ISSN 1990-7079
C o n t e n t s:
Letter of the Editor
Editorial - Preview
Wolfgang Pröpper
On Quadratics and Conics
Josef Böhm
A Conics Explorer and a Conics Trainer
Textbooks and the arctanh
Klaus Körner
Airplane and Conformal Mapping
Tensor Algebra and DERIVE for DOS
Tom Fowler
Tensor Algebra
Roger Folsom
Growth Rates and Comparative Statics
Walter Klinger
An Equation of Equations
September 2011
Proceedings of former DERIVE & CAS-TI and ACDCA-Conferences
You are invited to browse and to download lectures and workshops (all in df-format) of
(almost) all earlier Conferences starting with Krems 1992 and ending with Malaga 2010:
Krems 1992, Krems 1993, Derive Days Düsseldorf 1995, Bonn 1996, Gettysburg 1998,
Portoroz 2000, Liverpool 2000, Vienna 2002, Montreal 2004, Dresden 2006, Buffelspoort (South Africa) 2008, Malaga 2010
Some proceedings have been published originally in printed form only. I scanned them
all and interlinked them with the respective contents page. Many thanks to Peter Hofbauer who set all files on the web.
Happy Browsing and Happy Remembering for many of you - and don’t forget:
Attend TIME2012 in Tartu, Estonia
July 11-14, 2012
Browse a library of over 2,400 educational videos at
Nur in deutscher Sprache (in German only):
Franz Embachers Mathe-Online wurde erweitert und verbessert:
DUG-Mitglied Johannes Zerbs ist sehr aktiv. Gerne veröffentliche ich den Link zu seinen
Publikationen und wünsche weiterhin gutes Gelingen und viel Erfolg.
MATHE CHILLED –ONLINE - ein Buch, eine Sammlung, ein "Rätselblock" von
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Alle Infos mit der Buchbeschreibung unter Klick auf:
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→ UE-klasse-center → UE-klasse-Worddokus
Von der Homepage gelangt man auch zu einer mathematischen „Millionenshow“ und zu
einem Mathe-Quiz, Josef
Letter of the Editor
Liebe DUG-Mitglieder,
eigentlich wollte ich in diesem Newsletter
vor allem die Beiträge von Dietmar Oertel,
Piotr Trebisz und Duncan McDougall weiter bzw. zu Ende führen. Ich war auch
schon ziemlich weit mit den Übersetzungen der deutschen Beiträge, aber dann
erreichten mich einige Anfragen und
mails, die schließlich über weitere Kommunikation eine so heftige Eigendynamik
entwickelten, dass ich sie nicht länger
liegen lassen konnte. Nur Robert Setifs
Anfrage musste auf den nächsten DNL
verschoben werden.
Ich möchte Sie besonders darauf hinweisen, dass nun die Tagungsbände aller DERIVE- und ACDCA-Konferenzen im pdfFormat verfügbar sind. Beachten Sie bitte den Hinweis auf der Info-Seite.
Ich freue mich, wieder einen TI-NspireBeitrag vorstellen zu können. Mit Version
3 kann man sehr ordentlich arbeiten. Nun
sind wirklich schon einige von uns lange
gewünschten DERIVE-Möglichkeiten implementiert worden.
Besonders spannend für mich war die Beschäftigung mit der konformen Abbildung
bzw. deren Umkehrung, zu der mich ein
Brief von Klaus Körner gebracht hat.
Ich habe auch sehr wenig Ahnung über
Tensore gehabt. Dank Tom Fowler weiß ich
nun ein wenig mehr darüber. Roger Folsom
schreibt einiges über „Comparative Statics“ – wofür ich keine passende deutsche
Übersetzung finden konnte. Ich habe aber
ein paar sehr schöne Unterlagen im Web
gefunden, die zu einer weniger bekannten
Anwendung der Differentialrechung im
wirtschaftlichen Bereich führen.
Walter Klinger ist wieder zum Einsatz von
DERIVE im Unterricht zurückgekehrt und
erlebt mit der Version 6 immer wieder
Überraschungen, die er gerne mit mir
bespricht. Walter, vielen Dank für die
anregenden Gespräche.
p 1
Dear DUG Members,
Actually I intended to continue the contributions of Dietmar Oertel, Piotr Trebisz and Duncan McDougall in this newsletter. I was very busy translating the
German papers but then I received some
requests and emails which together with
the resulting communication got a strong
self dynamics and I could not leave them
on my to-do-list any longer. Only Robert
Setif’s request must be left for the next
I’d like to inform you that the proceedings of all earlier DERIVE and ACDCA
Conferences are available in pdf-format
and can be downloaded. More info on the
information page.
I am very pleased about being able to
present an TI-Nspire related article
again. Version 3 allows comfortable and
powerful working. Finally some of long
desired DERIVE features have been implemented
Extra exciting for me was dealing with
the conformal mapping and its inverse
provoked by Klaus Körner’s mail.
I must admit that I knew nothing whatsoever about tensors. Thanks Tom Fowler I
now have at least a weak idea about these
mathematical objects. Roger Folsom
writes about “Comparative Statics” – I
couldn’t find an appropriate German
translation. What I could find are some
fine resources on the web. They are informing about a not so well known application of calculus in economics.
Walter Klinger returned to use DERIVE in
secondary level math teaching. He is coming across some surprises with DERIVE 6
which he likes to discuss with me. Many
thanks for the inspiring talks.
With my best regards
Mit herzlichen Grüßen
Download all DNL-DERIVE- and TI-files from
p 2
The DERIVE-NEWSLETTER is the Bulletin of the DERIVE & CAS-TI User Group.
It is published at least four times a year
with a content of 40 pages minimum. The
goals of the DNL are to enable the exchange of experiences made with DERIVE,
TI-CAS and other CAS as well to create a
group to discuss the possibilities of new
methodical and didactical manners in
teaching mathematics.
Editor: Mag. Josef Böhm
D´Lust 1, A-3042 Würmla
++43-(0)660 3136365
[email protected]
Please send all contributions to the Editor.
Non-English speakers are encouraged to
write their contributions in English to reinforce the international touch of the DNL. It
must be said, though, that non-English
articles will be warmly welcomed nonetheless. Your contributions will be edited but
not assessed. By submitting articles the
author gives his consent for reprinting it in
the DNL. The more contributions you will
send, the more lively and richer in contents
the DERIVE & CAS-TI Newsletter will be.
Next issue:
December 2011
Contributions waiting to be published
Some simulations of Random Experiments, J. Böhm, AUT, Lorenz Kopp, GER
Wonderful World of Pedal Curves, J. Böhm, AUT
Tools for 3D-Problems, P. Lüke-Rosendahl, GER
Hill-Encription, J. Böhm, AUT
Simulating a Graphing Calculator in DERIVE, J. Böhm, AUT
Do you know this? Cabri & CAS on PC and Handheld, W. Wegscheider, AUT
An Interesting Problem with a Triangle, Steiner Point, P. Lüke-Rosendahl, GER
Overcoming Branch & Bound by Simulation, J. Böhm, AUT
Graphics World, Currency Change, P. Charland, CAN
Cubics, Quartics – Interesting features, T. Koller & J. Böhm, AUT
Logos of Companies as an Inspiration for Math Teaching
Exciting Surfaces in the FAZ / Pierre Charland´s Graphics Gallery
BooleanPlots.mth, P. Schofield, UK
Old traditional examples for a CAS – what´s new? J. Böhm, AUT
Truth Tables on the TI, M. R. Phillips, USA
Where oh Where is It? (GPS with CAS), C. & P. Leinbach, USA
Embroidery Patterns, H. Ludwig, GER
Mandelbrot and Newton with DERIVE, Roman Hašek, CZ & Rob Gough, UK
Tutorials for the NSpireCAS, G. Herweyers, BEL
Some Projects with Students, R. Schröder, GER
Dirac Algebra, Clifford Algebra, D. R. Lunsford, USA
Treating Differential Equations (M. Beaudin, G. Piccard, Ch. Trottier), CAN
A New Approach to Taylor Series, D. Oertel, GER
Statistics with TI-Nspire, G. Herweyers, BEL
Cesar Multiplication, G. Schödl, AUT
Henon & Co; Find your very own Strange Attractor, J. Böhm, AUT
Rational Hooks, J. Lechner, AUT
Cubus Simus, H. Ludwig, GER
Simulation of Dynamic Systems with various Tools, J. Böhm, AUT
and others
Medieninhaber: DERIVE User Group, A-3042 Würmla, D´Lust 1, AUSTRIA
Richtung: Fachzeitschrift
Herausgeber: Mag. Josef Böhm
Wolfgang Pröpper: On Quadratics and Conic Sections
p 3
On Quadratics and Conic Sections1
Wolfgang Pröpper, Nürnberg, Germany
[email protected]
Nowadays conic sections have nearly disappeared from textbooks. Nevertheless it is an interesting field containing a lot of nontrivial algebra.
1. (General 2-Variable) Quadratic Equations
The intersection of a circular cone with the equation z2 = x2 + y2 in R3 with a plane
α.x + β.y + γ.z = ε yields a so called quadratic which only has terms in x2, y2, x⋅y, x, y and
a constant term. It looks like
a⋅x2 + 2b⋅x⋅y + c⋅y2 + 2d⋅x + 2e⋅y + f = 0.
Due to its (geometrical) genesis its graph is called conic section or just conic.
The left side of this equation is elegantly expressed by W T ⋅ M ⋅ W with M a symmetric ma-
a b d 
trix M =  b c e  and W a vector W =  y  . This construction leads to the factors 2 in
d e f 
 1
 
the coefficients of the mixed and the linear terms [CS, 2.2].
With this contribution is connected an extended TI-Nspire document OnConicSections_NN.tns
(NN is 20 or 30 depending on which release of TI-Nspire CAS it should run. 20 is for release 2.x and
30 for the latest release 3.x). When referring to this document in this contribution we denote this by
[CS, Pr.Pg] where Pr means problem and Pg the page in this problem.
p 4
Wolfgang Pröpper: On Quadratics and Conic Sections
2. Classification of conic sections
At first we will investigate the case b = d = e = 0. Equation (1) is reduced to
a⋅x2 + c⋅y2 + f = 0.
In case a⋅c⋅f ≠ 0 the classification is done easily by analyzing the signs of these coefficients.
If a, c and f have equal signs then the solution set is empty. Otherwise we have ellipses or
hyperbolas, so called central curves with a centre point at the origin of the coordinate
system and their half-axes on the coordinate axes.
Hyperbola, symm. wrt y-axis
Hyperbola, symm. wrt x-axis
Empty set
Table 1
If however a⋅c⋅f = 0, - i.e. at least one of the coefficients disappears -, then we have 3 instances:
1. Pair of straight lines (exactly one coefficient is zero):
If f = 0 and a and c have different signs, the left side of equation (2) decomposes into
two linear factors. This results in two symmetric straight lines through the origin.
If a = 0 or c = 0 we get a pair of parallels with regard to the x- or the y-axis, if the signs
of a or c and f are different.
2. Double line (exactly two coefficients are zero):
In case a and f or c and f disappear then the x-axis or the y-axis is the only one solution of the corresponding equation.
3. Point:
If f = 0 and a and c have equal signs the equation from above is only fulfilled by the
In all other cases there will be no more genuine intersection figure, i.e. the solution set is
empty. Except if a = c = f = 0. Then equation (2) is fulfilled by the whole xy-plane.
The following table shows this in a clear form:
Wolfgang Pröpper: On Quadratics and Conic Sections
Pair of straight lines through the origin
Pair of parallel lines with respect to x-axis
Pair of parallel lines with respect to y-axis
Double line, the x-axis
Double line, the y-axis
Point, the origin
Empty set
Empty set
Empty set
Whole xy-plane
p 5
Table 2
The impact of coefficients b, d and e [CS, 2.3 and 2.4]
If just b ≠ 0 (and further on d = e = 0) then a rotation where x is replaced by
x⋅cos(θ) – y⋅sin(θ) and y is replaced by x⋅sin(θ) + y⋅cos(θ) can bring the coefficient of the
mixed term to zero without introducing linear terms in x or y. This means that we have
again one of the previous cases.
p 6
Wolfgang Pröpper: On Quadratics and Conic Sections
. (In case a = c the conic seca −c
tion is a circle which is invariant with respect to rotations around its centre.)
The condition for removing the mixed term is tan(2θ ) =
Likewise (at an arbitrary b) and d, e ≠ 0 the translation x := x + u and y := y + v may remove the coefficients of the linear terms in x and y. The conditions are u =
b ⋅e − c ⋅d
a ⋅ c − b2
b ⋅d − a ⋅e
. The coefficients of x2, x⋅y and y2 remain unchanged.
a ⋅c − b
There is a lot of algebra in the Calculator Window, isn’t it?
As one easily can see, these conditions are only valid if a⋅c – b2 ≠ 0. This case has not
been discussed up to now. Here we have a parabola, as we will see in the next section.
The parabola case [CS, 2.5]
If a⋅c – b2 = 0 and a ≠ 0 we can omit c in equation (1) and we get
a ⋅ x 2 + 2b ⋅ x ⋅ y +
b2 2
⋅ y + 2d ⋅ x + 2e ⋅ y + f = 0
This equation can be brought to the form
a'⋅x2 + 2e'⋅y = 0
with a' and e' only depending on the coefficients of equation (1'). (If in (1') the term with x2
should be omitted then (3) takes the form c'⋅y2 + 2d'⋅x = 0. The calculations are principally
the same.)
Wolfgang Pröpper: On Quadratics and Conic Sections
p 7
In both cases we have parabolas, so called non-central curves. They are symmetric with
respect to the y- or x-axis and they are passing the origin.
we can remove the mixed term and the
term which is quadratic in y in equation (1') as well. So it takes the form
Applying a rotation by angle θ with tan(θ ) =
a'⋅x2 + 2d'⋅x + 2e'⋅y + f' = 0.
And after a translation x := x + u and y := y + v with u = -
d ′ 2 − a′ ⋅ f ′
and v =
we finally
2a′ ⋅ e′
yield equation (3).
We can constrain the investigation of equation (1) to two cases:
It is sufficient that either b = d = e = 0 or that b = c = d = f = 0 (or a = b = e = f). The first
case yields ellipses and hyperbolas and their degenerations (straight lines or point). In the
other case the conic section is a parabola.
3. Classification using a matrix M [CS, 3.1-3]
By an operation HT⋅M⋅H, where matrix H is either a rotation matrix
 cos(θ ) −sin(θ ) 0 
R =  sin(θ ) cos (θ ) 0  or a translation matrix T =
 0
1 
1 0 u
a b d 
 0 1 v  , matrix M =  b c e 
 0 0 1
d e f 
p 8
Wolfgang Pröpper: On Quadratics and Conic Sections
 a′ 0 0 
can be transformed to  0 c ′ 0  , and, in case a⋅c – b2 = 0, to
 0 0 f ′
 a′ 0 0 
 0 0 e′  (or respec 0 e′ 0 
 0 0 d′
tively to  0 c ′ 0  ). As the kind of a conic section is neither changed by a translation
 d′ 0 0 
nor by a rotation, we can use matrices for the classifications provided in paragraph 2.
( )
For doing this we introduce new quantities: ∆ = det(M), δ = det a b and S = a + c.
c c
They are invariant with respect to the rotations and translations from above. So we get:
∆⋅S<0: Ellipse
Empty set
Intersecting pair of straight lines
Pair of parallel straight lines
Table 3
Again a lot of algebra!
4. Representation of Conic Sections
In this section some constructional ideas for conic sections are presented. All these ideas
use the ‘zeros’ command from TI-Nspire CAS. (So they can not be used in the nonCAS
version of TI-Nspire.)
Wolfgang Pröpper: On Quadratics and Conic Sections
p 9
If p(x,y) is a 2nd degree polynomial in x and y then zeros(p(x,y), y) yields a list of (root-)
expressions in x. The G&G application interprets this list as a family of functions in x with
two elements maximum. So a definition f1(x):=zeros(p(x,y),y) in G&G plots a graph with
two colors and labels f1_1 and f1_2.
Many thanks to Philippe Fortin (France) for providing most of the ideas realized in the following
4.1 The Conic Section Generator [CS, 4]
The classification of conic sections from paragraph 2 and 3 can easily be studied with
the ConicSectionGenerator (CSG).
The workspace is divided into two G&G windows, a (smaller) Geometry and a (wider)
Graphs window. In the left window we insert 6 sliders for variables a through f having
a range from -10 to 10, which can easily be changed. The step size is 0.1 for a, b, c,
and 0.5 for d, e, f and may also be changed as needed.
The right window has the definition
f1(x):=zeros(a⋅x2 + 2b⋅x⋅y + c⋅y2 +2d⋅x + 2e⋅y +f = 0, y), (in function mode) which plots
the (maximum) two components of the conic section.
( )
a b d
In addition the determinants ∆ = det  b c e  , δ = det a b and the sum S = a + c
c c
d e f 
are shown in the upper left corner.
In the upper right corner we see angle of inclination of the main axis and the coordinates of the centre point.
By modifying parameters a through f ellipses, hyperbolas, parabolas and straight lines
can be created and can be compared with the summaries given in tables 1, 2 and 3.
p 10
Wolfgang Pröpper: On Quadratics and Conic Sections
This is the screen after moving some sliders, i.e. changing some coefficients:
Below are the screens from above as they are presented on the handheld Nspire:
4.2 The Focus Feature [CS, 5]
For ellipses the sum of the focus rays, i.e. the line segments connecting one point on
the ellipse with the two focus points is constant. That means d(F1X) + d(F2X) = k
(= const.). For a hyperbola this condition is |d(F1X) – d(F2X)| = k. Both expressions on
the left side are containing roots. One can eliminate these roots by quite elaborate algebraic transformations [see CS, 5.2] in order to obtain a rather complicated expression in x and y which depends on the focus point coordinates f11,f12, f21, f22, and the
constant k.
The value of k can be varied by a slider. For the focus point coordinates we store the
coordinates of two arbitrary points in f11, f12 and f21, f22 respectively. Now the graph
can be plotted, using again the zeros function. By dragging the foci and changing k we
can create ellipses and hyperbolas [see CS 5.3].
In general it would also be possible to insert the focus point condition directly in G&G,
that means to define f1(x) = zeros(abs(norm([x;y]-[f11;f12])-norm([x;y]-[f21;f22]))-k,y).
Wolfgang Pröpper: On Quadratics and Conic Sections
p 11
If previously the coordinates and k have been created like mentioned above we should
get the graph of a conic section. However, it takes quite a long time (depending on the
performance of the computer) to receive a result. And if the foci are dragged around or
k is changed Nspire is blocked for several minutes and the dynamics of the application
gets lost.
A final remark: As for this example there are not so many objects on the screen, it can
also run on a TI-Nspire CAS Handheld device (with reasonable visibility). The complicated calculations from [CS, 5.2] are omitted and just the result is inserted. The file
Focus_NN.tns, where again NN=20 indicates a document for Nspire version 2.x and
NN=30 is for version 3.x is among the files connected with this DNL.
Two handheld-screenshots of FOCUS
4.3 The Five Point Feature [CS, 6]
The quadratic a⋅x2 + 2b⋅x⋅y + c⋅y2 + 2d⋅x + 2e⋅y + f = 0 in x and y contains 6 coefficients. With the coordinates of five different points inserted for x and y we get five
equations. An equation like a + 2b + c = 1 avoids that a, b and c are zero at a time and
we call it normalization condition.
This system yields a unique solution for a through f [1].
For the implementation we define a function fivepoint(P1, P2, P3, P4, P5) where
P1,…, P5 are 2-dimensional column vectors with components p1,…, p5 and q1,…, q5.
This function solves the system mentioned above and assigns the solution to
a⋅x2 + 2b⋅x⋅y + c⋅y2 + 2d⋅x + 2e⋅y + f.
We only have to draw five different points in a G&G window, assign their coordinates
to p1,…, p5 and q1,…, q5 and define f1(x):=zeros(fivepoint(P1,P2,P3,P4,P5),y). A
conic section through those five points appears. Now we can grab one or another
point and drag it around observing what is happening.
I have provided a stand alone document to run on handhelds. It is called
FivePoint_NN.tns and consists of only two pages: a short introduction and the construction in G&G. The function fivepoint is contained in the document but no longer directly visible.
Two screenshots of fivepoint are given on the next page.
p 12
Wolfgang Pröpper: On Quadratics and Conic Sections
Additional Comments of the Editor:
It might be a good question for students, how to explain the fact that there are 6 unknown coefficients
in the quadratic but only five points are sufficient to define a conic?
Dividing (1) by f yields a′x 2 + b′x y + c ′y 2 + d ′x + e′y + 1= 0 with a′ =
, b′ =
, ... . Now it is easy to see
that 5 points are sufficient. (One could also divide by a, …).
TI-Nspire has implemented a long desired DERIVE-feature: Background pictures.
I load a picture of a bridge (jpg, bmp, …) not far from my home and would like to analyze the form of
the arc. Ask students (and teachers!!) and there first answer will be: This is a parabola.
Now drag the five points as accurate as possible on one of the arcs and you will find a wonderful
curve. Go back to the Calculator App and ask for f1(x). As Wolfgang mentioned above you will find two
branches, take one of them, do a little algebra and get the quadratic. Then follow the recipe given in
CS for classifying and analyzing the curve. Or you write a program which does the job for you!
You could also use the CSG (page 10) to model the bridge (or other pictures). This is a little bit more
sophisticated but the students might find out the impact of the various parameters.
Finally, there is a German version of this contribution atöpper.pdf
Wolfgang Pröpper: On Quadratics and Conic Sections
Let’s do a little algebra to find the quadratic in its implicit form:
The next contribution will do this job for you!
p 13
p 14
Josef Böhm: A Conics Explorer and a Conics Trainer
A Conics Explorer and a Conics Trainer
Josef Böhm, Würmla, Austria
As promised on page 13 here is the analysis of the “bridge – conic”.
It seems to be a hyperbola. I say “it seems to be” because unlike to translation and rotation the conics
can change their type after applying a perspective projection which is given by taking a picture of the
object with a camera. So it could be indeed a parabola!!
In fact, the program would not give the answer parabola - even if it would be one - because of
numerical reasons. As Wolfgang explained on page 6 the discriminant must become zero – and I am
quite sure that the pixel coordinates would not allow this.
This example was the introduction.
The Conics Explorer is the core part of my Conics Trainer:
Load conics.mth as a Utility file and simply Simplify start
Josef Böhm: A Conics Explorer and a Conics Trainer
p 15
Let’s assume that your students should train the conics of type 2 (shifted origin and not rotated):
delivers a randomly generated quadratic.
The students do their investigation with or without technology and then they can check their solution.
Finally they can plot the curve together with all the important details.
con3= gives the most complicated cases – shifted and rotated.
p 16
Josef Böhm: A Conics Explorer and a Conics Trainer
The quadratic in expression #6 turns out as an imaginary conic, but in expression #9 we find a rotated
and shifted ellipse.
The program code:
Josef Böhm: A Conics Explorer and a Conics Trainer
p 17
Expression #11 (not printed here) is the main function ch_conic(). It investigates the random
conic con from #10 which is also stored as global variable task and stores the results in ans and ant
(for the German users!).
Let’s try finding a hyperbola:
p 18
Josef Böhm: A Conics Explorer and a Conics Trainer
The next task is not so easy.
Model the arc as a conic using the slider bars:
You can find conics everywhere. I used the sliders to model the first three ellipses. Three more are
remaining waiting to be plotted!
Textbooks and the arctanh
p 19
Textbooks and the arctanh
An email-discussion
From: Michel Beaudin
To: Albert Rich, David Jeffrey, David Stoutemyer, Josef Boehm
Subject: Textbooks and the arctanh
Hello Albert and my friends -- it is always exciting for me when I am writing an email to the
"fathers of Derive", to my best ROC friend ("ROC" stands for Rest of Canada: Quebecers love
this word) and to my best friend in Europe.
My question is: do you know why textbooks (at least calculus ones) are so quiet about the
arctanh function? I understand that arctanh(x) has the open interval -1 < x < 1 as domain
and this can cause a problem for students when they will be integrating 1/(1 - x^2) if the
bounds of their definite integral are, for example, 3 to 5. But it is quite easier for them to
use the antiderivative arctanh(x) instead of 2 logs (or one log with quotients) because of the
symmetry with the antiderivative arctan(x) for 1/(1 + x^2). Moreover, when textbooks will
start to use arctanh(x) for the antiderivative of 1/(1-x^2), it will be natural to perform the
integral of sec(x): sec(x) = 1/cos(x) = cos(x)/cos^2(x) = cos(x)/(1 - sin^2(x)), so the substitution u = sin(x) leads to 1/(1-u^2) and that will be a good reason to use arctanh(sin(x))
instead of log (abs(sec(x) + tan(x))) -- what textbooks are still using -- as an antiderivative
of sec(x).
Looking back to an old copy of the DNL journal, David Stoutemyer wrote: "Calculus students
are introduced to complex numbers in high-school algebra, and they are taught that
antiderivatives can differ by arbitrary constants. Why not exploit logarithms of negative
numbers to reinforce both ideas? This is not to say that we should generalize all elementary
calculus to complex numbers -- only that if a complex number is most correct and natural in
a few places, exploit it rather than fight it." (DNL #26, page 5).
Things do not change as fast as I would like!
Von: Albert Rich
An: Michel.Beaudin, David Jeffrey, David Stoutemyer, Josef Boehm
Betreff: Re: Textbooks and the arctanh
Hello Michel,
Not being an educator, I am probably least qualified of your addressees to know “why textbooks are
so quiet about the arctanh function?” Or, more generally, why the inverse trigonometric functions get
so much more "ink” than the inverse hyperbolic ones? This is despite the undeniable symmetry between these two sets of functions. This symmetry is evidenced by the identical, parallel treatments
they receive in Chapter 4 of the recently released NIST Handbook of Mathematical Functions, which
is the authoritative successor to Abramowitz and Stegun.
p 20
Textbooks and the arctanh
So good luck on your quest to make the inverse hyperbolic functions (and hyperbolic functions?) first
class objects of study.
The father of Rubi,
De : David Jeffrey
Objet : Re: Textbooks and the arctanh
My guess is that trigonometric functions are introduced early in high school because of the
geometric application.
Hyperbolic functions, on the other hand, have to wait until the exponential function has been
covered, and thereafter they never catch up to trig functions in the mathematical mind.
One advantage of arctanh(sin x) - or invtanh(sin x) as I prefer to write it - is that it is always a
real value, unlike ln(sec x + tan x).
For the same reason arctanh( (x^2-1)/(x^2+1) ) is a better integral of 1/x than ln(x) but I don't
think it will catch on.
From: Michel Beaudin
Subject: RE: Textbooks and the arctanh
David, if the first goal when choosing an antiderivative is to have a real value, then I “prefer”
ln(abs(x)) instead of arctanh((x^2-1)/(x^2+1)) for the integral of 1/x (these 2 expressions
are identical). In the same idea, ln(abs(sec x + tan x)) is always real, so those who are writings calculus textbooks will continue to use it instead of arctanh(sin x) for the integral of
sec x.
Von: David R Stoutemyer
Betreff: Re: RE: Textbooks and the arctanh
Hi Michel,
I can remain quiet no longer.
The absolute value:
1. unnecessarily makes the antiderivative incorrect for non-real endpoints (and some people
will eventually use non-real endpoints not realizing the incorrectness),
Textbooks and the arctanh
p 21
2. completely hides the non-integrable singularity if someone integrates from a negative to a
positive value. (With ln(x) you get a some warning from the i pi term.)
3. makes it very difficult to simplify subsequent expressions that use the antiderivative -- for
examples, mixtures of ln(x) and ln(abs(x)). As another example, iterated integrals become
very difficult if an inner integral generates a ln(abs(x)).
With all of its extra closure, the complex domain is so much easier than the real domain. I
don't know why we put these extra hurdle constraints on students who have been exposed to
complex numbers in earlier courses.
-- aloha, david s.
Von: Michel.Beaudin
Betreff:* RE: RE: Textbooks and the arctanh
Hi David,
You are absolutely right and I will add more!
After their calculus course, my students will take the ODE course. They will learn how to
solve by hand "easy" ODEs and also they will learn how to use the "desolve" command of
their CAS calculator (Voyage 200 and, starting in September, Nspire CX CAS). Those who
have set the "complex format" to "Real" will sometimes get a "Non-real result" message on
their screen when trying to solve some ODE. With complex format set to "Rectangular", we
get a solution involving the number "i" but, choosing the appropriate domain -- and this is a
nice opportunity to recall the existence and uniqueness of solution --, the "i" will "disappear"
and students will obtain a real solution!
In my talk at ACA 2007 (“Using ln(abs(x)) as an Antiderivative for 1/x is a Bad Choice!”), I
used the separable ODE y ‘ = 4*y/(x^2-9), y(0) = 5. In order to remove the imaginary unit
“i” from the solution found by the device (and also Derive, Maple, …), students have to use
the fact that we are looking for a solution defined in the strip – 3 < x <3 because the initial
condition is at the point (0, 5) : according to the existence and uniqueness theorem, this
ODE has a unique solution defined for -3 < x <3 because both f and dif(f, y ) are continuous
in some rectangle centred at the point (0, 5) with -3 < x < 3 (here f stands for 4*y/(x^2 –
9)). And restricting the domain gives this unique and real solution!
Consequently, we are doing more and not less mathematics if we avoid absolute values in
the integral of 1/x.
Best regards,
p 22
Textbooks and the arctanh
On 11-05-17 03:00 PM, Josef Böhm wrote:
Dear friends,
I am very fascinated by your high level discussion. Unfortunately I can not contribute because I never had to do with hyperbolic functions and their inverse. BTW, in
German the inverse hyperbolic functions are called “Area-functions”, eg. Area sinus
hyperbolicus, etc. I never taught on a technical school where the hyperbolic functions might be part of the curriculum.
What I can contribute is the fact that in Austria secans and cosecans are widely
unknown and unused. I am quite sure if I’d ask some math teachers about secans
and cosecans – they would have never heard about them. They don’t appear in our
textbooks. I never heard about sec and cosec during my math study on the Technical University …
Many thanks and I will follow your discussion with high interest – and I will not
hesitate including it into the next DERIVE newsletter. It is a nice coincidence that I
am just revising DNL#26 with David’s paper “ln x or ln |x|?”
Best regards to you all,
From: David Jeffrey
Subject: Re: AW: RE: Textbooks and the arctanh
Dear All,
My Russian wife has been teaching North American calculus for the first time, and also commented that she was not taught secant and cosecant. I guess in Europe you are taught to write
1+tan^2 = 1/cos^2 instead of our 1+tan^2=sec^2.
We always boast that mathematics is a universal language, but maybe not quite as universal as
we hope.
Some years ago, James Davenport was working on the open math project and reminded us of
even more obscure functions such as versin(x)=1-cos(x) See
for an attractive diagram.
David J
Textbooks and the arctanh
p 23
Von: David R Stoutemyer
Betreff: Re: AW: RE: Textbooks and the arctanh
Tables of csc, sec and cot usefully replaced a division with a multiplication in the days of
manual computation. However, those days are long gone, and most students quickly forget
even whether csc is the reciprocal of cos or sin. Therefore we should stop wasting valuable
curriculum time torturing students with these secondary trigonometric functions and their hyperbolic counterparts, just as we no longer torture students with other trivially related trigonometric functions such the haversine, which was convenient for manual celestial navigation
computations. We don't have a special name for the reciprocal of ln(...), abs(...), etc.
Also, I think that hyperbolic functions are way less important than trigonometric, because
exponentials are often a reasonable alternative to hyperbolics, whereas complex exponentials are a less attractive alternative to sinusoids. (Yes, even I prefer results absent i when i
offers no strong advantage.)
-- aloha, david s.
Von: [email protected]
Gesendet: Mittwoch, 18. Mai 2011 02:22
An: [email protected]
Betreff: RE : RE: Textbooks and the arctanh
Dear Josef, this email is sent only to you because, some months ago, I sent my "new integral
table" to David Jeffrey and Albert Rich but not to you (David Stoutemyer also received it). In
fact, this "high level discussion" started last November when I was teaching single variable
calculus (see the abstract of my Derive session) and when I started to be fascinated by Rubi!
So, here is the table of integrals that I would like to use and also here is the table actually
used at ETS in the single variable calculus course. If you can't make it to Houston, this table
of integrals will give you an idea of what I will be talking about in my lecture!
Best regards,
Do you know this solid?
Find more on page 38!
p 24
Textbooks and the arctanh
This is Michel Beaudin’s very individual table of integrals (which he is finding to be
Klaus Körner: Airplane and Conformal Mapping
p 25
An email came in, which led to an interesting application (translation of the
most important parts of the mail next page).
Sehr geehrter Herr Böhm,
durch Zufall bin ich auf Ihre e-mail Adresse gestoßen und möchte Sie, wenn es für Sie keine
allzu große Mühe macht, um eine mathematische Hilfe bitten. Ich habe gehört, dass die Lösung meines Problems mit DERIVE möglich wäre, doch kann ich selbst damit nicht umgehen.
Einige Sätze zu meiner Person: Ich bin 1939 geboren, habe in München an der TU Maschinenbau studiert, war dann mein ganzes Berufsleben in einer Turbinenfabrik in Nürnberg beschäftigt. In relativ jungen Jahren wurde ich zum Leiter der technischen Berechnung und der
technischen EDV-Abteilung gemacht und hatte mit Thermodynamik, Strömungsmechanik,
Festigkeitslehre, Schwingungslehre, Materialfragen, Lagerauslegung, Wellenschwingungen
und Wuchttechnik zu tun. Mein ganzes Berufsleben hat mir viel Spaß gemacht. Als "technisches Gewissen" unserer Vertriebsleute hatte ich die Möglichkeit fast die ganze Welt zu bereisen doch gesehen hatte ich dabei relativ wenig weil nie Zeit war.
Nun im Ruhestand denke ich manchmal über Probleme nach die mich während meines Berufslebens irgendwie begleitet haben, doch deren Lösung aus Zeitmangel immer verschoben
Es geht um das Wissensgebiet "konforme Abbildungen". Wenn Sie sich auch damit auskennen, dann überlesen Sie bitte meine nachfolgenden kurzen einführenden Worte.
Mittels konformer Abbildung kann man einen Kreis in eine tragflügelförmige Figur verwandeln. Diese Figur sieht dann auch so ähnlich aus wie ein Schaufelprofil. Meine Idee ist es
nun ein gegebenes Schaufelprofil zurück zu transformieren und zu sehen, ob die Rücktransformation eine "schöne, ebenmäßige" Figur ergibt oder so etwas wie eine "Kartoffel". In diesem Fall würde ich daraus schließen, dass dieses vorgegebene Profil strömungstechnisch
nicht optimal ist.
Gegeben sind die Koordinatenwerte eines Kreises mit vielen x =... und y = ... als Funktion
von Radius und den Mittelpunktskoordinaten.
Mit den Formeln u = x * (1 + k/(x² + y²)) und v = y * (1 - k/(x² + y²)) ergibt sich als transformierte Figur eine Schaufelprofil in einem Koordinatensystem u,v.
Mein Problem liegt nun darin, aus diesen beiden vorgenannten Formeln u = ... und v = ... die
Werte x und y als Funktion von u und v darzustellen. Ich habe mir schon die Finger krumm
gerechnet und bin zu keiner Lösung gekommen.
Man sagte mir allerdings, dass eine solche Formelumkehr mit DERIVE möglich wäre.
Da Sie sich offensichtlich mit diesem System auskennen, wage ich es, Sie hier um eine fachliche Hilfe zu bitten.
Ich würde mich sehr über eine Antwort freuen, gegebenenfalls auch auf einen Hinweis wie
ich hier weiterkommen könnte.
Viele Grüße aus Nürnberg
Klaus Körner
p 26
Klaus Körner: Airplane and Conformal Mapping
Klaus Körner’s problem – Inverse Conformal mappings
… My problem deals with the field of knowledge “conformal mappings“.
Applying a conformal mapping it is possible to transform a circle into a figure similar to a wing
of an airplane. This figure is also looking similar to a coop profile. My idea is to transform a
given coop profile back in order to check if the back transformation results in a “beautiful regular“ figure or something like a “potato“: in the latter case I would conclude that the given
profile is not optimal in fluidic regard, it could be a “bluff body“.
Given is a circle by its radius and the coordinates of its centre.
Applying the formulae u = x * (1 + k/(x² + y²)) and v = y * (1 – k/(x² + y²)) we obtain as transformed figure the profile of a wing in a uv-coordinate system.
My problem is to express the variables x and y as functions of u and v. I have calculated my
“fingers awry” but I could not get a solution.
I was told that DERIVE should be able to solve the problem.
It would be great if you could provide any advice or the solution of the problem.
My (Josef’s) answer was:
First of all I’d like to present the airplane wing profile as result of the conformal mapping given in
Mr Körner’s mail.
Klaus Körner: Airplane and Conformal Mapping
p 27
I generalized the mapping and introduced a slider for k. It was interesting to observe that only for
k = 25 a wing profile appeared.
Next step was dealing with Mr Körner’s problem, i.e. solving the system of nonlinear equations for x
and y. I assumed that DERIVE – with Groebner bases implemented – should be able to do the job.
This did not work because the Groebner algorithm can be applied on systems of polynomials only. So
I rewrote both equations free of fractions and then I tried again.
I kept the system as general as possible and replaced 25 by a variable k:
And I received a solution, not only one, but five including the trivial solution (x = y = 0). But the others were bulky enough with two of them containing complex expressions.
p 28
Klaus Körner: Airplane and Conformal Mapping
I sent my results to Mr Körner and he – he does not use DERIVE – transferred the expressions to
MS-Excel – and then had some satisfying results. He wrote (the most important parts of his letter will
follow in a translation):
Sehr geehrter Herr Böhm,
nun habe ich endlich Zeit gefunden um Ihnen nochmals Dank zu sagen für die übermittelten
Formeln als Ergebnis der DERIVE-Rechnungen.
Zunächst zu Ihrer Frage woher ich Ihre Adresse bekommen habe:
Nachdem ich vor mehreren Jahren von einem Geschäftsfreund in Südafrika auf DERIVE
hingewiesen wurde (er hatte eine höhere Position in einer Raffinerie und war nebenbei „Freizeitmathematiker“) ist mir die Idee zur Nutzung dieses Programms für mein anstehendes
Problem in den Sinn gekommen. Im Internet habe ich dann bei der „DERIVE USER GROUP“
Ihren Namen gefunden.
Im Internet habe ich dann auch (irgendwo) Ihren Artikel über die moderne Unterrichtung des
Faches Mathematik gelesen und finde ihn echt zukunftsweisend (leider finde ich ihn nicht
mehr…. Bitte um Hilfe).
Zu meiner Zeit und wahrscheinlich auch noch zu Ihrer Zeit wurde Mathematik sehr trocken
gelehrt und dadurch hatte dieses Fach auch wenig Freude für die Schüler. Ich erinnere mich
noch an meine Oberstufe wo Kurvendiskussion besprochen, doch niemals ein Beispiel aus
dem täglichen Leben angefügt wurde. Dafür wurden wir mit sphärischer Trigonometrie
(Himmelskunde) gequält.
Beim Studium (allgem. Maschinenbau an der TH München) waren die Mathe-Vorlesungen
auch sehr schwer und oft undurchsichtig. Ein ganzes Semester beschäftigte sich mit Funktionentheorie, konformen Abbildungen (auch 3-dimensional) und auch mit geschlossenen
und offenen Torusformen. Im Nachhinein halte ich das für ein Maschinenbaustudium für völlig übertrieben!
Doch nun zu unserem Rechenbeispiel:
Ich habe mir die Beschreibung des DERIVE5 – Programms von Frau Maria Koth, Universität
Wien, angesehen und festgestellt welche traumhaften Möglichkeiten dieses Programm bietet.
Aus Ihren Formeln habe ich die 3. Lösung und die 4. Lösung programmiert und ganz eigenartigerweise festgestellt, dass teilweise die Ergebnisse der 3. Lösung richtig sind und teilweise die der 4. Lösung.
Damit Sie eventuell mein Beispiel mit den Zeichenmethoden von DERIVE nachempfinden
können möchte ich folgende Daten nennen:
Der Kreis hat den Mittelpunkt M(-1;+1) und einen Radius von Wurzel aus 37. Damit kann
man schrittweise (ich habe immer 5° genommen) die Umfangskoordinaten berechnen.
Mit den Ihnen bekannten Formeln u = … und v = … ergibt sich dann eine Form, die einem
Tragflügel ähnelt.
Klaus Körner: Airplane and Conformal Mapping
p 29
Die berechneten u- und v-Werte habe ich nun verwendet, um mit den langen Gleichungen
der 3. und 4. Lösung eine „Rück“-Transformation in Richtung Kreis zu erreichen.
Eigenartigerweise geht das dann nicht ganz ohne Probleme. Ich habe nämlich festgestellt,
dass es am Vorzeichen des jeweiligen y-Werts des Ausgangskreises liegt. Im Bereich positiver y-Werte stimmt das Ergebnis der 4. Lösung, sonst das Ergebnis der 3. Lösung. Beim
Vorzeichenwechsel von y machen die Ergebnisse der 3. und der 4. Lösung einen Sprung!
Da ich aber eine bestimmte Profilform vorgeben möchte um dann mit der Rücktransformation
zu sehen welche Form herauskommt, kann ich ja nicht vorausahnen, ob y einen Vorzeichenwechsel macht oder nicht!
Ich hoffe doch und glaube auch, dass ich bei der Umsetzung dieser komplizierten Formeln
nach EXCEL keinen Fehler gemacht habe!?!
Bitte probieren Sie es vielleicht einmal mit DERIVE.
Es gibt ja mehrere Versionen von DERIVE. Ich habe auch gehört, dass die Version DERIVE5 besser sein soll als DERIVE6.
Mit welcher Fassung arbeiten Sie?
Sollten Sie mit den Derive-Plot Funktionen auch die eigenartigen Ergebnisse wie ich mit
meinen EXCEL-Methoden erhalten, dann bitte ich Sie doch mit Mathematica nochmals einen
Versuch zu starten. Vielleicht kommt dann irgendeine andere Lösung heraus.
Als Anhang schicke ich Ihnen meine EXCEL-Ergebnisse. Darin können Sie ändern oder erweitern wie Sie möchten.
Für heute viele Grüße und nochmals „Danke“.
Klaus Körner
Ein Teil der Excel-Datei und die zugehörigen Grafiken folgen der englischen Übersetzung der wichtigen Teile des Briefes.
… finally I found time to thank once more for the sent formulae which resulted from the DERIVE calculations.
Using your expressions I took the real solutions (#3 and #4) and found out that partially results based on #3 and partially based on #4 are correct.
If you would like to reproduce my example with DERIVE I’ll give the following data:
The circle is given by its centre M(-1;+1) and its radius √37. I calculated the points of the circumference in 5°-steps. Then using the formulae u = … and v = … one obtains the figure
which is similar to an airplane wing.
These u- and v-values were used applying the extended expressions of solutions #3 and #4
for achieving a „back transformation“ to get the circle.
Curiously this does not work without problems. I found out that the problems are connected
with the sign of the respective y-values of the initial circle. For y > 0 one has to take the 4th
solution otherwise the 3rd one. When y changes its sign then the results of #3 and #4 show a
p 30
Klaus Körner: Airplane and Conformal Mapping
As I want to start with a certain given wing profile in order to investigate the form resulting
from the back transformation I cannot guess in advance whether y changes its sign or not.
I hope – and I am pretty sure – that I didn’t make a mistake by transferring the bulky expressions to EXCEL!?
Please try with DERIVE. There are several versions of DERIVE. I’ve heard that DERIVE 5
should be better than version 6. Which version do you use?
I attach my Excel-results. You are free to change or to extend the file as you like. Many
thanks again.
Klaus Körner
Parts of Klaus Körner’s Excel worksheet
Klaus Körner: Airplane and Conformal Mapping
p 31
p 32
Klaus Körner: Airplane and Conformal Mapping
I define four functions for the nontrivial solutions (sol1 and sol2 are the complex ones, sol3 and
sol4 the real ones).
I try the functions with u and v being the parameter form of the wing – and k = 25 – and hope to come
back to the circle:
Klaus Körner: Airplane and Conformal Mapping
p 33
This is interesting and surprising as well. I don’t receive the expected circle but I receive two circles,
one of them being the expected one. At the first view it seems to be that I am not facing Klaus
Körner’s sign-problem.
But see what is happening when plotting the circles in steps:
There are strange arcs belonging to the other circle which I don’t know how to explain. (I must admit
that I am not an expert in conformal mappings.)
Now I’ll deal with the second circle. It’s centre seems to lay on y = x and it passes (5,0) and (0,5). This
is not enough to find its parameters. I didn’t know better than to use sliders for estimating the coordinates of the centre and the radius.
Centre: (≈0.73, ≈0.73)
Radius ≈4.36
What is the result when applying the conformal mapping from above to this (second) circle?
Let’s see:
p 34
Klaus Körner: Airplane and Conformal Mapping
Obviously we receive the same profile as result of the mapping applied to two different circles (which
are surely related in any form, but which one? Who knows?)
(It is also possible to generate trochoids in two different ways.)
I am much more surprised by the two little pieces of the circles. Do you have any explanation for this?
Who is an expert?
Just now, finalizing this DNL, I performed the calculations with MATHEMATICA and received the
same results – some in another form. I found that the DERIVE results are much more compact. You
can find parts of the MATHEMATICA treatment after Mr Körner’s next mail.
And I had one more question which I tried to answer for myself – and for you, too, of course.
But see first Klaus Körner’s response:
Klaus Körner: Airplane and Conformal Mapping
p 35
Hello Mr Böhm,
I would like to inform you about an interesting link:
You can find numerous meaningful (and also less meaningful) EXCEL-applications (free of
charge). Mr Klaus Kühnlein is the main organizer of this publication.
Finally I’d like to thank for your efforts. Your results agree completely with my Excelsolutions. I will try to “back transform” the profile of a real turbine blade in order to find out
whether it results as a more or less regular figure.
Best regards
Klaus Körner
These are the MATHEMATICA-calculations and plots.
p 36
Klaus Körner: Airplane and Conformal Mapping
At first I performed the MATHEMATICA-plots using u_ and v_ from page 32. Then I asked
MATHEMATICA to do the substitution and received quite other expressions (c and d in In[16]).
But the resulting plot is pretty the same.
It might be a good exercise for students proofing the identity of c and u_ (d and v_). For us it is easier,
let’s ask DERIVE!
Solutions #1 and #2 are complex, back transformation applying them gives also complex expressions
and we cannot obtain any plot.
I wondered what the plot of the real part (and the imaginary part, too, of course) would look like and
I was presented a big surprise:
The real part was again the basic wing profile! Solution #2 gave also the wing, the imaginary part is
mirrored wrt to the origin:
From the eDUG Forum: Tensors and DERIVE for DOS
p 37
From the eDUG-Forum: Tensor Algebra and DERIVE DOS
Thomas Fowler:
I posted two new versions of the Tensor Algebra file, updating the fine work done by Hans Dudler.
The version for Derive 5 & 6 is TensorAlgebraExtd.mth. The version for Derive 3 (last DOS release)
is TENSORNE.MTH. I also posted an updated and expanded explanation of the operations, TensorAlgebra.doc. The extensions are primarily for functions in use in the study of general relativity, and
include the Riemann Curvature function K.
(See pages 39 – 41)
Rick Nungester
I have Derive for DOS v3.04 working on Windows XP, but it always starts full-screen instead of in a
small terminal-like text window. Right-clicking on the shortcut and seeing its Properties shows tabs
General, Program, Font, Memory, Screen, Misc, Compatibility, and Summary. I've played with various settings, but it always starts full-screen.
Can Derive for DOS be made to open in a small window instead of full-screen?
I also have Derive for Windows v5.06, but use Derive for DOS v3.04 on my circa-1994 HP 200LX
palmtop and want the same functionality on Windows.
Thomas Fowler
Until TI decides to issue a version of DERIVE for Android devices (which would be a cash cow for
them), you can still get most of the functionality if you have your old DOS release. The last DOS release was DERIVE 3. I have it running on an Android tablet under aDOSBox. It works well, but you
really need an external keyboard. You can speed it up by going into the menu option (from the F1
key), selecting DOSBox, and increasing the CPU cycle variable. I am running mine at 87350, about
20x faster than the default, which naturally emulates and 8088.
Aleksey Tetyorko
IMHO The last DOS release was DERIVE 4.11 with the extended syntax.
Sorry, one question: can you switch to a windowed mode by pressing Alt-Enter? I cannot! My DOS
Derives are 3.11 and 4.11. But when I've changed the Derive Display Mode to Text (Option, Display,
Mode:Text), I made my Derive into a window by Alt-Enter. Do your Derive and XP behave in this
p 38
From the eDUG Forum: Tensors and DERIVE for DOS
Rick Nungester
Thank you Aleksey, that is just what I needed. Now I can start Derive for DOS 3.04 from a Desktop
icon directly into a window without using Alt-Enter at all. Here is what I did:
In Derive, make Text mode the start-up setting:
* Start Derive (no other changes before the next steps)
* Option, Display, Mode:Text
* Transfer, Save, State:DERIVE.INI (Y, overwrite)
* Quit
In Windows XP, make "window" the start-up choice:
* Desktop icon, right-click, Properties, Screen tab, Usage:Window, OK
Now whenever I double-click the Desktop icon, Derive for DOS 3.04 starts in a text window. Plotting
still works, drawing lines as a series of "." text characters instead of using graphics. But then switching
to Option, Display, Mode:Graphics switches to full-screen and good-looking graphics. If desired, then
switching back to text mode and doing Alt-Enter will go from nice full-screen graphics back to a
smaller text-only window. Great!
Aleksey Tetyorko
For your service. You may try using DosBox ( DOS emulator for XP), and you can do it without
switching to Derive text mode. I've test this method some hours before now. My DosBox version is
0.73, maybe not the last. I cannot recommend it because of slow operation.
When informing about tensors in E. Weisstein’s Concise Encyclopedia of Mathematics I came across
this pretty quartic surface, a TANGLECUBE, Josef
x 4 − 5 x 2 + y 4 − 5 y 2 + z 4 − 5 z 2 + 11.8 (15) = 0
Tom Fowler: Tensor Algebra
p 39
* The Derive Tensor Algebra and Analysis Package *
* Documentation for the utility file Tensor.mth *
Written by Hans A. Dudler (619 420-1787) 25 November 1994
Updated for Derive 5
10 November 2001
New functions added, documentation expanded by Tom Fowler, September 2011
This file describes the DERIVE utility file Tensor.mth which provides definitions and functions
for tensor algebra and analysis. Tensor.mth should normally be loaded as a utility file using
the File > Load > Utility File command. Note that Tensor.mth activates the case sensitive
and word input modes. As an introduction to the concepts described in this document the
demonstration files Tensor1.dmo and Tensor2.dmo are recommended. They should be loaded using the File > Load > Demo File command.
1. Tensor Representation in DERIVE.
From the algebraic point-of-view tensors are an extension of the vector/matrix concept to
higher order arrays. Thus a tensor of rank (order) 1 is a vector. A rank 2 tensor is a matrix,
which DERIVE treats as a vector of vectors. A rank 3 tensor is a vector of matrices, a rank 4
tensor is a matrix of matrices, and so on. The obvious way to handle tensors in DERIVE is
therefore to extend the vector concept to higher ranks (orders).
The equivalent of the ELEMENT function to extract an element from a tensor is the function
(see function descriptions below)
EL_T(A,iv) ,
where A is the name of the tensor and iv is the index 'vector'(quotes used to distinguish a
DERIVE vector from a rank 1 tensor). As an example consider the tensor of rank (order) 3
and dimension 3
 a b
A =   d e
  g h
f 
i 
 j k l   s t u
m n o  v w x  
 
 p q r   y z 0  
EL_T(A,[2,3,2]) returns q
EL_T(A,[3,1,3]) returns u
As is clear from the example, the first index (element 1 of the index vector) selects the matrix, the second index selects the row-vector in the selected matrix and the third index finally
selects the element in the row vector. Note that under DERIVE the indices must range from 1
to n (the space dimension), zero (often used in relativity) is not allowed. From the above example we deduce that
EL_T(A,[i,j,k]) <=> ELEMENT(ELEMENT(ELEMENT(A,i),j),k)
in this case, and the extension to higher ranks is obvious. Note that
<=> ELEMENT(A,i)
First page of the documentation, one example follows:
p 40
Tom Fowler: Tensor Algebra
CURVE_T(T,Γ,s) : vector components needed for curvature and geodesics, in terms of arc
length parameter s. Calculates components of b as
 d 2 xi
dx dx dt
i dx dx 
b =  2 + Γ pq
 , x = ( x (t )),
ds ds 
ds dt ds
 ds
And arc length parameter s=s(t) given by
s (t ) = ∫
dx i dx j
ε gij
du du
Easiest to understand with an example. First define G (metric tensor), T (curve whose curvature is desired), and coordinate vector x:
Next compute Christoffel Symbols of 2nd Kind:
Now convert G to parameterized coordinates by Simplify, Variable Substitution, and calculate
arc length parameter s. Replace u with t, then solve for t in terms of s:
Tom Fowler: Tensor Algebra
p 41
In this case we have chosen the last of the three possibilities for t, and assumed that a>0,
giving t = ATAN(s/a), and converting T to s variable by substituting this and invoking Simplify,
Basic. This allows us to get rid of the SIGN function, so we have
Use these values for xi’s in terms of s to substitute into G_:
Invoking CURVATURE on this will yield 0, which indicates that the original curve was a geodesic.
(There are problems with the “Γ” in the DERIVE-file because Γ is reserved for the Gamma-function. I
took G_ instead, Josef)
About Tensors (from CRC Concise Encyclopedia of Mathematics):
An nth-rank tensor is a mathematical object in m-dimensional space which has n indices and
mn components and obeys certain transformation rules. Each index of a tensor ranges over
the number of dimensions of space. If the components of any tensor of any rank vanish in
one particular coordinate system, they vanish in all coordinate systems.
Tensors of rank 0 are calles scalars, tensors of rank one are vectors and tensors of rank two
are matrices, …
Two recommended links to introductory papers on Tensors
An Introduction to Tensors for Students of Physics and Engineering (NASA):
A paper in GERMAN (Technische Universität Graz):
p 42
Roger Folsom: Growth Rates and Comparative Statics
Von: R.N. Folsom [[email protected]]
Gesendet: Donnerstag, 16. Dezember 2010 09:24
An: "Böhm, Josef"
Betreff: A Constrained Inequality's Truth
First, thank you very much for your message of 13 December. I
very much appreciate your work supporting the Derive User Group, and
I have attached a zipped file, "GrowthRates Summed vs Multiplied, N3 - Derive," that contains a dfw file of the same name,
which describes a problem I am having. I hope you can pass the file
on to someone who might know the answer, or that you decide it is
worth including in the next newsletter bulletin.
But I would be happy to revise this document if you think I
should do so.
Sending the Comparative Statics Technique information is definitely on my short term todo list.
Thanks again for your help.
As you will see below, I am trying to formally prove the circumstances under which a particular inequality is true --- and Derive's Simplify or Solve command actually says that it is true.
(For now, all I need is sufficient conditions.)
But I have not been able to figure out the Derive6.1 command or commands that will do that.
My sources are Bernhard Kutzler and Vlasta Kolol-Voljc's "Introduction to Derive 6, Advanced Mathematics for Your PC" (hereafter Derive6); and Albert and Joan Rich, Theresa
Shelby, and David Stoutemeyer's "Derive User Manual Version 6" (hereafter Derive3).
Derive6's equation solver (page 57), when used to determine whether a relationship is True
or False, doesn't apply to what I am trying to do because it doesn't allow conditions to be
imposed on the equation's (or, in my case, the inequality's) variables.
Derive6's discussion of the IF tool (on pages 166 and 227) didn't help because I didn't understand page 166 (I don't know any Analytic Geometry), and although I think that I did understand page 227 I couldn't figure out how to apply it to my problem.
Derive3's discussion of the IF tool (pages 309-310), made sense, but again I couldn't figure
out how to apply it to my problem.
My understanding is that the IF tool has four fields: a test condition, a "then" clause of results if the test condition is satisfied, an "else" clause of results if the test condition is NOT
satisfied, and an "unknown" clause.
Derive3's opening IF example (page 309) is a payroll problem: IF(h <= 40, 10h, 400 + 15(h40)) where h is hours worked and 10h and 15(h-40) are worker payment amounts. Unfortunately, the problem does not include the unknown clause, which if not included but is needed
turns out to be the entire IF(...) tool. An example that does include an unknown clause would
have been helpful.
Roger Folsom: Growth Rates and Comparative Statics
p 43
What I want is a tool, or set of tools, that let me state restrictions on my inequality's variables,
then state the inequality itself, and returns either "true" or "false", or ? (or the same object) if
the restrictions are too weak to determine whether the inequality is true or false.
---------------------------------------------------------------The r↓n variables below are annual (or some other time period) rates of growth, such as interest rates, inflation and deflation rates, real output growth or decline rates, etc. Rates of
growth r↓n can be either positive, zero, or negative.
The number N is the number of time periods being considered. Sometimes "total" rates of
growth over N time periods are calculated by summing the N periodic rates of change r↓n (n
= 1,2,...,N), which is less accurate than multiplying the N corresponding (1+r↓n) factors.
Summing the r↓n can give results that are either larger or smaller than the more accurate
multiplication of the (1 + r↓n).
However, a common situation is that over N time periods, more rates of growth r↓n are positive than negative. If that difference is sufficiently large, summing the N periodic rates of
change r↓n gives a lower total rate of growth over N periods than does multiplyng the corresponding (1 + r↓n) numbers, so that summation understates the true total rate of change
over N periods.
Mathematically, that "common situation" inequality can be written in three basic ways. See
statements #2, #3, and #4, below.
r↓n ≥ 0
If ALL of the r↓n are positive or zero (r↓n ≥ 0), the "common situation" inequality holds, because the left side's terms are smaller than the right side's factors. For an example, let N =
3, and see statements #5, #6, #7, and #8. Note that in #7 --- a rearrangement of #2 with N =
3 --- the left side is -1, and ALL of the variables on the right side are positive.
Therefore, a SUFFICIENT condition for the "usual condition" inequality is simply that all all
r↓n >= 0.
p 44
Roger Folsom: Growth Rates and Comparative Statics
In statement #6, multiplying out the (r↓n + 1) factors would eliminate the -1 - (r↓3 + r↓2 + r↓1)
expression. Doing that (by using Derive's Simplify-Factor command) gives statement #7.
Given the sufficient condition that all r↓n ≥ 0, statements #7 and #8 are obviously true because the left side is negative, and every term on the right side is positive.
And since statements #7 and #8 are merely a rearrangement of #5, which is an example of
statements #4, #3, and #2, statement #7 demonstrates that if all r↓n ≥ 0, the "usual situation"
inequality holds, so that summing N periodic rates of change r↓n (instead of multiplying corresponding 1 + r↓n numbers) understates the true total rate of change over N periods.
HOWEVER, I need to learn how to persuade Derive6.1 to do a formal proof (results either
true, false, or unknown) of the preceding paragraph's results, because my next step is to
experiment with less restrictive conditions, such as all r↓n >= -1.
TO DO: Figure out a formal proof --- for any statement #5 through #8 --- that when #10 (below) holds, comes up with "true" rather than the wrong answer of "false."
For example, Simplify-Basic of either IF(#10, #7) or IF(#7, #10) returns the same (written out)
IF statement, presumably meaning that Simplify could not determine whether the inequality
was true or false, despite the "all r↓n >= 0" variable conditions.
But Solve-Expression of either IF(#10, #7) or IF(#7, #10) concludes that the constrained inequality is false, when I know that it is true (from the analysis that follows #8).
I clearly am not using IF properly. It may not be intended for what I am trying to do. In that
case, I need to know what else to do.
Roger Folsom
Thanks for welcoming me to the Derive Users Group.
Actually, it turns out (prior to the Derive 6.1 CD) that although my latest MS-DOS diskettes were for Derive 4.0.7, I have a
Derive 3 manual (in addition to the Derive 6 manual by Bernhard
Kutzler and Vlasta Kokol-Voljc), so apparently my use of Derive goes
back at least to Derive 3.x. The Derive 3 manual is quite useful in
providing more detail about various Derive features and capabilities.
> Applications in Economics would be highly appreciated!! <
Roger Folsom: Growth Rates and Comparative Statics
p 45
You may already have some. I have, on paper, a message posted
by "R.N. Folsom" on the Derive BBS, 26 April 1993, about "Folsom's
Comparative Statics Technique." I was responding to a message
#2896, 19 April 1993, on "Derive/Mathware" (apparently a different
BBS) by "Harald Lang to Roger Folsom," with the title "Harald Lang's
Use of Taylor's Series to Specify the Chain Rule."
And that was preceded, again on "Derive/Mathware," by message #2893
by "Hadud to Roger Folsom about Response To Questions," which apparently was a response to a set of 7 questions posted by me. Unfortunately, I don't have the seven questions, but Hadud's answers give
pretty good clues about what my questions were. However, they apparently were about how Derive worked, and did not have anything to
do specifically with economics.
But "Comparative Statics" is something that economists do frequently. If you have a system of functions --- i.e., a model --describing a set of markets (as in microeconomics) or describing in
broad terms the entire economy (macroeconomics), the variables fall
into two categories: endogenous (whose values are determined by the
model's functions) and exogenous (whose values are determined outside of the model, e.g. Russia's awful wheat-destroying fires this
past summer). And you want the model to tell you what the exogenous
event will do to markets (not merely the market for wheat, but also
to markets for grain substitutes such as corn, other things that
grow such as cotton, markets for beef since cattle often are fed
corn, etc.). ln short, you want to know (more accurately, to estimate) the effects of an exogenous variable (or possibly multiple exogenous variables) on endogenous variables such as the prices and
quantities of corn, cotton, beef, etc.
Comparative statics usually (maybe always) involves the "chain
rule": derivatives (or finite difference quotients) of y = Y(x) and
x = X(a) where x and y are endogenous and a is exogenous, and where
I have used capital letters instead of "f" to indicate a function
(because I'll probably want to use f for something else, such as
fire severity!). In short, you want to know not merely dy/dx and
dx/da but also dy/da = [dY(x)/dx].[dX(a)/da], which in the U.S. and
maybe elsewhere is commonly known as the "chain rule."
But for a multi-function system, to calculate a full set of total derivatives of all of the endogenous variables with respect to
each of the exogenous variables, Derive wouldn't use the chain rule.
Harald Lang came up with one solution, and I used his notation to
come up with another solution (namely, not substituting any function
into any other function before taking derivatives and calculating
Jacobian matrices for the system).
If you do not already have the messages I have described above,
let me know and I will send them. I'd do so right now except that
they all are paper documents and the ink is not as dark as I wish it
were, so I would need some time to fuddle around with my scanner and
optical character recognition software to send them electronically.
I don't have time to do that right now because I am facing deadlines
for two entirely different projects, but I think I could get to it
in January.
My apologies for taking your time in trying to "explain" the
terms comparative statics, and endogenous and exogenous.
I did so
because I don't know if those terms are used by non-economist mathematicians, or in countries outside the U.S. and Canada.
p 46
Roger Folsom: Growth Rates and Comparative Statics
I briefly visited your Didactics of Computer Algebra, and tentatively concluded that it focuses
on teaching computer algebra. But the site's index frequently uses
the CAS acronym, where presumably C is for Computer and A is for Algebra, but I'm wondering what the S is for. Studies? Or a nonEnglish word? At your convenience, please satisfy my curiosity!
A couple of suggestions: After downloading all of the zipped
MTH files from the DUG website, one at a time, I then saw and followed the recommendation to look at the "back" of the PDF membership
file, which contained the statement "Disk of the year containing all
.MTH files is included." It's been a long day (almost 2:00 a.m.
California time), so I likely am operating in stupid mode, but: In
what is that "Disk" included?
With or without a disk, it might be useful if the DUG website had
three large zip files that included each of the three groups of numbered MTH zip files.
Similarly, since the newsletters/bulletins apparently include articles that explain the use of those zipped mth files, it might be
useful if the DUG website had zipped packages of dnl*.pdf files, so
each pdf file wouldn't have to be downloaded individually.
I'll close with a very minor request: Mathematicians, and also
Derive, like to write polynomial functions "downhill," meaning with
the largest exponent term first, followed by successively smaller
exponent terms, and then a constant. I do understand that that sequence has mathematical advantages, for example by facilitating dividing one polynomial by another. But economists often like to
write polynomials "uphill" (constant term first, followed by terms
with successively larger exponents), because we often don't know how
big the polynomial needs to be until we use data to estimate its coefficients. And it is more convenient to add or delete a high-order
term if it is at the end of the polynomial rather than at its beginning, because if you add or delete a term at the beginning of the
polynomial the coefficients of the previous version (if identified
by subscripts) have to be renumbered.
I wouldn't want to change the way Derive works. But if you know of
a Derive operation (probably created by some user, since I'm pretty
sure it's not in Derive itself) that rearranges a downhill polynomial (and any polynomial subcomponents) and rewrites it (and its
subcomponents) to go uphill rather than downhill, please let me
know. That would be a lot faster than doing reordering the polynomial manually, particularly if it is in a factored form.
Thanks again for welcoming me into the Derive Users Group.
Cordially, Roger Folsom
BTW, it is not easy to find a meaningful translation of “Comparative Statics” into German, Josef
Interesting readings on this issue can be found among others at:
Walter Klinger: An Equation of Equations
p 47
Some days ago I was phoned by Walter Klinger, a DERIVIAN from the first days, about an interesting
behavior of DERIVE. It must be said that Walter has used several technologies in math teaching
(Lower and Upper Secondary) during the last years and finally came back now to his “first love”,
DERIVE. He explained the problem on the mobile phone and then sent a short DERIVE file to explain
his problem more detailed:
An Equation of Equations
Walter Klinger, Stockerau, Austria
p 48
Walter Klinger: An Equation of Equations
My first answer via phone was, that I cold imagine that DERIVE is interpreting h and g as statements
and then treats expression h = g as a Boolean expression.
But I was not really sure and asked myself for an explanation for #4.
What to do? Simply ask DERIVE applying its really unique Stepwise simplification:
The first steps confirmed my conjecture about treating g = h as a Boolean expression. Some steps are
following until coming to an end which is given in #16:
I tried with two other equations of parallel lines and found out that the “non-plotting” of the
“forbidden” points seems in fact to depend on the resolution of the screen and connected with this on
the scaling of the plane.
See the next example:
Walter Klinger: An Equation of Equations
p 49
p 50
Walter Klinger: An Equation of Equations
My next question: What will happen in case of a solution? Take equations h and j.
Any comments on this interesting behavior are welcome. I tried other Computer Algebra Systems.
Most of them do not accept the “equation of equations”.
I tried with changed resolution of the screen (1024 × 768 instead of 1280 × 1024) and received other
patterns in the shaded plane.
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