package, using the commands polygonplot and polygonplot3d.

**rainbow.mw**

The Calculus of Rainbows from James Stewart: Calculus Concepts and Contexts 7e Applied Project Section 4.1, Page 282. Evaluating the various rainbow angles.- raindrop.mw

Method of Gradient Descent, example of numerically solving a system of ODEs and plotting its solutions, orthogonal trajectories to a contourplot. - ReverseReductionofOrder.mw
reversing the reduction of order of a 1 order linear constant coefficient DE
system to a single higher order decoupled DE.
- RGB_Hue_and_HSV.mw

Understanding the color parameters for nice effects in 3-D graphics especially - rootseq.mw fast root finding using iteration
- roxanasystem.mw [pdf]

Exploring square linear systems whose augmented matrices have rows which are all either arithmetic or geometric sequences. The latter is only clear for 2x2 systems. Higher dimensions remain a mystery. The worksheet goes further than the PDF.

- stewart-movieviewing.mw

James Stewart Calculus:, Applied Project 6: Where to Sit at the Movies. - SEQPTPLT.MWS

Generating Sequences of Pairs of Points for Plotting. - series.mw

For those undergraduates who jump into multivariable calculus as freshmen and miss power series, a short review. - sfps3-16.mw

James Stewart Calculus: Concepts and Contexts, Focus on Problem Solving 3.16, p 265. Locus of points where pair of tangent lines to ellipse are perpendicular. - sfps5-1.mw Calculus Early Trancendentals 83

James Stewart Calculus: Concepts and Contexts, Focus on Problem Solving 5.1, p 444. Divide a 14 inch diameter pizza into (equal area) thirds by 2 parallel cuts. - [1998 version from solution by Fritz Hartmann: sfps5-6.mw] [bob version 2015 Stewart 8e: s5plus-14.mw]

James Stewart Calculus: Concepts and Contexts, Focus on Problem Solving 5.6, p 445. A circular disk of radius r is used in an evaporator and is rotated in a vertical plane. If it is to be partially submerged in the liquid so as to maximize the exposed wetted area of the disk, show that the center of the disk should be positioned at a height r/sqrt(1+Pi^2) above the surface of the liquid. (Solution by Fritz Hartmann)

- s5plus-16.mw

James Stewart Calculus 8e. Archimedes parabola sector area theorem in general and by his method of exhaustion. -
../mat1505/handouts/s6plus-4a.mw

James Stewart Calculus 8e. Tilt a class of water till the water level touches the top of the rim of the bottom base. What is the volume of water? Going further, how much work would it take to pump all the water to the rim. - shadein.mw

Shading in the region between two function graphs in the plane over a specified interval, by Tim Feeman. This is partially outdated since the Maple plotting has gotten more sophisticated, but when the region between two curves also crosses the horizontal axis, it is more efficient than the following plot option which requires dividing the interval up into further subintervals in which the two curves are ordered on each subinterval, since the "filled = true" option shades between the curve and the horizontal axis.

Here is a nice example using the simpler "filled = true" option which does not have this complication: cubicareaputnam.mw

However, subtracting 1/2 from these functions being graphed leads to the additional problem. - displaysidebysidevideo.mw how to synchronize side by side videos to play together
- sound.mw

Uses a telephone number as the coefficients for a Fourier Series (=sound). Shows how the telephone number can be retrieved via Fourier analysis. (part of CSC1024 demonstrations)

- squarewheel.mw inverted catenary road
profile allows a square wheel to ride with its center moving horizontally
only.

- trianglecentroid.mw

Derives the centroid of a triangle and extends it to a polygon. - transfercurve.mw

Stewart Calculus 5e applications of curvature: transfer curve connecting straight pieces of railroad track. - umbtorus.mw [more
about this]

From the Larson et al Calculus (6th Edition) by Houghton Mifflin (Prentice Hall): The Umbilic Torus (an interesting 3-D plot) - valleep_bob8.mw

Critical points of the Vallee-Poussin polynomials plotted in the complex plane for high n. - earthcentricvenus.mw

Venus Earthcentric orbits make a beautiful polar coordinate curve design

**OUTDATED.**

The nonlinear curve fitting files below were described in
The
Maple Reporter
Summer
1998 Edition (pp.6-7 article: *Fit for Anything* in this PDF file,
now lost). It took a decade for Maple to finally address this problem with a new
Statistics Fit command. See below.

**Logistic and nonlinear curve fitting [maple 6? recall the change maple5: " to maple6: % for the ditto operator;
require updating to maple13]**

- LGTRUDSM.MWS

True Least Squares Logistic Regression with Downhill Simplex Minimization by Les Wright using the MAPLE routine:

DOWNHILL.M

extracted from the worksheet

DOWNHILL.MWS

the downhill simplex minimization method in multidimensions, implementation by Dr Francis J. Wright [new] [seems to work better than Alshom's procedure for this problem] - LOGDOWN.MWS

Solving Logistic Regression Problems Using Downhill Simplex Minimization - LOGISTI2.MWS

Preliminary worksheet to the next one: - LOGISTIC.MWS

Logistic Curve Fitting via Linear Least Squares Applied to Transformed (linearized) Data. - LOGISTRU.MWS
**[now outdated by Statistics Fit command! see:**logistru.mw]

True Least-Squares Logistic Regression by Les Wright using the following nonlinear regression procedure: - NNLINFIT.MWS

A nonlinear regression procedure by Preben Alsholm. - LOGFIT.MWS

An application of both the nonlinfit procedure and the downhill simplex minimization method to fitting a 3-parameter family of logarithmic curves to some data. - MTC1093.HTML

Interesting SIAM web article on the Levenberg-Marquardt nonlinear regression problem. - NLINFIT3.MWS

NONLINFIT3.MWS: The damped Gauss-Newton-fit procedure by Preben Alsholm, September 1, 1998 (original version: December 12, 1996) - FITODES2.MWS

FITODESYS2.MWS: Estimates of parameters in differential equations Preben Alsholm, August 28 1998 - LSQDDEM1.MWS

Least Squares Nonlinear Regression and Downhill Simplex Minimization: A Marriage of the Two by Les Wright, January 1, 1999 - LSQD.M

Accompanying Least Squares Nonlinear Regression .M file by Les Wright, January 1, 1999 - NLFIT.MWS

Single Independent Variable Nonlinear Regression with Levenberg-Marquardt procedure by David Holmgren, November 29, 1999 - MNLFIT.MWS

Multivariable Nonlinear Regression with Levenberg-Marquardt procedure by David Holmgren et al, November 29, 1999