A jolly info :)
Henri Girard <[email protected]> Fri, 25 Apr 2025 14:19:07 +0200
| Newsgroups | gmane.editors.texmacs.user |
|---|---|
| Message-ID | <[email protected]> |
Ola amigos, I think I started learning texmacs about 30 agos, I wrote a lot of individual files with it, learned latex then jupyter noteboo came and as I used texmacs for maths session I changed editor but still try it from times to times ! But I have never been able to write a book ! My mind absolutly didn't understand how to do it ! Yesterday I thaught of using copilot/gpt and miracle they gave me the key so simply ! It's not yet perfect but surely will become because I understood the way to do it ! They just made an exemple an it did it ! Thank you mes amis it's my texmacs gift for my 80 years ! Best Henri
pressure.tm
(text/plain, 82.1 KB)
<TeXmacs|1.0.7.18>
<style|article>
<\body>
<doc-data|<doc-title|Understanding Functions By Example>|<doc-subtitle|The
Standard Atmosphere>|||<doc-author|<author-data|<author-name|David E.
Miller>|<\author-email>
[email protected]
</author-email>|<author-homepage|mathboxvm.org>>>|<\doc-date>
01 February 2013
</doc-date>>
<no-indent>The author is <name|David E. Miller>. He is a graduate of the
<name|University of Cincinnati> with a BS degree in Aerospace Engineering
and a graduate of <name|The Ohio State University> with an MS degree in
Systems Engineering. He lives in <name|Pickerington, Ohio>.
<\abstract>
The purpose of this article is to provide a practical example of
functions and related concepts in order to demonstrate the technical
details of the concepts in a non-rigorous way. The value lies in having
to relate the details of the example to the technical matters of the
function concept using a model of the standard atmosphere. Specifically,
the example involves thr relation between pressure and altitude. The
<name|Maxima> program does the ``heavy lifting'' for symbolic snd
numerical manipulations of the expressions involved. This allows the
focus to be the ideas involved rather than the drudgery of routine
mathematics tasks.
</abstract>
<\framed>
\;
<\with|par-mode|center>
This work is licensed under the Creative Commons
Attribution-Noncommercial 3.0 Unported License. To view a copy of this
license, visit
<with|font-family|tt|http://creativecommons.org/licenses/by-nc/3.0/>
or send a letter to
Creative Commons, 171 Second Street, Suite 300, San Francisco,
California, 94105, USA.
</with>
\;
<\with|par-mode|center>
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</with>
\;
<with|par-mode|justify|<no-indent>Attributed quotations from copyrighted
works may appear in this document under the ``fair use'' provision of
Section 107 of the United States Copyright Act (Title 17 of the United
States Code). The license of this document is not applicable to those
quotations.>
\;
</framed>
<page-break>
<\table-of-contents|toc>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|1<space|2spc>Introduction>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-1><vspace|0.5fn>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|2<space|2spc>Geopotential
and Geometric Altitudes> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-2><vspace|0.5fn>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|3<space|2spc>Pressure
As a Function of Altitude> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-3><vspace|0.5fn>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|4<space|2spc>Summary>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-6><vspace|0.5fn>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|Acknowledgment>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-7><vspace|0.5fn>
</table-of-contents>
\;
<page-break*><section|Introduction>
\;
A standard atmosphere is a mathematical model that intends to represent
typical values of some of the attributes of the actual atmosphere of the
Earth. Its primary purpose is to serve as a standard for comparision
between results obtained that depend on the properties of the atmosphere.
Otherwise, results would each be conditioned on the varying specific
atmospheric properties that were assumed or existing at the time the
results were obtained. This could make comparisons difficult or impossible.
By referencing results in terms of the values of the properties of a
standard atmosphere, comparison and standardization are possible.
An example is the calibration of aircraft instrumentation. These
instruments such as altimeters and airspeed indicators are calibrated using
values obtained from a standard model. This provides pilots with altitiudes
and airspeeds that are, within manageable error, the same for different
aircraft. Clearly, it is essential that airplanes flying at the same
altitude do not indicate different altitudes to the pilots. Such would be
the case if each aircraft used an altimeter that was calibrated according
to different non-standard methods.
<section|Geopotential and Geometric Altitudes>
\;
<math|H<rsub|g>> is the name used here for a list of <strong|geometric
altitudes> which are physical altitudes referenced from the surface of the
Earth. Actually, since the surface of the earth is not uniform this is an
idealization of the earth as a sphere with a radius that is a constant.
Hence, the term geometric altitudes which are the differences of straight
line distances from the center of the Earth to points in the atmosphere
above the surface and a constant radius which is an acceptable mean value.
If the distances from the center of the Earth to some point in the
atmoshere above the surface are named <math|h<rsub|a>> and the radius of
the Earth to the surface is taken to be a fixed constant named <math|r>
then geometric altitudes are defined as:
<\equation*>
h<rsub|g>=h<rsub|a>- r
</equation*>
\
Create a list named <math|H<rsub|g>> of geometric altitudes from 0 to
11,000 meters as follows:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>2) >
<|unfolded-io>
display(H[g]:makelist (i+1000, i, -1000, 10000, 1000))$
<|unfolded-io>
<math|<with|math-display|true|H<rsub|g>=<around*|[|0,1000,2000,3000,4000,5000,6000,7000,8000,9000,10000,11000|]>>>
\;
</unfolded-io>
</session>
The extent of this list is approximately the part of atmosphere defined as
the <em|troposphere>. G<em|eopotential altitude>s <math|h> are "fictitious"
values related to geometric altitudes by the expression:
<math|h=G(H<rsub|g>)> where the expression <math|G(H<rsub|g>) >is defined
as:
<\eqnarray*>
<tformat|<table|<row|<cell|G<around*|(|h<rsub|g>|)>\<triangleq\>>|<cell|<dfrac|r
\<cdot\>h<rsub|g>|r+h<rsub|g>>>|<cell|>>>>
</eqnarray*>
<\equation*>
h=G<around*|(|h<rsub|g>|)>
</equation*>
\;
This expression for this relation between values of <math|h> and values of
<math|h<rsub|g>> is defined using the expression shown below which is
intended to be used with the list object <math|H<rsub|g>> above instead of
individual values of <math|h<rsub|g>>. In this way the entire list of
values of geometric altitudes can be used as a variable to this function to
find a resulting list named <math|H> of geopotential altitiudes directly
without having to involve individual altitude values as objects. This is
merely a matter of convenience made possible by the capabilities of
<name|Maxima> to process lists of objects.
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>8) >
<|unfolded-io>
G(Hg):=r*Hg/(r+Hg);
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o8>)
>>G<around*|(|<math-up|Hg>|)>\<assign\><frac|r*<math-up|Hg>|r+<math-up|Hg>>>>
</unfolded-io>
</session>
The name <verbatim|r> is a constant that is the value for the radius of the
Earth. Note that geopotential altitudes are only important due to the fact
that their use makes the mathematics of the model used easier to solve.
Hence, while the geometric altitudes are the values of interest, the
geopotential altitudes are used for determining values of quantities such
as pressure, temperature, and density that are related to the geometric
altitude. These values are related to geometric altitude by way of
geopotential altitudes by the above relation of
<math|h=G<around*|(|h<rsub|g>|)>>. Assume that the value of the radius of
the Earth named <verbatim|<math|r>> is <math|6 356 766> meters:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>3) >
<|unfolded-io>
display(r:6356766)$
<|unfolded-io>
<math|<with|math-display|true|r=6356766>>
</unfolded-io>
</session>
Then the above expression named <math|G(H<rsub|g>)> relating the geometric
altitude to the geopotential altitude can be used to convert the list of
geometric altitudes named <math|H<rsub|g>> above to a list of geopotential
altitudes named <math|H> to be used for the calculations which follow. This
list is converted to matrix column form by the following expression for
display purposes that allows easier reference :
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>11) >
<|unfolded-io>
disp('H=transpose(matrix((H:float(G(H[g]))))))$
<|unfolded-io>
<math|<with|math-display|true|H=<matrix|<tformat|<table|<row|<cell|0.0>>|<row|<cell|999.8427120469675>>|<row|<cell|1999.370947130308>>|<row|<cell|2998.58485359367>>|<row|<cell|3997.484579687415>>|<row|<cell|4996.070273568692>>|<row|<cell|5994.342083301508>>|<row|<cell|6992.300156856804>>|<row|<cell|7989.94464211253>>|<row|<cell|8987.275686853711>>|<row|<cell|9984.293438772525>>|<row|<cell|10980.99804546838>>>>>>>
\;
</unfolded-io>
</session>
The lists of values of geometric altitudes <math|H<rsub|g>> and
geopotential <math|H> altitudes above, and their differences can be
displayed for comparison as shown below with geometric altitude values
listed as the first column and the difference as the third column. These
lists of altitudes are displayed in this column matrix form for comparison
of corresponding values and easier reference by the expression:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>12) >
<|unfolded-io>
disp(transpose(matrix(H[g],H,H[g]-H)))$
<|unfolded-io>
<math|<with|math-display|true|<matrix|<tformat|<table|<row|<cell|0>|<cell|0.0>|<cell|0.0>>|<row|<cell|1000>|<cell|999.8427120469675>|<cell|0.15728795303255>>|<row|<cell|2000>|<cell|1999.370947130308>|<cell|0.62905286969203>>|<row|<cell|3000>|<cell|2998.58485359367>|<cell|1.415146406330223>>|<row|<cell|4000>|<cell|3997.484579687415>|<cell|2.515420312584865>>|<row|<cell|5000>|<cell|4996.070273568692>|<cell|3.929726431308154>>|<row|<cell|6000>|<cell|5994.342083301508>|<cell|5.657916698492045>>|<row|<cell|7000>|<cell|6992.300156856804>|<cell|7.699843143195722>>|<row|<cell|8000>|<cell|7989.94464211253>|<cell|10.05535788746965>>|<row|<cell|9000>|<cell|8987.275686853711>|<cell|12.72431314628921>>|<row|<cell|10000>|<cell|9984.293438772525>|<cell|15.70656122747459>>|<row|<cell|11000>|<cell|10980.99804546838>|<cell|19.00195453162087>>>>>>>
\;
</unfolded-io>
</session>
It is evident that geopotential altitiudes are only slightly less than the
corresponding geometric altitudes with the difference in meters shown as
the third column increasing with altitude. The greatest difference is less
than 20 meters at 11,000 meters. However, this difference is large enough
to make a difference in numerical values of quantities related to these
altitudes. The second column <math|H> of the listing above is the list of
"fictitous" altitudes which are used to calculate the values of interest
such as pressure as a function of geopotential altitude. Assume that the
variable that names values of geopotential altitude to be <math|h>.
<section|Pressure As a Function of Altitude>
\;
That pressure in the standard atmosphere is a function of <math|h> is
expressed as <math|p=P(h)>. The expression which defines <math|P(h)> is as
follows<\footnote>
The derivation of this expression is not provided here. It is based on
solving a differential equation that is a consequence of the hydrostatic
equation as it applies to a differential volume of a gas.
</footnote>:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>13) >
<|unfolded-io>
P(h):=P[0]*(1-alpha*h/T[0])^(g[0]/(alpha*R));
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o13>)
>>P<around*|(|h|)>\<assign\>P<rsub|0>*<around*|(|1-<frac|\<alpha\>*h|T<rsub|0>>|)><rsup|<frac|g<rsub|0>|\<alpha\>*R>>>>
</unfolded-io>
</session>
Notice that the only variable involved in the expression that defines
<math|P(h)> is <math|h>. All the other names are intended to name fixed
numerical values (constants) which are yet to be specified. The numbers
that are these constants depend on the system of units under consideration.
Since the altitudes are expressed in meters, SI is the consistent system of
units in this case.
Hence, the above implies that values of pressure <math|p> in the standard
atmosphere are related to values of geopotential altitude <math|h>
according to the following expression:
<\equation*>
p=P<rsub|0>*<around*|(|1-<frac|\<alpha\>*h|T<rsub|0>>|)><rsup|<frac|g<rsub|0>|\<alpha\>*R>>
</equation*>
\;
<no-indent>This relation is assumed to be a function at this point. Whether
this is the case or not depends on the values of the variables <math|h>,
<math|p>, and the nature of the expression relating <math|p> and <math|h>
named <math|P(h)> above.
Values consistent with SI units can now be assigned to the names of the
constants of the expression <math|P(h)> above. The first expression below
creates a list of the names of these constants. The second expression below
creates a list of the values by assigning values to the names of the
constants:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>15) >
<|unfolded-io>
C:[P[0],T[0],alpha,g[0],R]; L:[P[0]:1013.25,T[0]:288.15,alpha:0.0065,g[0]:9.80665,R:287.053];
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o15>)
>><around*|[|P<rsub|0>,T<rsub|0>,\<alpha\>,g<rsub|0>,R|]>>>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o16>)
>><around*|[|1013.25,288.15,0.0065,9.806649999999999,287.053|]>>>
</unfolded-io>
</session>
<\framed>
<\note*>
Pressure is in <em|hectopascals> (<math|h Pa>) which is <math|Pa/100>
<emdash> 100's of pascals . A pascal is 1 newton per square meter. It
is a relatively small unit of pressure, so using hectopascals instead
of pascals is more convenient. This represents a minor variation in
that pascals would otherwise typically be used<\footnote>
\;
</footnote>.
</note*>
</framed>
\;
Values of these physical constants may be displayed for reference:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>17) >
<|unfolded-io>
for c in C do disp(c=ev(c))$
<|unfolded-io>
<math|<with|math-display|true|P<rsub|0>=1013.25>>
\;
<math|<with|math-display|true|T<rsub|0>=288.15>>
\;
<math|<with|math-display|true|\<alpha\>=0.0065>>
\;
<math|<with|math-display|true|g<rsub|0>=9.806649999999999>>
\;
<math|<with|math-display|true|R=287.053>>
</unfolded-io>
</session>
The expression defining the function of variation pressure with altitude
can be checked for consistency. The value of pressure <math|p> in
hectocpascals (<math|h Pa>) at <math|h> is 0 meters altitude is:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>13) >
<|unfolded-io>
P(0);
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o13>)
>>1013.25>>
</unfolded-io>
</session>
The value of pressure at <math|h> is 11,000 m is:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>14) >
<|unfolded-io>
P(11000);
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o14>)
>>226.3205545875172>>
</unfolded-io>
</session>
So the value of pressure <math|p> decreases from 1013.25 <math|h Pa> at
<math|h> is 0 to a value of about 226 <math|h Pa> at <math|h> is 11,000 m.
The assumption is that the relation <math|p=P(h)> defines a function. A
graph of the pairs of related values <math|(h,p)> of pressure and altitude
serves to provide a visual picture of the nature of this relation:
<\session|maxima|default>
<\input>
<with|color|red|(<with|math-font-family|rm|%i>21) >
<|input>
plot2d([P], [h,0,11000], [plot_format, gnuplot])$
</input>
</session>
<\big-figure>
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|png>|660px|469px||>
</big-figure|Graph of values defined by <math|p=P(h)>>
\;
Now it is true that a graph does not prove that the relation <math|p=P(h)>
is a function, but unless there is something unexpected occurring, the
graph provides strong evidence that the above graph is of <math|p> as a
function of <math|h> for the intervals of values of the variables <math|h>
and <math|p>.
There are several options for expressing the pairs of values that are
defined by the relation <math|p=P(h)>. A convenient object for this purpose
is to use the rows and columns of a matrix where the rows are the pairs of
values of pressure <math|p> and altitude <math|h> in that column order. The
following expression accomplishes this task and displays the results:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>18) >
<|unfolded-io>
display(t:transpose(matrix(H[g],P(H))))$
<|unfolded-io>
<math|<with|math-display|true|t=<matrix|<tformat|<table|<row|<cell|0>|<cell|1013.25>>|<row|<cell|1000>|<cell|898.7628248259629>>|<row|<cell|2000>|<cell|795.0141980166974>>|<row|<cell|3000>|<cell|701.2115575785965>>|<row|<cell|4000>|<cell|616.6043644364495>>|<row|<cell|5000>|<cell|540.4827762044484>>|<row|<cell|6000>|<cell|472.1763342680731>>|<row|<cell|7000>|<cell|411.0526642441395>>|<row|<cell|8000>|<cell|356.5161898913356>>|<row|<cell|9000>|<cell|308.0068605489772>>|<row|<cell|10000>|<cell|264.9988921874964>>|<row|<cell|11000>|<cell|226.9995221604465>>>>>>>
\;
</unfolded-io>
</session>
These values may also be displayed as a comma separated value (<abbr|CSV>)
list using the following expression for the <verbatim|printf()> function:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>19) >
<|unfolded-io>
printf(true,"~{~{~d,~,3f ~}~%~}",t)$
<|unfolded-io>
0,1013.250\
1000,898.763\
2000,795.014\
3000,701.212\
4000,616.604\
5000,540.483\
6000,472.176\
7000,411.053\
8000,356.516\
9000,308.007\
10000,264.999\
11000,227.000\
</unfolded-io>
</session>
A <abbr|CSV> file of these data is useful for importing to a program (e.g.,
a spreadsheet program) that may be used to publish a formatted table of
these values. For this purpose the following expressions are used:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>44) >
<|unfolded-io>
outfile:openw("maxout.csv");
printf(outfile,"~{~{~d,~,3f~}~%~}",t)$
close(outfile);
<|unfolded-io>
\;
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o46>)
>>>>Stream [STRING-CHAR]
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o46>)
>>>>true
</unfolded-io>
</session>
<\framed>
<\note*>
The first expression above opens a "stream", that is, the file
referenced by the name <verbatim|outfile>. The <verbatim|printf()>
function prints the comma separated data to the file. Then the stream
is closed by the <verbatim|close()> function. The user must provide the
actual path and file name to be used to save the data as a string
object that is the argument to the <verbatim|openw()> function. The
path and file name ("<verbatim|maxout.csv>") used here is for example
purposes only.
</note*>
</framed>
Recall that the expression relating <math|p> and <math|h> was defined as:
<math| ><math|<with|math-display|true|p=P<around*|(|h|)>>>. Using the
values of the constants this expression evaluates to:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>21) >
<|unfolded-io>
disp(p=P(h))$
<|unfolded-io>
<math|<with|math-display|true|p=1013.25*<around*|(|1-2.2557695644629534\<times\>10<rsup|-5>*h|)><rsup|5.255877432444129>>>
</unfolded-io>
</session>
The right-hand side of this equality is the expression <math|P(h)> with the
values of the constants after simplification. The form of this expression
is:\
<\equation*>
y=a*<around*|(|1-b*x<rsup|c>|)>
</equation*>
where <math|a>, <math|b> and <math|c> are real number constants. Thus in
the end this expression simply involves the form of a power of a variable
<math|x> with a real number exponent. This type of expression is known to
define a function given non-negative values of <math|x>. Hence, this
provides analytic evidence that <math|p=P(h)> defines a function.
The question now is does this same expression define a function of values
<math|(p,h)>? That is, if in the pairs <math|p> and <math|h> the values of
<math|p> are considered independent and the values of <math|h> are
considered dependent, does <math|p=P(h)> also define <math|h> as a function
of <math|p>? This is one version of this question. Another version is the
question: "Are the pairs <math|(p,h)> defined by the expression
<math|p=P(h)> the inverse function of the pairs <math|(h,p)> also defined
by the expression <math|p=P(h)>?"
There are several approaches to the answer to this question. The expression
<math|p=P(h)> is an explicit relation between <math|p> and <math|P(h)>. The
approach to be used here is to attempt to express <math|h> as an explicit
expression in terms of <math|p>. First, remove the values of all constants
so that all expressions are again symbolic<\footnote>
It is easier to use Maxima to manipulate expressions without numerical
values and then when the final result is obtained, evaluate this
expression using relevant numerical values.
</footnote>:
<\session|maxima|default>
<\input>
<with|color|red|(<with|math-font-family|rm|%i>34) >
<|input>
remvalue( P[0], T[0], alpha, g[0], R)$
</input>
</session>
Name an expression <verbatim|eq> that is the expression of the relation
between <math|p> and <math|h>:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>35) >
<|unfolded-io>
eq:p=P(h);
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o35>)
>>p=P<rsub|0>*<around*|(|1-<frac|\<alpha\>*h|T<rsub|0>>|)><rsup|<frac|g<rsub|0>|\<alpha\>*R>>>>
</unfolded-io>
</session>
<\framed>
<\note*>
Naming this expression makes its algebraic manipulation simpler. The
name for the expression can be used in the place of the expression.
</note*>
</framed>
The following steps are used to express <math|h> in terms of <math|p>.
First apply the exponent
<\equation*>
<frac|\<alpha\>*R|g<rsub|0>>
</equation*>
to both sides of the equation:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>36) >
<|unfolded-io>
eq^((alpha*R/g[0]));
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o36>)
>>p<rsup|<frac|\<alpha\>*R|g<rsub|0>>>=<around*|(|P<rsub|0>*<around*|(|1-<frac|\<alpha\>*h|T<rsub|0>>|)><rsup|<frac|g<rsub|0>|\<alpha\>*R>>|)><rsup|<frac|\<alpha\>*R|g<rsub|0>>>>>
</unfolded-io>
</session>
The above is merely a power of both sides of the equality expression as
shown. Now, simplify this resultin eauation between these two powers:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>37) >
<|unfolded-io>
radcan(%);
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o37>)
>>p<rsup|<frac|\<alpha\>*R|g<rsub|0>>>=-<frac|P<rsub|0><rsup|<frac|\<alpha\>*R|g<rsub|0>>>*\<alpha\>*h-P<rsub|0><rsup|<frac|\<alpha\>*R|g<rsub|0>>>*T<rsub|0>|T<rsub|0>>>>
</unfolded-io>
</session>
The above resolves the powers of each side of the equation to an equivalent
form. Now solve the resulting expression above for <math|h>:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>38) >
<|unfolded-io>
solve([%], [h]);
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o38>)
>><around*|[|h=-<frac|T<rsub|0>*p<rsup|<frac|\<alpha\>*R|g<rsub|0>>>-P<rsub|0><rsup|<frac|\<alpha\>*R|g<rsub|0>>>*T<rsub|0>|P<rsub|0><rsup|<frac|\<alpha\>*R|g<rsub|0>>>*\<alpha\>>|]>>>
</unfolded-io>
<\input>
<with|color|red|(<with|math-font-family|rm|%i>39) >
<|input>
\;
</input>
</session>
The above solves the expression for <math|h> in terms of <math|p>. It
creates a list object with all possible solutions. There is only one
solution in this case. Simplify this expression by factoring:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>39) >
<|unfolded-io>
factor(%);
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o39>)
>><around*|[|h=-<frac|T<rsub|0>*<around*|(|p<rsup|<frac|\<alpha\>*R|g<rsub|0>>>-P<rsub|0><rsup|<frac|\<alpha\>*R|g<rsub|0>>>|)>|P<rsub|0><rsup|<frac|\<alpha\>*R|g<rsub|0>>>*\<alpha\>>|]>>>
</unfolded-io>
<\input>
<with|color|red|(<with|math-font-family|rm|%i>40) >
<|input>
\;
</input>
</session>
The above factors the expression resulting in an equivalent form with
common factors resolved. Now get the expression for the right-hand side
(RHS) of the equation of the list expression above:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>40) >
<|unfolded-io>
rhs(%[1]);
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o40>)
>>-<frac|T<rsub|0>*<around*|(|p<rsup|<frac|\<alpha\>*R|g<rsub|0>>>-P<rsub|0><rsup|<frac|\<alpha\>*R|g<rsub|0>>>|)>|P<rsub|0><rsup|<frac|\<alpha\>*R|g<rsub|0>>>*\<alpha\>>>>
</unfolded-io>
</session>
The above expression is merely the right-hand side (RHS) of the expression
of the solution list. Define the value of the expression above as
<verbatim|H(p)>:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>41) >
<|unfolded-io>
H(p):=''%;
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o41>)
>>H<around*|(|p|)>\<assign\>-<frac|T<rsub|0>*<around*|(|p<rsup|<frac|\<alpha\>*R|g<rsub|0>>>-P<rsub|0><rsup|<frac|\<alpha\>*R|g<rsub|0>>>|)>|P<rsub|0><rsup|<frac|\<alpha\>*R|g<rsub|0>>>*\<alpha\>>>>
</unfolded-io>
</session>
The above expression defines <math|H(p)> as the RHS of the list of
solutions:
<\equation*>
<with|math-display|true|h=H<around*|(|p|)>>
</equation*>
Above is the expression of the relation between <math|h> and <math|H(p)>.
With this now determined, the two expressions that are the relations that
define pressure as a function of altitude and altitude as a function of
pressure are as shown below:
<\eqnarray*>
<tformat|<table|<row|<cell|p>|<cell|=>|<cell|P<rsub|0>*<around*|(|1-<frac|\<alpha\>*h|T<rsub|0>>|)><rsup|<frac|g<rsub|0>|\<alpha\>*R>>>>|<row|<cell|>|<cell|>|<cell|>>|<row|<cell|h>|<cell|=>|<cell|-<frac|T<rsub|0>*<around*|(|p<rsup|<frac|\<alpha\>*R|g<rsub|0>>>-P<rsub|0><rsup|<frac|\<alpha\>*R|g<rsub|0>>>|)>|P<rsub|0><rsup|<frac|\<alpha\>*R|g<rsub|0>>>*\<alpha\>>>>>>
</eqnarray*>
\;
In order to get numerical values reassign the values to the names of the
constants as before:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>42) >
<|unfolded-io>
L:[P[0]:1013.25,T[0]:288.15,alpha:0.0065,g[0]:9.80665,R:287.053];
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o42>)
>><around*|[|1013.25,288.15,0.0065,9.806649999999999,287.053|]>>>
</unfolded-io>
</session>
A value for the pressure in <math|h Pa> at an altitude of 11,000 meters was
found above and is repeated below:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>43) >
<|unfolded-io>
display(P(11000))$
<|unfolded-io>
<math|<with|math-display|true|P<around*|(|11000|)>=226.3205545875172>>
</unfolded-io>
</session>
The expression <math|H(p)> above can be checked for consistency by using
this value of pressure to find the related altitude as follows:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>43) >
<|unfolded-io>
display(H(226.3205545875172))$
<|unfolded-io>
<math|<with|math-display|true|H<around*|(|226.3205545875172|)>=11000.0>>
</unfolded-io>
</session>
Likewise the same expression may be checked using the value of pressure of
1013.25 for zero altitude as:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>45) >
<|unfolded-io>
display(H(1013.25))$
<|unfolded-io>
<math|<with|math-display|true|H<around*|(|1013.25|)>=0.0>>
</unfolded-io>
</session>
The relation <math|h=H(p)> can be graphed to help clarify the nature of
this relation:
<\session|maxima|default>
<\input>
<with|color|red|(<with|math-font-family|rm|%i>69) >
<|input>
plot2d(H(p), [p,226,1013],[plot_format, gnuplot])$
</input>
</session>
<\big-figure>
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|png>|684px|469px||>
</big-figure|<math|h=H(p)> <emdash> altitude in meters as related to
pressure in <math|h Pa>.>
As expected, this graph is evidence that this represents the inverse
function of pressure as a function of altitude. In this case it would be
expected that for all values of <math|h> that <math|H(P(h)) = h> is true.
Also, for all values of <math|p> that <math|P(H(p))= p> is true. This is
equivalent to asserting that the pairs of values of <math|h=H(p)> may be
obtained by forming <math|(h,p)> from the pairs of the function defined by
<math|p=P(h)> of the form <math|(p,h)> merely by changing which values are
the first of the ordered pairs. For <math|h> is 0 the following shows that
<math|P(H(p))= p> is true for <math|p> is 1013.25 <math|h Pa>:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>46) >
<|unfolded-io>
P(0);
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o46>)
>>1013.25>>
</unfolded-io>
</session>
This is the same as:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>47) >
<|unfolded-io>
P(H(1013.25));
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o47>)
>>1013.25>>
</unfolded-io>
</session>
For a geometric altitude of 11,000 m the geopotential altitude is:
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>48) >
<|unfolded-io>
float(G(11000));
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o48>)
>>10980.99804546838>>
</unfolded-io>
</session>
For <math|h> is 10980.99804546838 m, the following shows that
<math|P(H(p))= p> is true to 12 decimal places.
<\session|maxima|default>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>49) >
<|unfolded-io>
P(10980.99804546838);
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o49>)
>>226.9995221604465>>
</unfolded-io>
<\unfolded-io>
<with|color|red|(<with|math-font-family|rm|%i>50) >
<|unfolded-io>
P(H(226.9995221604465));
<|unfolded-io>
<math|<with|math-display|true|<text|<with|font-family|tt|color|red|(<with|math-font-family|rm|%o50>)
>>226.9995221604463>>
</unfolded-io>
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ <space|0.6spc>
</session>
As expected <math|p=P(h)> and <math|h=H(p)> appear to define inverse
functions on the above evidence. Inverse functions therefore "undo" each
other. A proof of this assertion is beyond the intended scope considered
here.
<section|Summary>
There is much that can be learned from this example in terms of applying
the concept of functions to "practical" problems:
<\itemize-dot>
<item>The functions of this example are all defined by expressions of the
form of the relation<math| y=f(x)>.
<item>Functions are a particular type of relation. This is what it means
to say that functions are are a constrained relation. In general, there
are no restrictions on the values of the ordered pairs of a relation. For
a relation to be a function it must be the case that for no value of the
first value of the ordered pairs is it the case that more than one value
for the second value of the ordered pair exists. In other words, for each
instance <math|x> is a of the domain of a function it is the case there
exists one and only one value for <math|f(a)>. So, if<math| f(a)=f(b)>
then <math|a> is <math|b> is <samp|false> if <math|y=f(x)> defines a
function. This is the logical constraint that determines if a relation is
a function.
<item>Graphs provide visual evidence that a relation is a function. This
evidence is not proof, however. For practical situations, this is seldom
an issue, but it cannot be ruled out by a picture alone that over some
values of the variables there is behavior that is not detectable, given
the visual scale of the graph, that calls into question the assertion
that the relation is a function. As noted, the mathematical expressions
typical of practical problems can be complex and unfamiliar. The complete
picture of the "behavior" of the values of these expressions may not be
obvious from a graph alone.
<item>An expression of the form <math|y=f(x)> may define a function that
has as instances ordered pairs of the form <math|(x,y) >or <math|(y,x)>.
Whether this is the case or not depends on the values of the variables
<math|x> and <math|y> and the nature of the expression <math|y=f(x)>. If
so, the functions are related as inverses.
<item>When the concept of functions is brought to bear on the expressions
of "real" problems (physics, engineering, economics, science, etc.)
notation generally may not follow conventions. Also, the expressions
involved may be complex combinations of the simpler expressions typical
of the study of mathematics. For these reasons, in the context of the
publications typical of these areas of endeavor, it is incumbent on the
reader to interpret the mathematical models and expressions involved with
care.
<item>In this example, there were instances of functions, composite
functions, and inverse functions utilized for the purpose of
investigating some basic implications of a model for a standard
atmoshere. That there was a composite function utilized may not be
obvious. Recall that the altitudes of interest were geometric altitudes.
For the sake of mathematical simplifcation, geopotential altitudes were
used instead as a varible in order to find values of pressure. Geometric
and geopotential altitudes were related. Hence, the values of pressure
were obtained using in effect the expression: <math|p=P(G(Hg))>. This is
the form of a composite function. This is an example of what was meant in
the above about interpreting with care. Sometimes functions, composite
functions, and inverse functions require some effort to recognize the
forms and to interpret when applied to mathematical models and
descriptions of "real world" processes and phenomena.
<item>There is no unique or standard form for expressions used to define
relations in general and functions in particular. Any expression form may
be used to define relations and functions as long as a well-defined set
of ordered pairs results. Different expressions may define equal
relations or functions, that is, the same set of ordered pairs. The same
expression or an equivalent may define a function and its inverse if an
inverse exists. This is because all functions are relations, but not all
relations are functions. In other words, the inverse of a function may be
a relation that is not a function. In these cases, some restriction for
the sets of values of the variables may be necessary to circumvent this
issue.
<item>The term " relation" is used in two related (Sorry! There is no way
around this.), but different ways. First, it is used to describe the fact
that two variables are related to each other in a particular way. Thus
<math|x=y> expresses that y and x are related by equality. That
<math|x\<less\>y> is also a relation between values of <math|x> and
<math|y>. In this case, if the value of <math|x> is less than the value
of <math|y>, then the expression <math|x\<less\>y> is <samp|true>. In
this sense of the word, a relation between objects <math|x> and <math|y>
is what it is defined to be. In the related second sense of the term, a
relation is a well-defined set of ordered pairs of the values of two
variables.So as to make this distinction clear. a relation (a set of
ordered pairs) in the second sense, may be defined by a relation in the
first sense. In fact, this is commonly the case. It should be easy to see
how "relation" used in the second sense as a set of ordered pairs came by
its name from "relation" used in the first sense. The word "function" is
used to describe a type of relation used in the second sense as a set of
ordered pairs. So a function is a relation (a set) that may be defined by
an expression that is in the form of a defined relation between
variables. Mathematical notation and terminology is often "overloaded"
and this is an example. By "overloaded" it is meant that terms and
symbols often mean different things depending on the context.
Fortunately, the context more often than not provides the clues necessary
to resolve these relatively infrequent ambiguities. Even functions can be
related in the first sense as discussed above. Specifically it was
claimed that the equality relations (in the first sense) <math|p=P(h)>
and <math|h=H(p)> defined functions that were inverses of each other. So
"is the inverse of" asserts that a relation exists between two sets
(e.g., <math|\<b-P\>> and <math|\<b-H\>>) of ordered pairs such that one
is the inverse of the other. This relation is symmetric, that is, if
<math|\<b-H\>> is the inverse of <math|\<b-P\>>, then <math|\<b-P\>> is
the inverse of <math|\<b-H\>>. In the end, this distinction arises due to
the fact that "is related to" involves a predicate while "is a relation"
involves an adjective. In common usage, these two senses of the word
relation are often used synonymously.
</itemize-dot>
<section*|Acknowledgment>
This article was created using GNU <TeXmacs> with the <name|Maxima> session
plug-in. The websites for this project are
<verbatim|http://www.texmacs.org> and <verbatim|http://www.gnu.org/software/texmacs>.<page-break>
</body>
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<associate|auto-10|<tuple|<with|mode|<quote|math>|\<bullet\>>|14>>
<associate|auto-11|<tuple|<with|mode|<quote|math>|\<bullet\>>|15>>
<associate|auto-12|<tuple|<with|mode|<quote|math>|\<bullet\>>|16>>
<associate|auto-13|<tuple|<with|mode|<quote|math>|\<bullet\>>|17>>
<associate|auto-2|<tuple|2|3>>
<associate|auto-3|<tuple|3|5>>
<associate|auto-4|<tuple|1|6>>
<associate|auto-5|<tuple|2|10>>
<associate|auto-6|<tuple|4|11>>
<associate|auto-7|<tuple|<with|mode|<quote|math>|\<bullet\>>|12>>
<associate|auto-8|<tuple|<with|mode|<quote|math>|\<bullet\>>|12>>
<associate|auto-9|<tuple|<with|mode|<quote|math>|\<bullet\>>|13>>
<associate|footnote-|<tuple|?|1>>
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<associate|footnote-2|<tuple|2|?>>
<associate|footnote-3|<tuple|3|8>>
<associate|footnr-1|<tuple|1|5>>
<associate|footnr-2|<tuple|2|5>>
<associate|footnr-3|<tuple|3|8>>
</collection>
</references>
<\auxiliary>
<\collection>
<\associate|figure>
<tuple|normal|Graph of values defined by
<with|mode|<quote|math>|p=P(h)>|<pageref|auto-4>>
<tuple|normal|<with|mode|<quote|math>|h=H(p)>
<with|font|<quote|roman>|\V> altitude in meters as related to pressure
in <with|mode|<quote|math>|h Pa>.|<pageref|auto-5>>
</associate>
<\associate|toc>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|1<space|2spc>Introduction>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-1><vspace|0.5fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|2<space|2spc>Geopotential
and Geometric Altitudes> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-2><vspace|0.5fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|3<space|2spc>Pressure
As a Function of Altitude> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-3><vspace|0.5fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|4<space|2spc>Summary>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-6><vspace|0.5fn>
<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|Acknowledgment>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-7><vspace|0.5fn>
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article-template.tm
(text/plain, 16.4 KB)
<TeXmacs|1.0.7.18>
<style|article>
<\body>
<doc-data|<doc-title|My <TeXmacs> Article Template>|<doc-subtitle|A General
Example>|||<doc-author|<author-data|<author-name|David E.
Miller>|<\author-email>
[email protected]
</author-email>|<author-homepage|quaoar.us>>>|<\doc-date>
01 February 2013
</doc-date>>
The author is <name|David E. Miller>. He is a graduate of the
<name|University of Cincinnati> with a BS degree in Aerospace Engineering
and a graduate of <name|The Ohio State University> with an MS degree in
Systems Engineering. He lives in <name|Pickerington, Ohio>.
<\abstract>
This purpose of this document is to provide a template for use in
producing a technical article.
</abstract>
<\framed>
\;
<\with|par-mode|center>
This work is licensed under the Creative Commons
Attribution-Noncommercial 3.0 Unported License. To view a copy of this
license, visit
<with|font-family|tt|http://creativecommons.org/licenses/by-nc/3.0/>
or send a letter to
Creative Commons, 171 Second Street, Suite 300, San Francisco,
California, 94105, USA.
</with>
<\with|par-mode|center>
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</with>
<with|par-mode|justify|<no-indent>Attributed quotations from copyrighted
works may appear in this document under the ``fair use'' provision of
Section 107 of the United States Copyright Act (Title 17 of the United
States Code). The license of this document is not applicable to those
quotations.>
\;
</framed>
<page-break>
<\table-of-contents|toc>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|1<space|2spc>Section>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-1><vspace|0.5fn>
<with|par-left|1.5fn|1.1<space|2spc>Subsection
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-2>>
<with|par-left|3fn|1.1.1<space|2spc>Subsubsection
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-3>>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|2<space|2spc>Another
Section> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-4><vspace|0.5fn>
<with|par-left|1.5fn|2.1<space|2spc> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-5>>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|Acknowledgment>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-6><vspace|0.5fn>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|Bibliography>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-7><vspace|0.5fn>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|Glossary>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-8><vspace|0.5fn>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|List of
figures> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-9><vspace|0.5fn>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|List of tables>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-10><vspace|0.5fn>
<vspace*|1fn><with|font-series|bold|math-font-series|bold|Index>
<datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
<no-break><pageref|auto-11><vspace|0.5fn>
</table-of-contents>
<page-break*><section|Section>
\;
<subsection|Subsection>
<subsubsection|Subsubsection>
<section|Another Section>
<subsection|>
<page-break>
<section*|Acknowledgment>
This article was created using GNU <TeXmacs> with the <name|Maxima> session
plug-in. The websites for this project are
<verbatim|http://www.texmacs.org> and <verbatim|http://www.gnu.org/software/texmacs>.<page-break>
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<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|1<space|2spc>Section>
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<with|par-left|<quote|1.5fn>|1.1<space|2spc>Subsection
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<with|par-left|<quote|3fn>|1.1.1<space|2spc>Subsubsection
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<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|2<space|2spc>Another
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<with|par-left|<quote|1.5fn>|2.1<space|2spc>
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<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|Acknowledgment>
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<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|Bibliography>
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<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|Glossary>
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<vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|Index>
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