Maxima Book Chapter 2

8/13/2019 Maxima Book Chapter 2

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Real numbers, functions, and units in MaximaIn this Chapter we introduce calculations with real and complexnumbers using Maxima. We also introduce the use of functions,conditional statements, and logical particles. This chapter alsoincludes examples of calculations using units of measurements.

Symbolic and floating-point results with real numbersIn this section we present simple calculations with real numbers and introduce conversionfrom symbolic to floating-point results. ymbolic results represent the results that onewould obtain by wor!ing by hand, and producing simplification of operations with numbers."y default, Maxima produces symbolic results when operating with real #and complex$numbers. To produce floating-point values it is necessary to use functionfloat asillustrated in the examples shown below. %otice that the first example shows a simplefraction of integer numbers, while the second example shows a s&uare root calculation.

'ractions involving floating-point numbers produce floating-point results, e.g.,

Changing the default format(se the menu item Numeric > Toggle numeric output to change the default format ofcalculations with real numbers. 'or example, activating this item once, changes the

default output format to floating point as illustrated below.

2-1 Gilberto E. Urroz, 2008N


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) second application of the menu item Numeric > Toggle numeric outputwill return thedefault output format to symbolic*

In this specific example the output %e-2is the symbolic representation of the number e-2,while 0.1353...is the corresponding floating-point result.

+ther format changesThe Numeric menu includes also the menu items Numeric > To float and Numeric > Tobigfloat that can be used to convert a symbolic result into either a simple floating-pointformat, or to a double-precision floating-point format #referred to as bigfloatin Maxima$.These two menu items are e&uivalent to the functionsfloatand bfloat, respectively. When

applying any of these two menu items, the result refers to the last result available #i.e., to%$. The following two examples show the application of To floatand To bigfloatto thenumber e-2.

%otice that the last result is inpower-of-ten format with bindicating the power of ten.While the output shown uses the default number of digits #$, the value is stored in adouble-precision location.

ower-of-ten formatTo enter power-of-ten floating-point values use eto indicate the power of ten, someexamples are shown below*



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If entering a power-of-ten floating-point number using binstead of e, the number will bestored as a bigfloatnumber. ome examples are shown below*

Combiningfloat and bigfloatnumbers results in a bigfloat number, e.g.,

etting floating-point precision"y default, Maxima shows digits in a floating-point number. (sing the menu itemNumeric > Set Precision ... produces a dialog form where you can enter the precision#number of digits$ to show in your floating-point results, e.g.,

) simpler way to change the precision is to redefine parameterfpprec. In the followingexamples we changefpprecto values of , /0, 1, and 02, and then display the value ofp 3pi$ using function bfloat*



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Calculations with real numbersIn this sections we present calculations with real numbers involving not only simplearithmetic operations #4, -, 5, 6, 7$, but also s&uare roots #sqrt$, other roots, absolutevalues #abs$, trigonometric functions and their inverses #sin, cos, asin, acos, etc.$,hyperbolic functions and their inverses #sin, cos, asin, acos, etc.$, exponential #exp$,natural logarithms #log8 note* not ln$, ceiling,floor,fix, andfloat. "y default symbolic

results will be provided. (se functionfloatto obtain floating-point results. lease noticethat the arguments of trigonometric functions are in radians, the natural angular unit. Toconvert from degrees to radians use the factor %pi/180. Try some of the followingexamples #see the results in your own Maximainstallation$*

2+(1/(2+1/(2+1/2)));4./3.+3./4.+1./6.;sqrt(1+(3/2)^3);abs(-2.5+1/2.5);

sin(%pi/3+cos(%pi/3));sqrt(exp(-2)+lo(abs(-2+1/2)));

ceilin(3.25); "loor(3.25); "ix(3.25);3.25-"ix(3.25); sin#(2.5);

9valuation of formulas9valuation of formulas is a common application of real number calculations. I suggest usinga three-step approach*

. 9nter formula #remember to use * instead of :$0. 9nter list of values !nown, separated by ; to avoid ta!ing space in the window/. (se the command history to repeat the formula expression, which is now evaluated

If need be, usefloat#%$ to obtain a floating-point value. ome examples are shown below.9xample 0.*



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9xample 0.0*

%ote* in this last example, !ill"all#was used to clear all existing values. )lso, theintermediate e&uation$2float"%pi&'2(2)*#is calculated before calculating the value of +.

Defining functionsTo define a function write the function name and arguments, e.g.,f"x#,g"#, "x,#,followed by *: and by the function definition. 9valuation of the functions is performed byreplacing the un!nowns with variable names or numerical values.


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%otice that the definition of the function includes a couple of groupings of expressions,namely, #x2*$ and #xx$, that appear twice in the expression forf"x#. It could bepossible to calculate the function in three steps, namely,

a : x2* b / xx

fx =2a bab2

In Maxima, we can define a function such as this by using a se&uence of expressionsseparated by commas. 'or example,

Chec! the full expression of the function by using, for example,f"t#*

9valuation of the function is straightforward as illustrated in the following examples*

0sing function ratsimp - 'unction ratsimp#rational simplification$ can be used to simplifyrational expressions such as fractions, polynomials, etc. This function will be presented ina later chapter in reference to simplification of algebraic expressions. It is introduced hereto show an alternative way to define the function presented above*



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The function definition is now*

In this case, evaluation of the function with algebraic arguments will produce a fullyexpanded expression, e.g.,


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0sing ectors- To illustrate the operation of the function we will generate a vector of databy evaluating the function at pointsx / -2, 3, 4, 2, , *, and 5, i.e.,

Te if-then and if-then-else constructs 8 In the definition of function f#x$ shown above, weused both an if-tenand an if-ten-elseconstructs. The if-tenconstruct has the generalform*

i" &conition* t#en &action*

The result from this construct is to perform the =action> if the =condition> is true, or donothing otherwise.

+n the other hand, the if-ten-else construct has the general form*

i" &conition* t#en &action 1* else &action 2*

The result from this construct is to perform =action > if the =condition> is true, or perform=action 0> if the condition is false.

6on7itional statements8 Conditional statements are statements that result in a trueorfalseoutcome. %umerical comparisons are common forms of conditional statements. Inthe definition of functionf"x#used conditional statements such as*

2&x an x&4

This is a combined conditional statement including the simple conditional statements

2&x, x&4.

The following comparison operators can be used to produce conditional statements*

= less than =: less than or e&ual to : e&ual to not : not e&ual to > greater than >: greater than or e&ual to

8ogical particles8 ?ogical particles, such as an7, are used to combine simple conditionalstatements. The following logical particles are available in Maxima*

and combined statement is true only if both composing statements are true or combined statement is true if one of the two composing statements is true not resulting statement is true if original statement is false, and vice versa



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The numbers at the lower left corner are the coordinates of a point in the graph. Thus, theresultinggnuplot grapsin Maximaare interactive in the sense that you can place thecursor anywhere in the plot and get the coordinates of that particular point.

)s mentioned earlier, the graph can be exported to a drawing #or graphics manipulation$software for editing. The following is an edited version of this plot*

The following example shows a case where the user controls not only the range of valuesofx, but also those of . @a!e sure to clic! off thegnuplot grapwindow before enteringa new command in the wxMaxima 1NP0Tline. In this example we plot the function*

The graph to the left was generated using*

"ecause the function evaluates to atx / -4, the range of values of is extremely large,and the variation of the function near / 3is not clearly defined. To produce a moredetailed graph, use the command*



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The following example shows a bi-variate function plotted using functionplot7*

The graph is shown below*

)dditional information on the use of graphics functions in Maximawill be presented on asubse&uent chapter on graphics.

Calculations using units

+perations with units in Maximare&uires us to load the unit pac!age. ) pac!age in Maximais a collection of functions related to the pac!ageAs theme. To see a list of the pac!agesavailable in Maxima, launch the Maxima Manual by using the menu item 9elp > Maximaelp, and clic! on the 6ontents tab. Then, scroll down until you pass all the chapters thatstart with an upper-case letter. The Maxima pac!ages correspond to those chapters thatstart with a lower-case letter, as listed below*



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If you clic! on the B4 icon next to the unitpac!age, you will find the following items withinthe boo! icon*

%ext, clic! on the 1ntro7uction to 0nitsitem to get information on the use of units inMaxima. Read the text on 1ntro7uction to 0nitsin the Maxima Manual. )s you scroll downyou will get a listing of the functions available, namely*

setunits - select preferred units for different dimensions uforget - clears any preferred units set with setunits conert - convert a unit to a different set of units usersetunits - overules default units with units selected by the user metricexpan7all - allows selection of metric prefixes according to*

D - none. +nly base units

- !ilo, centi, milli0 - giga, mega, !ilo, hecto, de!a, deci, centi, milli, micro, nano Bdefault value/ - peta, tera, giga, mega, !ilo, hecto, de!a, deci, centi, milli, micro, nano, pico, femto1 8 all

%unitexpan7 - value of the argument for metricexpan7all

To get started using units, one needs to load the unitpac!age*

The International ystem of units #ysteme International, .I.$The basis of the unitpac!age is the International ystem #.I.$ of units. 'or detailedinformation on the .I. please visit the 0.S. National 1nstitute of Stan7ar7s an7 Tecnolog#N1ST$ web page on the .I.*http*66www.physics.nist.gov6cuu6(nits6units.html

http://www.physics.nist.gov/cuu/Units/units.htmlhttp://www.physics.nist.gov/cuu/Units/units.html


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The .I. defines seven basic units corresponding to the following base &uantities*

length meter #m$ mass !ilogram #!g$ time second #s$ electric current ampere #)$

thermodynamic pressure !elvin #E$ amount of substance mole #mol$ luminous intensity candela #cd$

Combinations of these units produce .I. derived units such as area #m0$, velocity #m6s$,and so on Bsee Table 0 in the %IT I web page. ome of these derived &uantities havespecial names and symbols, e.g., Bsee Table / in the %IT I web page*

force newton #%$ pressure, stress pascal #a$ energy, wor!, &uantity of heat Foule #G$

power watt #W$ electric charge coulomb #C$ electric potential difference volt #H$ capacitance farad #'$ electric resistance ohm #$ magnetic flux tesla #T$ inductance henry #$ luminous flux lumen #lm$ illuminance lux #lx$

The .I. in Maxima

(sing the unitpac!age in Maximawe can chec! the reduction of these derived units to itsbasic units. Compare the results of the following derived units as shown in Maximawiththose in Table / in the %IT I web page. To save space, we will create a vector of derivedunits and chec! their expressions in terms of the basic units*



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We can attach units to values and operate with them as illustrated by this simple examplein which we calculate the force re&uired to accelerate a mass of !gto a constant value of4.5 m)s2*

(se of function conert%otice that the result will be given in terms of the basic units, thus, instead of showing theresult in newtons it is shown in !gm)s2 . (sing function conertit is possible to convert thisresult to N*

ere is another example, in which we calculate the power developed by a particle movingat a velocity of 2.5 m)sunder the effect of a force of *.5 N*

The following example shows the calculation of the !inetic energy of a particle of 43 !gofmass moving at 43 m)s*

etting default unitsuppose that we are wor!ing with units commonly used in fluid mechanics, and that wewould li!e to reduce all unit results to combinations of newtons #%$, pascals #a$, Foules#G$, and watts #W$. In that case, we can force the default units to be*



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Thus, the following calculations will default to these units*

Resetting the default unitsTo reset the result to the basic units #!g, m, s, etc.$, use function uforget*

With this command, repeating the calculations shown above will produce the followingresults*

(sing prefixes in the .I.Jou are probably familiar with some of the most commonly used prefixes in the .I., e.g.,

nano #as in nanometer, or nm / 43 -:m$ micro #as in micrometer, orm / 43-;m$ milli #as in millimeter, or mm 43 -m$ centi #as in centimeter, or cm / 43 -2m$ !ilo #as in !ilometer, or !m / 43 m$ mega #as in megawatts, or M< / 43 ;


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J #yotta : D01$ L #Metta : D0$ 9 #exa : D2$ #peta : DN$T #tera : D0$ K #giga : DO$ @ #mega : D$ ! #!ilo : D/$h #hecto : D0$ da #de!a : D$ d #deci : D-$ c #centi : D-0$33m #milli : D-/$ 3mu #micro : D-$ n #nano : D-O$ p #pico : D-0$f #femto : D-N$ a #atto : D-2$ M #Mepto : D-0$ y #yocto : D-01$.

%otice that the only PstrangeQ characters are %%for milliand %for micro. The otherletters are the same as written.

Calculations involving .I. prefixes are shown below*

'unction conertcan be used to convert results to prefixed units #e.g., mmor !Pa$, asillustrated below*



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The M=S and cgs systems of units"efore the .I. system standardiMe metric units, there were several PmetricQ systems usedin physics calculations. Two of the best !now were the M=S#meter-!ilogram-secon7$ andthe cgs#centimeter-gram-secon7$ given in terms of their basic units of length, mass, andtime. The @E system is basically the same as the .I. system. owever, since the units inthe cgssystem are much smaller in magnitude than the corresponding unit in the M=S

system, the cgssystem is still used when dealing with small masses and lengths.The conversion of the basic units of the cgs to the @E system is straightforward*

The unit of force in the cgs system is the 7ne #abbreviated 7n$, such that 4 7n / 4 g4cm)s2*

The unit of wor! or energy in the cgs system is the erg#I thin! this is an abbreviation of theword energ$, and it is defined as 4 erg / 4 7n 4 cm. owever, the Maxima unitpac!agedoes not contain the ergas one of its units. We can still find a conversion from ergs to

oules#?$ by using*

)lternatively, you can add a variable called ergby using*

Converting tooules #?$ we get*



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The electron-@olt #e@$ince we are dealing with small &uantities, we should mention that the unit pac!age doescontain the electron-@olt#e@$ as a unit of energy. In electrical fields, a charge q#typicallygiven in coulombs, or 6$ affected by an electric potential difference @ #also referred to asoltage, and given in olts or @$ has a potential energy e&ual to the product of the chargeand the voltage. The electron-volt is a very small unit of energy e&uivalent to the charge

of electron affected by a voltage of volt. Converted tooules #?$, e@is e&ual to*

To see that conversion factor as a floating-point result it is necessary to type the following#this can be accomplished by selecting the denominator, and cutting and pasting into theI%(T line$*

Thus, 4 e@ / 4.;3243-4:?.

+ther metric units available in the unit pac!ageome of the metric units included in the unitspac!age that we havenAt dealt with beforeinclude the following*

mass grain #gr$ time minute #3min$ fre&uency hertM #M$ pressure torricelli #torr$ electric conductance siemens #$ volume liter #?$, basic unit : m/

absorbed dose gray #Ky$ catalytic activity !atal #!at$

The conversions for these units are shown below*

The torricelli#torr$ can be converted to a more commonly used unit, thepascal#Pa$*



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)lso, we should point out that electric conductance is the inverse of electric resistance,therefore, the siemens#S$ and the om#$ are inverse units, or S/ 4. ere we useMaximato chec! that result*

@etric units not available in the unit pac!age

This list includes other metric units that are not standard .I. units, but are commonly usedin practice. The list also includes units such as the nautical mile and the !now, that,although not metric in origim, are accepted for use in the .I. 'or conversion factors, seeTable in the %IT .I. web page

length* angstrom #S$, astronomical unit #ua$, nautical mile, hectare #ha$ area* are #a$, hectare #ha$, barn #b$ time* hour #h$, day #d$ velocity* !not mass* metric ton #t$, unified atomic mass unit #u$ pressure* bar

If needed in Maxima, you can define these units as done for the erg, above.

The 9nglish ystem #9..$ of unitsWhile the International ystem #.I.$ has been adopted throughout the world, the 9nglishystem of units, also !nown as the "ritish ystem or the Imperial ystem, is still very muchprevalent in the (nited tates of )merica0. The unit pac!age includes only two of the 9..units, namely,

length inch #3in$ mass slug #slug$

The corresponding conversions to .I. basic units are the following*

2 )ccor*i#$ to t+eWikipediae#try o# t+eInternational System, !s o !y 6, 2008, /t+ree #!tio#s +!e #ot oici!lly

!*ote* t+e S.. !s t+eir ri(!ry or sole syste( o (e!s&re(e#t iberi!, y!#(!r, !#* t+e U#ite* St!tes.



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If you want to do calculations with units of the 9nglish ystem, you could define thefollowing additional unit conversions*

length foot #ft$ : 0 in, mile #mi$ : N02D ft force pound #lb$ : slug ft6s0

energy "ritish thermal unit #"T($ : .DN1ND/G power horsepower #hp$ : NND lbft6s

(sing Maxima*

In terms of the basic .I. units the units defined above are*

Converting to related units #e.g., ft to m, mile to !m, lb to %, etc.$*

The fractions shown above, written in floating-point format, are*

'unctions unitinfo and a77unitThe unitpac!age is still in development, therefore, it still has some functions that are notyet #@ay, 0DD2$ implemented such as*

unitinfo provides definition for a specific unit a77units allows user to add other units.html0



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When the a77units function finally becomes available, it will be possible to add the unitsdefined above #e.g., erg, ft, mi, lb, AT0, p$ as PatomQ units. Thus, while a conversionsuch as*

is possible, the inverse conversion will not be available until AT0is added using a77units.Currently, an attempt to produce that conversion fails*

The only possibility is to force the conversion using an expression such as*

) result li!e this shows only the magnitude of the conversion factor. The user will have tointerpret this result, for example, as 43 ? / 3.33:*B AT0.

The unit mac file

In Chapter , 1ntro7uction to Maxima, we learned that a batch file is typically saved using asuffix .mac. In the failed conversion attempt shown above, namely, con'ert(10,),the error message provided by Maximashows two references to a file called unit.mac*

This file, unit.mac, is the batch file that is loaded when we first invo!e the unitpac!age#using loa(nit)$. The response from Maxima, as shown in page 0-0, above, includes theoutput*

This output represents the path of the unit.macfile in a


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folder names where allowed, but many programs still !ept the 2-letter restriction inreporting file paths, as in the case above. Ta!ing into account such abbreviations, we willrecogniMe the path to the unit.macfile as*

c7rora "ilesaxia-5.14.0s#areaxia5.14.0s#arecontribnitnit.ac

With this information you can navigate to the corresponding file and open it using a texteditor #e.g.,


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(se of function 7imension in dimensional analysis'unction 7imension can be useful to determine the dimensions of a given &uantity, as in theexample shown above for watts #


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surface tension force6length : %6m pressure, stress2 force6area : %6m0: a wor!, energy, heat force length : %m : G power wor!6time or force velocity : G6s : W viscosityO shear stress6velocity gradient : a6#6s$ : as !inematic viscosity viscosity6density : area6time : m06s

Enowing these units, then we can find the dimensions of all these &uantities*

mass, length, time, area, volume, velocity, and acceleration*

force, volumetric discharge, mass discharge, weight discharge, density, specificweight*

surface tension, pressure or stress, wor! or energy or heat, power*

viscosity, !inematic viscosity*

While these examples illustrate the use of dimensional analysis with the unitpac!age, youcan find a more detailed set of functions for dimensional analysis in thepsics)7imen

pac!age. This dimensional analysis pac!age is discussed in the next section.

7 S&r!ce te#sio# is ! roerty o ! li;&i* ree-s&r!ce !#* is *ei#e* !s t+e orce er it le#$t+ t+!t t+e s&r!ce e%erts o#

! bo*y i# co#t!ct it+ t+e s&r!ce.

8 )lso, t+e (o*&l&s o el!sticity o ! l&i* +!s its o ress&re or stress

9

e;&!tio# =du

dy, +ere is s+e!r stress !#* du/dyis ! elocity $r!*ie#t /elocity=le#$t+.



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Using thephysics package for units, physical constants, and dimensional analysisThepsicspac!age is a contributed Maximapac!age available in the following folder ordirectory in a


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)t this point, Fust continue pressing B9nter until the end of the demo file is reached. Jouwill recogniMe the end of the file when the pause prompt no longer appears.

To understand what these examples mean you will have to open the demo file using a texteditor. The demo file is typically well documented, but the comment lines are not shown inthe wx


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(sing the units pac!age(se a text editor to open the file units.macto chec! the listing of units provided whenloading the unitspac!age*

neton, >ole, att, acre, apres, o#,anstro, are, astronoicalnit, atosp#ere,bar, barrelr, barrel, barreloil, barl, bolt, bt, bs#el, canle, centira,centiliter, centieter, centieter o" ercr, c#ain, circlaril, cor, colobs,

cp, ccle, alton, a, ecira, eciliter, ecieter, eree, eree", ereec,ereer, e$ara,e$aliter, e$aeter, ne, ell, epica, er, "aras, "araa,"at#o, "eet, "eeto"ater, "lionce, "ootcanle, "rlon, allon, alloniperial,ill, rae, ra, racalorie, #an, #ectoaras, #ectoliter, #ectoeter,#ectoatt, #enries, #os#ea, #orsepoer, #or, inc#, inc#o"ercr, >oles,$iloras, $iloeter, $iloatts, $not, leae, li#tear, lin$, lin$sr'eor,liter, len, lix, eo#, eters, icro"ara, icrora, icro#, icroliter,icron, icrosecon, ilenatical, ile, illier, illiicron, illira,illi#enr, illiliter, illieter, illisecon, il, inte, inersinc#, inis,riara, riaeter, riaatt, neper, netons, o#s, parsec, pec$, pint,ponal, poise, pon, lb, lb", ponass, lb, once, onces, oncea'oirpois,oncetro, qartr, qart, qt, raian, ra, re'oltion, ro, sl, secons,sp#ere, sto$e, statcolob,steraians, tablespoon, teaspoon, 'olts, atts, ee$,ar.

?oading the units pac!age allows the user to find the corresponding .I. units for the myriadof units listed above. ome examples are shown below*

%ote*


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(sing thepsconst pac!ageThepsconstpac!age is used to load the definitions of many standard physical constants.(sing a text editor, open the filepsconst.macto see the definitions and symbols for theseconstants. The first three universal constants listed in that file are*

/spee o" li#t in 'ac/ ner'al(%%c, 2??@?2458/s)A

ner'al(%cB0, 2??@?2458/s)A/anetic constant/ ner'al(%B0, 4%pi1C-@:/(

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Maxima Book Chapter 2