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## my Maple cheat sheet

April 17, 2023   Compiled on April 17, 2023 at 12:37 Noon  [public]

### 1 How to ﬁnd a particular solution to ODE?

restart;
ode:=diff(y(x),x)+y(x)^2*sin(x)-2*sin(x)/cos(x)^2 = 0;
yp:=DETools:-particularsol(ode);



To step into the code, do

restart;
ode:=diff(y(x),x)+y(x)^2*sin(x)-2*sin(x)/cos(x)^2 = 0;
stopat(DEtools/particularsol);
DETools:-particularsol(ode);



To print it do

print(DEtools/particularsol);



### 2 How to ﬁnd basis solutions for homogeneous ode?

Use the output=basis option

ode:=diff(y(x),x$2)-x*diff(y(x),x)-x*y(x)=0; dsolve(ode,output=basis);  ### 3 How to convert Mathematica expression to Maple? restart; with(MmaTranslator); #load the package FromMma(Integrate[Cos[x],x]);  Or restart; with(MmaTranslator); #load the package convert(Integrate[Cos[x],x], FromMma);  ### 4 How to debug internal procedures, such as dsolve? f:=proc() eq:=x*diff(y(x),x)+y(x)=exp(2*x); dsolve(eq,y(x)); end proc;  Then used the command stopat(f); then called the procedure f(); and now the debugger comes up. Did step command and now it steps inside dsolve ### 5 How to display source code of a function? For integration use infolevel[evalf/int]:=5;infolevel[int]:=5;  Another option restart; interface(verboseproc=3) #(try 2 also)  then print(procedure); or eval(procedure_name); for example restart: interface(verboseproc=3): print(LinearAlgebra:-GramSchmidt); print(lcm);  Also can use showstat, in this case interface(verboseproc=3) is not needed. Also showstat gives line numbers and I think it is easier to read. Some examples showstat(odsolve/2nd_order) showstat(evalf/hypergeom); showstat(evalf/exp/general); showstat(evalf/Psi); showstat(evalf/int); showstat(dsolve/SERIES); showstat(odeadv/dAlembert); showstat(odsolve/dAlembert); showstat(odsolve/dAlembert/integrate); showstat(ODEtools/odeadv); showstat(DEtools:-odeadvisor); showstat(dsolve/series/froben/inhom) showstat(dsolve/series/froben)  To stop at anyone of these functions in debugger do stopat(dsolve/series/froben/inhom) #code here, say dsolve command.  The above will stop in the debugger in the above function. There is also a function by Joe Riel here here is the post by Joe Riel: "A disadvantage of showstat, particularly if you want to cut and paste the output, is that it includes line numbers. Here is a simple procedure I threw together to remove the line numbers." PrintProc := proc(p::name,lines::{posint,posint..posint}) local width; option Copyright (C) 2004 by Joseph S. Riel. All rights reserved.; description "Print like showstat, but without line numbers"; width := interface('screenwidth'=200); try printf("%s", StringTools:-RegSubs( "\n ...." = "\n" ,debugopts('procdump'= if(nargs=1,p,[args])))) catch "procedure name expected": error "%1 is not a procedure name",p finally interface('screenwidth'=width) end try; NULL end:  To print source code to ﬁle using the above, do the following currentdir("C:\\data"); interface('prettyprint'=1): interface('verboseproc'=3): writeto("listing.txt") PrintProc('singular'); writeto('terminal'):  Now the output will show up in the ﬁle "listing.txt" and also no line wrapping. The above I found is the best solution so far to do this. ### 6 How to display trace of a function as it runs in maple? trace(foo); untrace(foo);  also see debug(foo); Also infolevel[all]:=5: printlevel:=10:  Also look at kernelopts(opaquemodules=true) Here is a useful post by Carl Love from Maple prime forum that summarizes all of these Here are four things that you can do to get more information. I have listed them in order by how structured the information is, with the most structured ﬁrst. 1. Set infolevel[all]:= 5;  That will cause programs to print out additional information of the programmers’ choosing. You can use higher or lower numbers for more or less information. Most programs don’t use levels higher than 5. 2. Print the code of procedures with showstat: showstat(int); showstat(sin); showstat(cos);  3. Trace the execution of particular procedures with trace: trace(int); trace(sin);  4. Trace the execution of everything with printlevel: printlevel:= 10000:  You can use higher or lower numbers for more or less information. ### 7 How to display a build in function code? interface(verboseproc=3); print(DEtools)  Or to see line numbers interface(verboseproc=3); showstat(dsolve)  Or can use the Browse(); command with(LibraryTools); Browse();  Another option I found is s:=debugopts(procdump=showstat);  Then the above produces listing that can be copied as string with line wrapping ok. ### 8 How to build a LIST or a SET on the ﬂy? One way L:=[]: for i from 1 to 3 do : L:=[op(L),i]; end do;  But a better way is to use seq if one knows the length L:=[seq(i,i=1..3)]; L := [1, 2, 3]  Since list is unmutable, a more eﬃcient method, for long lists, is to use Array, and then convert the result back to list at the end since Array can grow dynamically without preallocation each time something is inserted as follows L:=Array(): for i from 1 to 3 do : L(i):=i; end do; for i from 1 to numelems(L) do : print(L[i]); end do; L := convert(L,list)  Which wil print  L :=  L := [1, 2] L := [1, 2, 3] 1 2 3 L := [1, 2, 3]  Notice that to add to an Array, () is used. But to access an entry in an array [] is used. And ﬁnally, using Array also, it can be done without using any indexing as follows L:=Array(1..0): for i from 1 to 3 do : L ,= i; end do; L := convert(L,list)  For the above to work, the array must be declared using Array(1..0). The new syntax A ,= i will append to the array, and there is no need to write A(i) := i ### 9 How to make function display more information of what it is doing? By Carol Devore on the net: Use infolevel. For example, to show what logic dsolve uses, do this: First try > infolevel[all]:= 5; That will probably give more information than you want, but if not, then try > printlevel:= 1000; If you want information about a specific procedure, you can use debug. For example, restart; debug(int/int); int(p, x= 0..1); To find out what procedures are being called without getting too much extra information, use excallgraph. Trying on dsolve infolevel[dsolve]:= 3; dsolve({eq1},y(x)); Methods for second order ODEs: Trying to isolate the derivative d^2y/dx^2... Successful isolation of d^2y/dx^2 --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 2; linear nonhomogeneous with symmetry [0,1] trying a double symmetry of the form [xi=0, eta=F(x)] <- double symmetry of the form [xi=0, eta=F(x)] successful  ### 10 How to solve a diﬀerential equation with initial conditions? To solve $y''-3y'+2y=10 e^{5 x}$ with $$y(0)=1,y'(0)=5$$ do eq1:= diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=10*exp(5*x);
dsolve({eq1,y(0)=1,D(y)(0)=5},y(x));

Methods for second order ODEs:
Trying to isolate the derivative d^2y/dx^2...
Successful isolation of d^2y/dx^2
--- Trying classification methods ---
trying high order exact linear fully integrable
trying differential order: 2; linear nonhomogeneous with symmetry [0,1]
trying a double symmetry of the form [xi=0, eta=F(x)]
<- double symmetry of the form [xi=0, eta=F(x)] successful
....



The above can also be written using D@@ notation, like this

eq:= (D@@2)(y)(x) - 3*D(y)(x) +2*y(x) = 10*exp(5*x);
IC := y(0)=1,D(y)(0)=5;
dsolve({eq,IC},y(x));



### 11 How to verify that the ODE solution given is correct?

use odetest and check if it gives zero.

eq1:= diff(diff(y(x),x),x)-3*diff(y(x),x)+2*y(x)=10*exp(5*x);
ans:=dsolve({eq1,IC},y(x));
odetest(ans,eq1);

0



### 12 How to know the type of ODE?

Maple can classify the ODE.

eq1:= diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=10*exp(5*x); R0 := DEtools['odeadvisor'](eq1,y(x)); R0 := [[_2nd_order, _with_linear_symmetries]]  To get help on this type of ODE, do DEtools['odeadvisor'](eq1,'help');  ### 13 What packages to load for diﬀerential equations? Use with(DEtools); ### 14 How to plot solution of diﬀerential equations? restart; eq1:= diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=10*exp(5*x);
DEtools[DEplot](eq1,y(x),x=-2..5, [ [y(0)=0, D(y)(0)=0]], y=-3..3,linecolor=red); To get a better plot, change the stepsize and independent variable range

restart;
eq1:= diff(y(x),x$2)-3*diff(y(x),x)+2*y(x)=10*exp(5*x); DEtools[DEplot](eq1,y(x),x=-1..1,[[y(0)=0,D(y)(0)=0]],y=-3..3,stepsize=0.001,linecolor=red); ### 15 How to plot a function? Here, I am looking at fouries series expansion of $$f(x)=0$$ between $$–\pi$$ and 0, and $$f(x)=1$$ between 0 and $$\pi$$. The Fouries series expansion is worked out to be as below. This shows that the series approximate the above $$f(x)$$ as more terms are added restart; f:=(x)-> 1/2 + (1/Pi)*(sin(x)+sin(3*x)/3+sin(5*x)/5+sin(7*x)/7); plot(f(x),x=-10..10); ### 16 How to run maple from command line? From DOS, point to where your cmaple is >"C:\Program Files\Maple 7\BIN.WNT\"cmaple To make it execute maple commands use the < foo.txt to pipe maple commands in the ﬁle to it. ### 17 How to use matrices in maple? A:= Matrix( [ [1, 2, 3] , [3, 6, 7] , [5, 6, 9] , [7, 7, 7] ]); whattype(A); Matrix size:=LinearAlgebra:-Dimension(A); size := 4, 3 row:=size; row := 4 col:=size; col := 3  You can extract any part of the matrix like this: B:=A[1..3,2..2];  $\left [ \begin {array}{c} 2\\ 6\\ 6\end {array} \right ]$ By Carl Devore http://mathforum.org/kb/message.jspa?messageID=1570678 Maple list and sequence structures are more flexible than Matrices, which are highly structured. A Maple list of lists (called a listlist in Maplese) is akin to a matrix in some other languages. Many matrix operations can be performed directly on the listlist form, but to do serious linear algebra, you should convert to a Matrix. Of course, it is trivial to convert a listlist to Matrix: LL:= [[1,2], [3,4]]; M:= Matrix(LL); So here is another solution in line with your original wishes. This is "index free", but the table-based solution I gave earlier should be faster. (It is usually considered bad form to repeatedly append to a list or sequence.) L:= [][]; # Create a NULL sequence do line:= readline(file); if line::string then if line contains valid data then Z:= a list of that data; L:= L, Z fi else break fi od A:= Matrix([L]); # Note []: seq -> list. To move move a column into a matrix: Here, I want to copy 2nd column to the 3rd column: A; $\left [ \begin {array}{ccc} 1&2&3\\ 3&6&7\\ 5&6&9\\ 7&7&7 \end {array} \right ]$ B:=A[1..row,2]; $\left [ \begin {array}{c} 2\\ 6\\ 6\\ 7 \end {array} \right ]$ A[1..row,3]:=B: A; $\left [ \begin {array}{ccc} 1&2&2\\ 3&6&6\\ 5&6&6\\ 7&7&7 \end {array} \right ]$ ### 18 How can maple return more than value from a procedure call? Maple can return multiple values. Make sure to use the comma "," in the body of the procedure to separate each return value. Example: size_matrix:=proc(x) 3*x, 4*x; end proc; row,col :=size_matrix(5);  ### 19 How does maple handle procedure arguments? When passing a variable to maple procesure, the variable VALUE is passed to the procedure (This is diﬀerent from say Fortran where the default is pass by reference). But this is the same as with Mathematica. For example, if a variable X had value 10, then you call a procedure FOO passing it X, then inside FOO, X will be the number 10, not the argument variable X. So, this means one can not have X on the left hand side inside FOO. Like this x:=1 The only way to assign new value to the input and return new value, is to use a local variable, like this: one:= proc(x) local y; print(x); y:=x+ 1; print(x); y; end proc; z:='z'; z:=5; f:=one(z); f := 6  ### 20 How to deﬁne your own data types? Use type/name to deﬁne new type name.  type/char:= x-> x::string and length(x)=1; P:= proc(c::char) print(c) end proc: P("x"); "x" P("xy"); Error, invalid input: P expects its 1st argument, c, to be of type char, but received xy > type/byte:= x-> x::integer and (x>= 0 and x<256); will define a byte (unsigned integer)  ### 21 How to ﬁnd max element in a matrix and its position as same time? Code from net by Carl Devore: MMax:= proc(M::{Matrix,matrix}) local C,r,c,mx,L,p; C:= op(if(M::Matrix, [1,2], [2,2,2]), eval(M)); L:= map(op, convert(M, listlist)); mx:= max(L[]); member(mx,L,'p'); r:= iquo(p, C, 'c'); mx, if(c=0, [r,C], [r+1,c]) end;  Code below from C W A:=matrix(12,12,rand(100)); Ao:=array((proc(E) local i; [seq(i=(rhs=lhs)(E[i]),i=1..nops(E))]end) (sort(op(3,eval(A)),proc(E1,E2) if rhs(E1)>rhs(E2) then true else false fi end))); Ao;  ### 22 How to create a package? First create the module: restart; nma:= module() option package; export getMaxMatrix; getMaxMatrix := proc (M::{matrix, Matrix}) local C, r, c, mx, L, p; C := op(if(M::Matrix,[1, 2],[2,2,2]),eval(M)); L := map(op,convert(M,listlist)); mx := max(L[]); member(mx,L,'p'); r := iquo(p,C,'c'); mx, if(c = 0,[r, C],[r+1, c]) end proc; end module; A:= Matrix( [ [1, 2, 3] , [3, 6, 7] , [5, 6, 9] , [7, 7, 7] ]); nma[getMaxMatrix](A);|  Gives 9, [3, 3]. Now save the module. savelibname := "C:/MAPLE_PACKAES"; march('create', savelibname, 20);  now save the library to disk. savelib(nma); Now we can test everything by reinitialize everything and reload the library. >restart #Add my library to LIBNAME >libname:="C:/MAPLE_PACKAGES",libname; > A:=matrix( [ [1,2,3],[4,6,9] ]); >with(nma); >nma[getMaxMatrix](A);  Now to print a proc() in the package, do >interface(verboseproc=3); > print(nma[getMaxMatrix]);  Now you can list what packages exist in the archive: march('list',savelibname); march('extract',savelibname,":-1.m","C:MAPLE_PACKAGES/t.m")  Some notes. need to clean later > module1lib:=module1\\lib; > system("md "||module1lib); > march('create',module1lib,100); > makehelp(module1,module1/module1.mws,module1lib): > makehelp(module1/export1,module1/export1.mws,module1lib): > savelibname:=module1lib: ### doesn't affect current libname > savelib(module1); ### no error message > restart; > module1lib:="module1\\lib": > libname:=module1lib,libname; ### now Maple will find module1 > with(module1); > ?module1  Also there is a long thread here on Maple prime on making personal packages in Maple How-To-Create-A-Personal-Package ### 23 How to convert from ﬂoating point to Hex? From: Robert Israel (israel@math.ubc.ca) Subject: Re: Getting non-integral results in hex Newsgroups: comp.soft-sys.math.maple Date: 2003-06-13 00:07:37 PST I assume you mean floating-point numbers. Note that Maple floats (as opposed to "hardware floats") are in fact stored in base 10. To convert a float to hex with n digits after the ".", you can use this: > convert/hexfloat:= proc(x::numeric, n::nonnegint) local A,B,ax,R; if nargs = 1 then return procname(x,round(Digits*log(10))) fi; if x = 0 then return cat(0.,0$n) fi;
ax:= abs(x);
A:= floor(ax);
B:= round(frac(ax)*16^n);
if B = 16^n then A:= A+1; B:= 0 fi;
R:= cat(convert(A,hex),.);
if x < 0 then R:= cat(-,R) fi;
cat(R,substring(convert(16^n+B,hex),2..-1));
end;

And then, e.g.:

> convert(1234.5678, hexfloat, 4);

4D2.915B

### 24 How to ﬁnd taylor series expansion of functions?

mtaylor(sin(x),[x],10);

$x-1/6\,{x}^{3}+{\frac {{x}^{5}}{120}}-{\frac {{x}^{7}}{5040}}+{\frac { {x}^{9}}{362880}}$

### 25 How to print elements of a matrix?

restart;
a:=Matrix([  [2,3,4],[4,5,6]   ]);
nRow,nCol :=LinearAlgebra[Dimension](a);
for i from 1 to nRow do
for j from 1 to nCol do
printf("a(%d,%d)=%d\n",i,j,a[i,j]);
end do;
end do;

a(1,1)=2
a(1,2)=3
a(1,3)=4
a(2,1)=4
a(2,2)=5
a(2,3)=6



### 26 How to ﬁnd determinant of matrix?

restart;
a:=Matrix([ [2,4],[5,7] ]);
LinearAlgebra:-Determinant(a);
-6



### 27 How to generate Hilber matrix?

H := LinearAlgebra:-HilbertMatrix(5);

\left [ \begin {array}{ccccc} 1&1/2&1/3&1/4&1/5\\ \noalign {\medskip }1/ 2&1/3&1/4&1/5&1/6\\ \noalign {\medskip }1/3&1/4&1/5&1/6&1/7 \\ \noalign {\medskip }1/4&1/5&1/6&1/7&1/8\\ \noalign {\medskip }1/5&1/6&1 /7&1/8&1/9\end {array} \right ]

### 28 How to plot matrix data?

Matlab is much easier here. In maple, need to covert the matrix to a list of list of points ﬁrst.

restart;
H := LinearAlgebra:-HilbertMatrix(5):
nRow,nCol :=LinearAlgebra[Dimension](H):
L:=[seq([seq( [i,j,H[i,j]], i=1..nRow) ], j=1..nCol)]:
plots:-surfdata(L); ### 29 How to catch an error from a proc()?

An error in maple raises an exception. So, use try catch to trap it as follows:

try
v,pos:=MMax(4);
catch:
printf("an error is cought\n");
end try;



### 30 How to convert 3456 to 3,456 ?

From the net, by Carl Devor:

print/commas:= proc(N::integer)
local n,s,i,b;
n:= ListTools:-Reverse(convert(abs(N), base, 1000));
if N<0 then n:= subsop(1= -n, n) fi;
nprintf("%s", sprintf(cat("%d", ",%03d" nops(n)-1), n[])) end proc: commas(456554); 456,554  To convert a string to array of chars use array(StringTools:-Explode(S)) s:="Nasser M. Abbasi": r:=array(StringTools:-Explode(s)); r:=["N" "a" "s" .......]  Now can use the string as normal array r; "s"  ### 31 How to use units ? Units[GetDimensions](base); amount_of_information, amount_of_substance, currency, electric_current, length, logarithmic_gain, luminous_intensity, mass, thermodynamic_temperature, time  ### 32 On High precision. Using taylor to solve ODE From: Robert Israel (israel@math.ubc.ca) Subject: Re: given precision in Maple Newsgroups: comp.soft-sys.math.maple Date: 2003-07-16 20:19:06 PST Set Digits:= n and all calculations from this point will be done with n digits. Mathematical functions will be correct to n digits as well (to the extent this is practical). If you want high-accuracy numerical ODE solutions, on the other hand, it's not so simple. I think the best way is using the taylorseries method. For example, consider the problem y' = y^2, y(1) = 1, where the exact solution y = 1/(2-x) has y(1.9) = 10. > Digits:= 30: sol:= dsolve({D(y)(x)=y(x)^2, y(1) = 1}, y(x), numeric, method=taylorseries, abserr=1e-25): sol(1.9); [x = 1.9, y(x) = 9.99999999999999999999999797691] > 10 - eval(y(x),%); -23 0.202309 10 The other methods (in particular the default rkf45) do not give results anywhere near this good. ### 33 How to evaluate catlan number and other sums? Use the Sum command. restart; expr:= (-1)^i/(2*i+1)^2; Sum(expr,i=0..infinity); evalf(%,50); 0.91596559417721901505460351493238411077414937428167  Notice, if I used the sum command instead of the Sum command I get this result: sum(expr,i=0..infinity); Catalan  ### 34 How to write a text ﬁle that contains a package, and load it and execute it? This shows how to do a simple package and use it without building a library. Just using a plain text ﬁle. Create this nma_pkg1.txt ﬁle:  nma_pkg1 := module() export f1; option package; f1:= proc() print("in pakcage nma_pkg1"); end proc; end module;  now save it, and from maple do >read("c:\\nma_pkg1.txt");  now execute f1() as this: >nma_pkg1[f1](); "in pakcage nma_pkg1"  now put it in a library (so that we can use with, instead of read) > savelibname:=("c:/maple"); > march('create', savelibname, 20); > savelib(nma_pkg1); >restart; > libname := "c:/maple",libname; > with(nma_pkg1); > f1(); "in pakcage nma_pkg1"  now make changes to the nma_pkg1.txt ﬁle and updated again as above. ### 35 How to ﬁnd what packages are included in maple ?index,package ### 36 How to plot the gradiant vector ﬁeld? restart; f:=3*x^2 + y* cos(x*y); the_grad :=linalg[grad](f,[x,y]); plots[fieldplot](the_grad,x=-2..2,y=-2..2); or or can do it in just one command: plots[gradplot](f,x=-2..2,y=-2..2); ### 37 How to put the digits of Pi into a list? Suppose you want the 100 digits of Pi put in a list. This is one way to do it: restart; L:=evalf(Pi,100); S:=convert(L,string); the_list:=[seq(parse(S[i]),i=3..length(S))]; the_list := [1, 4, 1, 5, 9, 2, 6, 5, 3, ..  This below now tells how many times each digits occurs. >stats[transform,tally](the_list); [Weight(0, 8), Weight(1, 8), Weight(2, 12), Weight(3, 11), Weight(4, 10), Weight(5, 8), Weight(6, 9), Weight(7, 7), Weight(8, 13), Weight(9, 13)]  ### 38 Digits of PI in maple and mma Written sometime in 2005? I should really record the time when I write something. I just run these now, Auust 2014, and now Maple 18 as very fast. So this all below is no longer valid. I will leave it here for now for reference until I update it all later I have written a few lines of code, which counts how many times each digit occurs after the decimal points of $$\pi$$ Written this in maple ﬁrst. Then did similar thin in mma 5.0. Both are run on the same PC. No other applications are running at the time when I run the code. The basic idea of the algorithm is to use evalf(Pi,digits) in maple to ﬁnd $$\pi$$ for any number of decimal digits, and to use N[Pi,digits] in mma for doing the same. (Where the variable digits above is the number of digits) Then in maple convert the above $$\pi$$ to a string, and generate a sequence of the characters to right of decimal point, then use stats[transform,tally] to do the actual counting. In mma, I use RealDigits[] to get a list of the digits, and then use Count[] to do the counting. This is result of some of the runs to ﬁnd Pi to some digits, and the total time (to ﬁnd Pi and do the counting) All times are in cpu seconds, machine is P4, 2.8 Ghz, 500 MB of RAM, single CPU, hyperthreading enabled, running XP home edition. Maple 9.03 student version, and mma 5.0 student version. Below is the result, and below that I show the maple code and the mma code. Because of this, before each run in mma, I exited the application and started it fresh. In maple, it does not matter for the above reason. 100,000 digits: Find_Pi Total Maple 9.0 55 84 Mma 5.0 0.9 1.54 Mma is 60 times faster in ﬁnding pi and about 56 times faster overall 300,000 digits: Find_Pi Total Maple 9.0 309 781 Mma 5.0 3.7 6 Mma is 300 times faster in ﬁnding Pi, and 130 times faster overall. 3,000,000 digits Find_Pi Total Maple 9.0 Mma 5.0 85 118 Maple time in hours ! Still running. Maple code > restart; startingTime :=time(); L:=evalf(Pi,100000): timeToFindPiInSecs:=time()-startingTime; S:=convert(L,string): the_list:=[seq(parse(S[i]),i=3..length(S))]: stats[transform,tally](the_list); endingTime :=time(): cpuTimeInSecs := endingTime - startingTime;  mma code Clear[] startingTime=TimeUsed[] t1=N[Pi,100000]; timeToFindPiInSecs=TimeUsed[]-startingTime {c,d}=RealDigits[t1]; theList=c[[Range[2,Length[c]]]]; f[digit_]:=Count[theList,digit]; r=Range[0,9]; Map[f,r] cpuTimeInSecs=TimeUsed[]-startingTime  update 12/25/03 Changed maple code on how to do the counting : To use StringTools[CharacterFrequencies](S)  Now the counting in maple is much faster. It is always hard to know which is the best function to use. restart; startingTime :=time(); L:=evalf(Pi,300000): timeToFindPiInSecs:=time()-startingTime; S:=convert(L,string): StringTools[CharacterFrequencies](S); endingTime :=time(): cpuTimeInSecs := endingTime - startingTime;  ### 39 How to ﬁnd where functions are? From: Ken Lin (maplemath@tp.edu.tw) Subject: Re: how to find which package a function belongs to? Newsgroups: comp.soft-sys.math.maple Date: 2003-12-04 03:49:26 PST When Maple first loaded, There are only two kinds of "internal" commands which can be called directly. One is the "kernal" commands coded in C, and the other includes many "internal" prodecures programmed by the kernal commands which lies in the "Main Library", There are also many other "external" procedures which were categorized into so called "packages", plots[display](...) for example, plots[] is a package(Library), and display() is the prodecure inside plots[]. All the packages can be loaded by with() command, like > with(plots); Because Different Packages include user library might have the same procedure name, Maple doesn't realize the "procedure_name" you type in, it took it for a "symbol". If you really want to know which packages provided by Maple the external procedure lies in, just mark the procedure_name and press F1 key, the Maple Help Browser will show you the packages you might be interested. By the way, plot3d() is a "internal" procedure lies in the Main Library. You can confirm that by: > op(0, eval(plot3d)); procedure or in Maple 9 > type( plot3d, 'std' ); #Is it internal? true > type( plot3d, 'stdlib' ); #Does is lie in "Standard(Main) Library"? true If you are interested the codes inside plot3d()... > interface(verboseproc=2): #Turn on verboseproc > print(plot3d); #eval() also works > interface(verboseproc=1): #Turn off verboseproc I hope this will give you some help. Have fun with Maple. Ken Lin ### 40 on maple data types ### 41 how to extract elements from a list based on some selection? use select. For example >restart; >my_list:=[1,3.4,3+I,5]; >select(x->evalb(Im(x)=0),my_list); [1, 3.4, 5]  ### 42 how to test if all elements of a matrix are integers? restart; m:=Matrix( [[1.3,2,3],[3,4,4] ]); matrixTestQ := proc(m::Matrix) local r,c,i,j; (r,c):=LinearAlgebra[Dimensions](m); for i from 1 to r do for j from 1 to c do if( not evalb( whattype(m[i,j]) = integer) ) then return(false); end if; end do; end do; return true; end proc; >matrixTestQ(m); false  I am sure there is a better way than the above. Need to ﬁnd out. ### 43 how to use laplace transform? restart; f:= t->sin(omega*t) ; L:=convert(inttrans[laplace](f(t),t,s),int);  ${\frac {\omega }{{\omega }^{2}+{s}^{2}}}$ To ﬁnd the inverse, do:  inttrans[invlaplace](L,s,t);  $\sin \left ( \omega \,t \right )$ ### 44 questions I have Any difference between using diffalg/Rosenfeld_Groebner(args) or diffalg[Rosenfeld_Groebner](args)  ### 45 3D plotting restart; f:= (x,y)->x^3-3*x*y^2; plot3d(f,-1..1,-1..1,numpoints=2500,style=patchcontour); ### 46 How to raise each element in a list to a power? Use map map(^,{1,2,3},3); {1, 8, 27}  ### 47 How to generate a sequence with any increment? incr:=.25; start:=0; last:=3; seq(start+i*incr,i=1..(last/incr));  ### 48 What shortcuts are there for matrix manipulation? read ?MVshortcut, ?MVassignment, and ?Mvextract and Transpose(R) can be shortened to R^%T ### 49 How to solve a set of equations for the derivative? Written feb 20, 2004 This is problem 7.4 chapter 4, in the Mary Boas book. Given \begin {align*} x s^2+y t^2 &= 1\\ x^2 s+y^2 t &= xy-4 \end {align*} Find $$\frac {dx}{dt}, \frac {dx}{ds}, \frac {dy}{dt}, \frac {dy}{ds}$$ at $$x=1,y=-3,s=2,t=-1$$ This is how I did it in maple: restart; alias(x=x(s,t)); alias(y=y(s,t)); alias(Xt= diff(x(s,t), t)); alias(Xs= diff(x(s,t), s)); alias(Yt= diff(y(s,t), t)); alias(Ys= diff(y(s,t), s)); eq1:= x*s^2+y*t^2=1; eq2:= x^2*s+y^2*t=x*y-4; r1:=diff(eq1,t); r2:=diff(eq1,s); r3:=diff(eq2,t); r4:=diff(eq2,s); sol:=solve({r1,r2,r3,r4},{Xt,Xs,Yt,Ys});  \begin {align*} {\frac {\partial }{\partial s}}x \left ( s,t \right ) &= -{\frac {x \left ( s,t \right ) \left ( x \left ( s,t \right ) {t}^{2}-4\,y \left ( s,t \right ) st+2\,x \left ( s,t \right ) s \right ) }{2\,x \left ( s,t \right ) s{t}^{2}-2\,y \left ( s,t \right ) t{s}^{2}+x \left ( s,t \right ) {s}^{2}-y \left ( s,t \right ) {t}^{2}}}\\ {\frac {\partial }{\partial t}}x \left ( s,t \right ) &=-{\frac {y \left ( s,t \right ) t \left ( -3\,y \left ( s,t \right ) t+2\,x \left ( s,t \right ) \right ) }{2\,x \left ( s,t \right ) s{t}^{2}-2\,y \left ( s,t \right ) t{ s}^{2}+x \left ( s,t \right ) {s}^{2}-y \left ( s,t \right ) {t}^{2}}}\\ {\frac {\partial }{\partial s}}y \left ( s,t \right ) &=-{\frac {x \left ( s,t \right ) \left ( 3\,x \left ( s,t \right ) s-2\,y \left ( s,t \right ) \right ) s}{2\,x \left ( s,t \right ) s{t}^{2}-2\,y \left ( s,t \right ) t {s}^{2}+x \left ( s,t \right ) {s}^{2}-y \left ( s,t \right ) {t}^{2}}}\\ {\frac {\partial }{\partial t}}y \left ( s,t \right ) &=-{\frac {y \left ( s,t \right ) \left ( 4\,x \left ( s,t \right ) st-y \left ( s,t \right ) {s }^{2}-2\,y \left ( s,t \right ) t \right ) }{2\,x \left ( s,t \right ) s{t} ^{2}-2\,y \left ( s,t \right ) t{s}^{2}+x \left ( s,t \right ) {s}^{2}-y \left ( s,t \right ) {t}^{2}}} \end {align*} points:= {x=1,y=-3,s=2,t=-1}; subs(points,sol);  ### 50 How to solve a set of equations for diﬀerentials? This is problem 7.15 chapter 4 in Boas: Given $$x^2 u-y^2 v=1$$ and $$x+y=uv$$ Find $$\frac {dx}{du},v$$ and $$\frac {dx}{du},y$$ This is the maple code to solve this: restart; eq1:=x^2*u-y^2*v=1; eq2:=x+y=u*v; r1:=D(eq1); r2:=D(eq2); r1_:=subs(D(v)=0,r1); r2_:=subs(D(v)=0,r2); sol:=solve({r1_,r2_},{D(x),D(u)}); print("dx/du,v="); rhs(sol)/rhs(sol); r1_:=subs(D(y)=0,r1); r2_:=subs(D(y)=0,r2); sol:=solve({r1_,r2_},{D(x),D(u)}); print("dx/du,y="); rhs(sol)/rhs(sol); ### 51 How to plot binary tree restart; t2 := proc(i, x, y) if i < 2 then [[x, y], [x, y - 1]], [[x, y], [x + 2^i, y - 1]] else [[x, y], [x, y - 1]], [[x, y], [x + 2^i, y - 1]], t2(i - 1, x, y - 1), t2(i - 1, x + 2^i, y - 1) end if end proc; PLOT(CURVES(t2(6,0,0))); ### 52 solving problem 12.4 chapter 4, Math 121A, Boas book. using maple restart; z:= Int( sin(t)/t, t=sin(x)..cos(x)); diff(z,x);  $-{\frac {\sin \left ( x \right ) \sin \left ( \cos \left ( x \right ) \right ) }{\cos \left ( x \right ) }}-{\frac {\cos \left ( x \right ) \sin \left ( \sin \left ( x \right ) \right ) }{\sin \left ( x \right ) }}$ ### 53 example of doing convergence test in maple restart; c:='c': C:='C': n:='n': P:='P': C := n -> ((n+2)/(3*n+1))^n: ### WARNING: calls to C for generating C code should be replaced by codegen[C] The general term is , c[n]= C(n);  ; The n-th root is:; ### WARNING: calls to C for generating C code should be replaced by codegen[C] P := C(n)^(1/n): abs(c[n])^(1/n) = P; P := simplify(P, assume=positive): abs(c[n])^(1/n) = P; ### 54 Solving problem math 121A, ch 14, 3.18, Boas book. contour integration restart; f:= 1/( (1-2*z)*(5*z-4) ); residue(f,z=4/5);  $\frac {-1}{3}$ ### 55 How to ﬁnd multiple roots to an equation such as $$sin(x) = 0$$ _EnvAllSolutions:=true; solve(sin(x)=0);  Pi _Z1~ ### 56 Dr Basti Associated Legendre Subject: Associated Legendre Author: Mehran Basti <Basti@worldnet.att.net> Organization: AT&T Worldnet Date: Mon, 25 Nov 2002 02:48:15 GMT Dear newsgroup: I had mentioned that my methods will solve classical equations without the use of inﬁnite series. The following is a Maple code of my old ﬁles. Those days I had Maple2 but the general idea is the same in the process and you see that we can also solve the integrals involved. It does not make sense how are the theory behind it but eventually it will come into light. Just read the procedures and you can see the solution of associated legendre AL at the end. > s1:=-diff(p(t),t)+p(t)^2; > > s2:=exp(2*int(p(t),t))*T(t); > s3:=s1+s2; > s4:=diff(T(t),t)/T(t); > s5:=-(1/2)*(diff(s4,t))+(1/4)*s4^2; > s6:=s5+s2; > p(t):=-1/t+(1)/(2-t); > s1:=simplify(s1); > s1:=collect(%,t); > s2:=simplify(s2); > s1+s2=(2*t^2-4*t+m^2-1)/(t*(-2+t))^2; > solve(%,T(t)); > T(t):=simplify(%); > s2:=simplify(s2); > s2+s1; > s3:=simplify(%); > > s6:=simplify(s6); > t*(-2+t); > simplify(%); > z:=(r3*t^3+r2*t^2+r1*t+r0)/(%); > > simplify(diff(z,t)+z^2-s6); > s7:=collect(numer(%),t); > > coeff(%,t,0); > solve(%,r0); > r0:=op(1,{%}); > coeff(s7,t,1); > solve(%,r1); > r1:=simplify(%); > coeff(s7,t,2); > solve(%,r2); > r2:=simplify(%); > coeff(s7,t,3); > solve(%,r3); > r3:=simplify(%); > simplify(s7); > s3:=simplify(s3); > s4:=simplify(s4); > s6:=simplify(s6); > T(t):=simplify(T(t)); > z:=simplify(z); > 1/2*s4+2*p(t)+z; > s8:=simplify(%); > exp(int(%,t)); > expand(%); > g:=(%); > simplify(g,power); > g:=%; > Int(%,t); > Integralg:=(%); > int(g1(t),t); > x1:=-p(t)+g1(t)/(%); > diff(x1,t)+x1^2-s3; > simplify(%); > s10:=numer(%); > solve(%,int(g1(t),t)); > Ing:=(%); > simplify(subs(g1(t)=g,%)); > > Ing:=(%); > expand(%); > Ing:=simplify(%); > simplify(diff(%,t)-g); > expand(%); > simplify(%); > x:=-p(t)+g/Ing; > simplify(diff(x,t)+x^2-s3); > int(x,t); > exp(%); > expand(%); > s11:=simplify(%); > ALT:=t*(2-t)*diff(u(t),t2)+2*(1-t)*diff(u(t),t)+(2-m^2/(1-(1-t)^2))*u(t);
> -2*(1-t)/(2*t*(2-t));
> int(%,t);
> exp(%);
> s12:=simplify(%,power);
>
> u1:=s12*s11;
> u1:=simplify(%,power);
>  simplify(subs(u(t)=u1,ALT));
> AL:=(1-nu^2)*diff(u(nu),nu$2)-2*nu*diff(u(nu),nu)+(2-m^2/(1-nu^2))*u(nu); > > u2:=subs(t=1-nu,u1); > simplify(subs(u(nu)=u2,AL)); > The advantage of these methods are that there are ample rooms for advances. Today my skills for solving classical equations such as Riccati is much advanced. Highly complicated and more general Riccati equations in its billions now possible. Sincerely Dr.M.Basti ### 57 Understanding conformal mapping in maple To plot mapping of complex function in maple, use [plots]conformal The trick is to how to specify the quadrant in the x-y plane. This example shows how. Suppose we want to map the ﬁrst quadrent. Then we specify the DIAGONAL points in the range, from the lower left corner to the upper right corner, which then should be 0..1+I Because 0 is the lower left corner, and $$(1,i)$$ is the upper right corner. Example: restart; assume(y,real); assume(x,real); #f:= z->I+z*exp(I*Pi/4); f:= z->z^2; w:=f(x+I*y); u:=Re(w); v:=Im(w); plots:-conformal(f(z),z=0..1+I,grid=[16,16],numxy=[16,16],scaling=constrained); This below uses the ﬁrst TWO quadents, i.e. the upper half of the x-y plane restart; assume(y,real); assume(x,real); #f:= z->I+z*exp(I*Pi/4); f:= z->z^2; w:=f(x+I*y); u:=Re(w); v:=Im(w); plots:-conformal(f(z),z=-1-I..1+I,grid=[16,16],numxy=[16,16],scaling=constrained); This below puts the plots next to each others so to see them restart; assume(y,real); assume(x,real); f:= z->I+z*exp(I*Pi/4); #f:= z->z^2; w:=f(x+I*y); u:=Re(w); v:=Im(w); A := array(1..2): A:=plots:-conformal(z,z=0..1+I/2,grid=[16,16],numxy=[16,16],scaling=constrained): A:=plots:-conformal(f(z),z=0..1+I/2,grid=[16,16],numxy=[16,16],scaling=constrained): plots:-display(A); ### 58 Is there a way to keep the assumptions but not see the tilda character show up? interface(showassumed=0) removes all tildas and interface(showassumed=1) adds the tildas. ### 59 Fourier series in maple I wrote this to generate FS in Maple for some HW I was doing. I think this was for Math 121A at UC Berkeley in 2003 restart; f:=x->piecewise(-Pi<x and x<Pi/2,-1, Pi/2<x and x<1,0,1); assume(n,integer); nmaFourier2:=proc(f,freq,from_,to_,maxN) local n::integer,denomC,denomS,a,b; denomC:=( to_ - from_ ) / 2; denomS:=( to_ - from_ ) / 2; a:=proc(n) int(f(x)*cos(n*freq*x),x=from_..to_) /denomC; end proc; b:=proc(n) int(f(x)*sin(n*freq*x),x=from_..to_) / denomS; end proc; evalf(denomC); 1/2*a(0) + sum( a(n) * cos(n*freq*x) ,n=1..maxN) + sum( b(n) * sin(n*freq*x) ,n=1..maxN) end proc; r:=[seq(nmaFourier2(f,1,-Pi,Pi,nIter),nIter=1..10)]; plot(r,x=-Pi..Pi); To animate do g:=n->plot(nmaFourier2(f,1,-Pi,Pi,n),x=-2*Pi..2*Pi); plots:-animate(g,[n],n=1..40);  Here is the animation from the Maple notebook: Another version restart; f:=x->piecewise(-Pi<x and x<Pi/2,-1, Pi/2<x and x<1,0,1); assume(n,integer); nmaFourier2:=proc(f,freq,from_,to_,maxN::integer) local n::integer,denomC,denomS,a,b; denomC:=( to_ - from_ ) / 2; denomS:=( to_ - from_ ) / 2; a:=proc(n) int(f(x)*cos(n*freq*x),x=from_..to_) /denomC; end proc; b:=proc(n) int(f(x)*sin(n*freq*x),x=from_..to_) / denomS; end proc; 1/2*a(0) + sum( a(n) * cos(n*freq*x) ,n=1..maxN) + sum( b(n) * sin(n*freq*x) ,n=1..maxN) end proc; plots[setoptions](title= , axesfont=[SYMBOL,8] ,font=[COURIER,1], xtickmarks=[seq(evalf(k*Pi/2)=sprintf("%a %s", k/2 ,"pi" ),k= -3..3)], ytickmarks=[-1.0="-1",-0.5="",0.0="0",0.5="",1.0="1"]); B:=array(1..3,1..3); k:=0; for i from 1 to 3 do for j from 1 to 3 do k:=k+1; B[i,j]:=plot({f(x),nmaFourier2(f,1,-Pi,Pi,k)},x=-Pi..Pi,size=[200,100]); end do; end do; plots:-display( B); ### 60 How to plot graphs next to each others in a grid like fashion restart; v:=1; B:=Matrix(3,3); for i from 1 to 3 do for j from 1 to 3 do v:=v+1; B[i,j]:= plot(x^v,x=-2..2,thickness=3,size=[200,100] ); end do; end do; plots:-display(B); ### 61 How to generate Pi on X-axis From book Maple animation by John Putz plot( sin(x), x=0..2*Pi, xtickmarks=evalf([Pi/2="p/2", Pi="p", 3*Pi/2="3p/2", 2*Pi="2p"]), ytickmarks=[-1,1], axesfont=[SYMBOL,16], labels=["",""] ); ### 62 How to make output from FunctionAdvisor look better? From Preben Alsholm res:=FunctionAdvisor(sin): res2:=op(2,eval(res)): map(print,res2);  or answer by Thomas Richard > FunctionAdvisor( display, sin );  ### 63 How to do partial fractions? Use convert(expr,parfrac) or convert(f,fullparfrac) ### 64 How to generate sequence sum symbolically n := 7; f:=sum('a[k]*b[k]','k'=1..n);  $a_{{1}}b_{{1}}+a_{{2}}b_{{2}}+a_{{3}}b_{{3}}+a_{{4}}b_{{4}}+a_{{5}}b_{ {5}}+a_{{6}}b_{{6}}+a_{{7}}b_{{7}}$ ### 65 Nice plot from Maple from Serge from the net: restart; with(geom3d): plane(OYZ,x=0,[x,y,z]): plane(OXZ,y=0,[x,y,z]): plane(OXY,z=0,[x,y,z]): c:=1/2:r:=1/4: L:=combinat[permute]([-c$3,c$3],3): S:=seq(sphere(s||i,[point(A||i,op(op(i,L))),r]),i=1..8): draw([OYZ,OXZ,OXY,S]); ### 66 How to check if 2 expressions are the same? Use evalb(). For example evalb(I*sinh(x)=sin(I*x)); gives true The above does not always work. Only sure way is to do this > m1 := exp(I*n*x); m2 := (cos(n*x)+I*sin(n*x)); simplify(m1-m2); simplify(m1-convert(m2,exp));  ### 67 converting series to factorials Function by Robert Israel from the net: restart; thefacts:= [seq(i!,i=2..20)]: getfacts:= proc(x::{algebraic,series}) local i; if type(x, {+,*,series}) then map(getfacts,x) elif type(x, fraction) then getfacts(numer(x))/getfacts(denom(x)) elif type(x,^) then getfacts(op(1,x))^op(2,x) elif type(x,negint) then -getfacts(-x) elif type(x,posint) then for i from 1 to 19 while irem(x, thefacts[i]) = 0 do od: if i = 1 then x elif thefacts[i-1] = x then (i)! else (i-1)!*getfacts(x/thefacts[i]) fi else x fi end; getfacts(series(sin(x),x));  $\text {series} \left ( x-{\frac {{x}^{3}}{ \left ( \left ( 3 \right ) \right ) !}}+{\frac {{x}^{5}}{ \left ( \left ( 5 \right ) \right ) !}}+O \left ( {x}^{7} \right ) ,x,7 \right )$ ### 68 How to ﬁnd what new additions made to Maple? ?updates,maple10 ### 69 Maple can’t solve laplace equation and numerically Maple 2020. restart; PDE := diff(u(x,y), y$2 ) + diff(u(x,y), x$2) = 0; BC:= u(x,0)=0, u(x,100)=100, u(0,y)=0, u(10,y)=0; sol:=pdsolve(PDE,[BC] ,numeric); Error, (in pdsolve/numeric) unable to handle elliptic PDEs  Compare to restart; PDE := diff(u(x,y), y$2 ) + diff(u(x,y), x2) = 0; BC:= u(x,0)=0, u(x,100)=100, u(0,y)=0, u(10,y)=0; sol:=pdsolve([PDE,BC]);  $u \left ( x,y \right ) =\sum _{n=1}^{\infty }-200\,{\frac { \left ( \left ( -1 \right ) ^{n}-1 \right ) {{\rm e}^{10\,\pi \,n}}\sin \left ( 1/10\,n\pi \,x \right ) \left ( {{\rm e}^{1/10\,n\pi \,y}}- {{\rm e}^{-1/10\,n\pi \,y}} \right ) }{\pi \,n \left ( {{\rm e}^{20\,\pi \,n}}-1 \right ) } }$ ### 70 Some Maple Matrix operations Create a new matrix, by appending some rows of one matrix to rows from another matrix: restart; with(LinearAlgebra): A:=< <1|2|3> , <4|5|6> >;  \left [ \begin {array}{ccc} 1&2&3\\ \noalign {\medskip }4&5&6 \end {array} \right ] B:=< <7|8|10> , <11|12|13> , <14|15|16> >;  \left [ \begin {array}{ccc} 7&8&10\\ \noalign {\medskip }11&12&13 \\ \noalign {\medskip }14&15&16\end {array} \right ] Now append ﬁrst row of A to last 2 rows of B C:=< A[1,1..-1] , B[2..-1,1..-1] >;  \left [ \begin {array}{ccc} 1&2&3\\ \noalign {\medskip }11&12&13 \\ \noalign {\medskip }14&15&16\end {array} \right ] # Now append first column of A to first 2 rows of B A[1..-1,1]; B[1..2,1..-1]; C:=< A[1..-1,1] | B[1..2,1..-1] >;  \left [ \begin {array}{cccc} 1&7&8&10\\ \noalign {\medskip }4&11&12&13 \end {array} \right ] #Now remove the middle row of B B; B:=<B[1,1..-1] , B[-1,1..-1] >;  \left [ \begin {array}{ccc} 7&8&10\\ \noalign {\medskip }14&15&16 \end {array} \right ] #now set the diagonal elements of B to be 0 B:=RandomMatrix(3); for i from 1 to 3 do B[i,i]:=0; end do: B;  B:=\left [ \begin {array}{ccc} 0&99&92\\ \noalign {\medskip }8&0&-31 \\ \noalign {\medskip }69&44&0\end {array} \right ] \left [ \begin {array}{ccc} 0&99&92\\ \noalign {\medskip }8&0&-31 \\ \noalign {\medskip }69&44&0\end {array} \right ] To ﬁnd inverse. restart; with(LinearAlgebra): A:=Matrix( [ [2,0],[4,2] ]); MatrixInverse(A);  \left [ \begin {array}{cc} 1/2&0\\ \noalign {\medskip }-1&1/2 \end {array} \right ] To check that for any matrix A, then A*transpose(A) is always a matrix which is symmetrical A:=RandomMatrix(2,3); A.Transpose(A);  A:=\left [ \begin {array}{ccc} 99&44&-31\\ \noalign {\medskip }29&92&67 \end {array} \right ] \left [ \begin {array}{ccc} 99&44&-31\\ \noalign {\medskip }29&92&67 \end {array} \right ] how to create a random lower triangular matrix? restart; with(LinearAlgebra); A:=RandomMatrix(4,4,outputoptions=[shape=triangular[lower]]);  \left [ \begin {array}{cccc} 67&0&0&0\\ \noalign {\medskip }-31&92&0&0 \\ \noalign {\medskip }44&29&99&0\\ \noalign {\medskip }69&8&27&-4 \end {array} \right ] ### 71 How set diagonal elements to some value, say 1? restart; with(LinearAlgebra); A:=RandomMatrix(5); LinearAlgebra:-Map[(i,j)->evalb(i=j)](x->1,A);  A:= \left [ \begin {array}{ccccc} 1&-98&-76&-4&29\\ \noalign {\medskip }-38& 1&-72&27&44\\ \noalign {\medskip }-18&57&1&8&92\\ \noalign {\medskip }87& 27&-32&1&-31\\ \noalign {\medskip }33&-93&-74&99&1\end {array} \right ] \left [ \begin {array}{ccccc} 1&-98&-76&-4&29\\ \noalign {\medskip }-38& 1&-72&27&44\\ \noalign {\medskip }-18&57&1&8&92\\ \noalign {\medskip }87& 27&-32&1&-31\\ \noalign {\medskip }33&-93&-74&99&1\end {array} \right ] ### 72 How to multiply roots of a polynomial? eq:=3*x^3+2*x^2+x+5=0; s:=[evalf(solve(eq,x))]; mul(s[i],i=1..nops(s));  Gives ### 73 How to plot a surface in 3D? restart; eq:=3*x+4*y+2*z=10; plot3d(solve(eq,z),x=-5..5,y=-5..5,axes=normal); One can also use impliticplot3d restart; with(plots): implicitplot3d(3*x+4*y+2*z=10, x=-5..5,y=-5..5, z=-20..20,axes=normal); ### 74 How to convert trigs to sinc function in an expression Maple doesn’t have a sinc function. If you mean the function sinc(x) = sin(x)/x, you could say something like > eval(expr, {sin = (x -> x*sinc(x)), cos = (x -> (x+Pi/2)*sinc(x+Pi/2)), tan = (x -> x*sinc(x)/(x+Pi/2)/sinc(x+Pi/2))});  ### 75 How to ﬁnd NullSpace and ColumnSpace of a matrix? restart; with(LinearAlgebra): A:=Matrix([[1,0,1,0,1],[0,1,0,1,0]]); NullSpace(A); ColumnSpace(A); ### 76 How to ﬁx the interface to using Maple notation for input? Go to tools->optiopn, and Display, and select Maple notation for input display. ### 77 How to ﬁnd all solutions using allvalues ? solve(x^2-sin(x),x); RootOf(-sin(_Z)+_Z^2) allvalues(%); RootOf(-sin(_Z)+_Z^2, 0.), RootOf(-sin(_Z)+_Z^2, .8767262154) evalf(%); 0., .8767262154  ### 78 How to add one to only the elements of the diagonal of a matrix? Use Map with ﬁlter A:=< 1,2,3;4,5,6;7,8,9>; LinearAlgebra:-Map[(i,j)->evalb(i=j)](x->x+1,A);  ### 79 How to search help for updates on some package and type say updates,Maple17,DE in the small box there. ### 80 How to work with groups in worksheet I use these special keystrokes constantly in my Maple worksheet typing: Ctrl-J: Insert execution group below cursor. Ctrl-K: Insert execution group above cursor. Ctrl-T: Switch from executable code mode to text mode (for entering extended formatted comments). Ctrl-M: Switch from text mode to executable code mode. Shift-Enter (or Shift-Return): Begin a new line in the same execution group. Func-3: Split execution group into two (at cursor). Func-4: Join cursor execution group with execution group below. ### 81 How to read code into worksheet? Use the read command, as in read "mycode.mpl" where mycode.mpl is plain text ﬁle that contains maple code ### 82 Code editors for Maple ### 83 How to ﬁnd if package is module or table? New packages are module, which allows using packageName:-function() since it is easier. Old packages use tables which needs packageName[function]() which is not common. To ﬁnd if package is based on module or not, use the command  type(combstruct,'module');  This will return true or false. To know if name is package use the command  type(combstruct,'package');  ### 84 How to replace a string? file_name :=StringTools:-SubstituteAll(file_name,":","-");  ### 85 How to use geometry and plottools ? restart; c:= i->([i/(1+i),0],1/(1+i)): d:= i->([1,1/i],1/i): geometry:-circle(c1,[geometry:-point(o,2/3,0),1/3],[x,y]): geometry:-circle(c2,[geometry:-point(o,1,1),1],[x,y]): geometry:-intersection(o,c1,c2,[u,v]): plots:-display(plottools:-circle(c(2)),plottools:-circle(d(1)),geometry:-draw(o));  To know more about the intersection, use this: geometry:-detail(o);  ### 86 How to simplify log expressions ? Use symbolic option restart; simplify(ln(3^x/2^y) =ln(n),symbolic);  ### 87 How to simplify hyperbolic expression ? How to convert $\frac {3+2\sinh (x)^2}{\sinh (x)^2\tanh (x)}$ to $3 \coth ^3(x)-\coth (x)$ restart; e := (3+2*sinh(x)^2)/(sinh(x)^2*tanh(x)); expand(student[changevar](sinh(x)^2=tanh(x)^2/(1-tanh(x)^2),e));  ### 88 How to create text ﬁle and append string to it? restart; try fd :=-1; fd := fopen("C:\\output3.txt",APPEND,TEXT); catch: print(Unable to open file, error is); print(StringTools:-FormatMessage(lastexception)); end try: if not(evalb(fd=-1)) then #file open ok str:="hello world"; try fprintf(fd,"%s\n",str); catch: print(failed to append to file, error is); print(StringTools:-FormatMessage(lastexception)); finally: close(fd); end try; fi:  ### 89 How to search packages and libraries? To ﬁnd in which library a command is do with(LibraryTools); FindLibrary('int',all); #find which library command int is in "C:\Program Files\Maple 18\lib\update.mla", "C:\Program Files\Maple 18\lib\DEsAndMathematicalFunctions18.mla", "C:\Program Files\Maple 18\lib\maple.mla"  To get content of library do restart; with(LibraryTools): LibLocation:=cat(kernelopts(mapledir),"/lib/maple.mla"); c:=ShowContents(LibLocation);  Then can use this to print the name of each symbol/command, and then use whattype command to ﬁnd its type seq(c[i,1],i=1..20);  To get list of Maple kernel builtin commands and symbols, use this. Written by Acer from Maple prime site: restart: interface(warnlevel=0): started := false: T := 'T': for i from 1 to 1000 do f := eval(parse(cat("proc() option builtin=",i,"; end proc"))); p := (s->StringTools:-Take(s,StringTools:-Search(";",s)-1))(convert(eval(f),string)[26..]); if not type(parse(p),posint) then T[i] := p; started := true; else if started then i:=1000; next; end if; end if; end do: i; [ entries(T,nolist) ]; nops(%);  The above gives on Maple 18.02 the following ["crinterp", "equation", "{}", "even", "debugopts", "embedded_imaginary", "define_external", "embedded_real", "coeff", "cx_zero", "coeffs", "embedded_axis", "conjugate", "constant", "convert", "cx_infinity", "dlclose", "identical", "divide", "hfloat", "done", "function", "", "fraction",
"denom", "float", "degree", "finite", "disassemble",
"extended_rational", "diff", "extended_numeric", "frem",
"union", "frontend", "upperbound", "exports", "writeto",
"factorial", "xor", "evalgf1", "type", "expand", "typematch",
"entries", "unames", "evalb", "unbind",
"evalf/hypergeom/kernel", "atomic", "hfarray", "anything",
"hastype", "complex", "has", "boolean", "goto", ":-",
"gmp_isprime", "!", "genpoly", "anyfunc", "gc", "algebraic",
"SFloatMantissa", "ssystem", "Scale10", "stop", "Scale2",
"sort", "SearchText", "[]", "~", "subset", "~Array",
"subsindets", "~Matrix", "streamcall", "~Vector", "subs",
"Unordered", "table", "ToInert", "system",
"_maplet", "trunc", "_jvm", "kernel/transpose", "_treeMatch",
"tcoeff", "_savelib", "taylor", "abs", "rtable_num_dims",
"_xml", "rtable_redim", "and", "rtable_scale", "andmap",
"rtable_scanblock", "alias", "rtable_size", "anames",
"rtable_sort_indices", "assign", "savelib", "assemble",
"rtable_zip", "array", "select", "appendto", "searchtext",
"cat", "series", "callback", "selectremove", "bind", "sign",
"attributes", "setattribute", "ormap", "ArrayOptions", "order",
"Array", "parse", "**", "overload", "*", "::", "numer",
"CopySign", "numelems", "^", "or", "||", "op", "nops",
"seq", "normal", "time", "not", "piecewise", "numboccur",
"?[]", "userinfo", "modp2", "inner", "mods", "timelimit",
"mvMultiply", "traperror", "negate", "rtable_normalize_index",
"call_external", "rtable_is_zero", "assigned", "rtable_indfns",
"evalf", "rtable_histogram", "eval", "evaln", "rtable_eval",
"truefalse", "evalhf", "rtable_convolution", "tabular", "mul",
"rtableInfo", "zppoly", "if", "rtable", "uneval", "remove",
"sfloat", "rhs", "specfunc", "readlib", "string", "reduce_opr",
"symbol", "ASSERT", "?()", "realcons", "TRACE", "quit",
"relation", "_local", "pointto", "sequential", "add", "print",
"int/series", "protected", "Record", "irem", "procedure",
"Re", "iquo", "poszero", "isqrt", "real_infinity", "RETURN",
"is_gmp", "ratpoly", "+", "lcoeff", "rational", "OrderedNE",
"kernelopts", "range", "Object", "NumericEventHandler",
"icontent", "numeric", "NumericStatus", "igcd", "odd",
"NumericClass", "ilog10", "nonpositive", "NumericEvent",
"ilog2", "nonreal", "implies", "posint", "NameSpace",
"indets", "positive", "NextAfter", "indices", "polynom",
"MPFloat", "intersect", "pos_infinity", "MorrBrilCull",
"<", "member", "neg_infinity", "Im", "maxnorm", "name",
"<>", "max", "negint", "<=", "map2", "negative", "modp1",
"nonnegative", "FromInert", "modp", "negzero",
"EqualStructure", "minus", "nonposint", ">=", "min",
"nonnegint", ">", "DefaultUnderflow", "lexorder",
"imaginary", "=", "lhs", "indexable", "ERROR", "ldegree",
"indexed", "EqualEntries", "length", "integer", "macro",
"list", "DEBUG", "map", "literal", "..", "lowerbound",
"module", "Default0", "lprint", "moduledefinition",
"DefaultOverflow"]
296


### 90 How to numerically solve a BVP ode and plot the solution?

This one has one solution

eq:=diff(u(z),z$2)+(k-1)*diff(u(z),z)/z+lambda*exp(u(z))=0; sol:=dsolve({subs({k=1,lambda=2},eq),u(0)=1,u(1)=0},numeric,u(z), method=bvp[midrich],'abserr'=0.001); plots[odeplot](sol);  This solved coupled ODE’s, so there are 2 solutions. Say $$x_1(t)$$ and $$x_2(r)$$, It is a little tricky to plot all solutions generated, but here is an example restart; R := 0.4; px := 32000; Mm := 0.1; Ds := 9; DO2 := 7.2; YXS := 0.3; KS := 10; Sp := 30; Cb := 8; KO2 := 0.2; R0 := 0.000001; YXO := 0.42857; Vs := px*1/YXS*(Mm*x2(r))/(KS + x2(r))*x1(r)/(KO2 + x1(r)); Vo := px*1/YXO*(Mm*x2(r))/(KS + x2(r))*x1(r)/(KO2 + x1(r)); eqs := diff(x1(r),r$2) + 2/r*diff(x1(r),r)= Vo/DO2,
diff(x2(r),r$2) + 2/r* diff(x2(r),r)= Vs/Ds; ic:=D(x1)(R0)=0,x1(R) = Cb,D(x2)(R0)= 0, x2(R) = Sp; sol:=dsolve({eqs,ic},numeric,{x1(r),x2(r)},'abserr'=.52,'maxmesh'=1000,output=listprocedure);  And now to plot do x1Sol:=rhs(sol); plot(x1Sol(r),r=0..0.4); x2Sol:=rhs(sol); plot(x2Sol(r),r=0..0.4);  ### 91 How to ﬁnd the indicial equation for an ODE? For say Bessel ode of order zero: eq:= x^2*diff(y(x),x$2)+x*diff(y(x),x)+x^2*y(x)=0;
DEtools[indicialeq](eq,x,0,y(x));
#x^2 = 0



The third argument above is the singularity point of interest. So we have two roots, both zero. These are now used for ﬁnding the power series solution $$y(x)$$ if needed.

Another example, is Bessel of order 1

eq:= x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-1)*y(x)=0; DEtools[indicialeq](eq,x,0,y(x)); #x^2-1 = 0  ### 92 How to display on screen for speciﬁc width? This below by Axel Vogt posted on sci.math.symbolic which does a nice job of formatting output to speciﬁc width. split_for_print:=proc(expr, len) # expr = some Maple expression # len = length to split with line breaks local L,s,tmp,j; s:=convert(expr, string); L:=[StringTools:-LengthSplit(s, len)]; for j from 1 to nops(L) do # if j = nops(L) then printf("%s ;", L[-1]) if j = nops(L) then printf("%s", L[-1]) else printf("%s\\\n", L[j]); end if; end do: end proc; evalf(Pi); split_for_print(%, 40); 3.14159265358979323846264338327950288419\ 7169399375105820974944592307816406286208\ 998628034825342117068  for VIM in vim, type set syntax=maple after putting the ﬁle maple.vim in ~/.vim/syntax/maple.vim. I found maple.vim in above link. For Maple IDE ### 94 loading, remove and ﬁnding what packages loaded use packages(); to ﬁnd what packages loaded. use unwith to remove package packages(); [] with(DynamicSystems): packages(); [DynamicSystems] unwith(DynamicSystems); packages(); []  ### 95 some rules of thumbs when using Maple 1. put restart in separate execution group 2. do not use with inside proc(). Use uses instead. ### 96 How to write derivative To write $$y'(x)=x$$, one way is diff(y(x),x)=x and another is D(y)(x)=x. To write $$y''(x)=x$$, one way is diff(y(x),x$2)=x and another is (D@@2)(y)(x)=x.

To convert from one form to another use convert(eq,diff) or convert(eq,D)

### 97 How to solve heat PDE in 1D in Maple 2017?

to solve $$\frac {\partial u(x,t)}{\partial t}=k \frac {\partial ^2 u(x,t)}{\partial x^2}$$ with homogeneous dirichlet boundary conditions $$u(0,t)=0,u(L,t)=0$$ the commands are

restart;
pde:=diff(u(x,t),t)=k*diff(u(x,t),x$2); bc:=u(0,t)=0,u(L,t)=0; sol:=pdsolve([pde,bc]) assuming 0<L:  Which gives $u \left ( x,t \right ) =\sum _{{\it \_Z1}=1}^{\infty }{\it \_C1} \left ( {\it \_Z1} \right ) \sin \left ( {\frac {\pi \,{\it \_Z1}\,x}{L}} \right ) {{\rm e}^{-{\frac {k{\pi }^{2}{{\it \_Z1}}^{2}t}{{L}^{2}}}}}$ Which can be made more readable as follows sol:=algsubs(_Z1=n,sol): sol:=algsubs(Pi*n/L=lambda(n),sol);  $u \left ( x,t \right ) =\sum _{n=1}^{\infty }{\it \_C1} \left ( n \right ) \sin \left ( x\lambda \left ( n \right ) \right ) {{\rm e}^{-kt \left ( \lambda \left ( n \right ) \right ) ^{2}}}$ For homogeneous Neumann B.C., at $$x=0$$, let $$\frac {\partial u}{\partial x}=0$$ and at $$x=L$$ let $$u(L,t)=0$$, the solution it gives looks diﬀerent than my hand solution restart; pde:=diff(u(x,t),t)=k*diff(u(x,t),x$2);
bc:=D(u)(0,t)=0,u(L,t)=0;
pdsolve([pde,bc]) assuming 0<L;



It gives

$u \left ( x,t \right ) ={\it \_C3}\,{\it \_C2}\, \left ( {{\rm e}^{1/4\,{\frac {2\,i\pi \,xL-k{\pi }^{2}t}{{L}^{2}}}}}+{{\rm e}^{-1/4\,{\frac {\pi \, \left ( 2\,ixL+k\pi \,t \right ) }{{L}^{2}}}}} \right )$

I need to look more into the above and see if this comes out to be the same as my hand solution.

Another example, with initial conditions now given

restart;
pde:=diff(u(x,t),t)=k*diff(u(x,t),x$2); bc:=D(u)(0,t)=0,u(L,t)=0; ic:=u(x,0)=f(x); sol:=pdsolve([pde,bc,ic],u(x,t)) assuming 0<L; sol1:=algsubs(_Z2=n,sol);  The result is $u \left ( x,t \right ) =\sum _{n=1}^{\infty } \left ( 2\,{\frac {1}{L}{{\rm e}^{-1/4\,{\frac {k{\pi }^{2}t \left ( 1+2\,n \right ) ^{2}}{{L}^{2}}}}}\cos \left ( 1/2\,{\frac {\pi \,x \left ( 1+2\,n \right ) }{L}} \right ) \int _{0}^{L}f \left ( x \right ) \cos \left ( 1/2\,{\frac {\pi \,x \left ( 1+2\,n \right ) }{L}} \right ) \,{\rm d}x} \right )$ Another example restart; pde:=diff(u(x,t),t)=k*diff(u(x,t),x$2);
bc:=D(u)(0,t)=0,u(L,t)=0;
ic:=u(x,0)=3*sin(Pi*x/L)-sin(3*Pi*x/L);
sol:=pdsolve([pde,bc,ic],u(x,t)) assuming 0<L;
sol1:=algsubs(_Z2=n,sol);



$u \left ( x,t \right ) =\sum _{n=1}^{\infty }768\,{\frac {1}{\pi \, \left ( 16\,{n}^{4}+32\,{n}^{3}-136\,{n}^{2}-152\,n+105 \right ) }{{\rm e}^{-1/4\,{\frac {k{\pi }^{2}t \left ( 1+2\,n \right ) ^{2}}{{L}^{2}}}}}\cos \left ( 1/2\,{\frac {\pi \,x \left ( 1+2\,n \right ) }{L}} \right ) }$

Another example

restart;
pde:=diff(u(x,t),t)=k*diff(u(x,t),x2); bc:=u(0,t)=0,u(L,t)=0; ic:=u(x,0)=3*sin(Pi*x/L)-sin(3*Pi*x/L); sol:=pdsolve([pde,bc,ic],u(x,t)) assuming 0<L;  $u \left ( x,t \right ) =\sin \left ( {\frac {\pi \,x}{L}} \right ) {{\rm e}^{-9\,{\frac {{\pi }^{2}kt}{{L}^{2}}}}} \left ( -2\,\cos \left ( 2\,{\frac {\pi \,x}{L}} \right ) +3\,{{\rm e}^{8\,{\frac {{\pi }^{2}kt}{{L}^{2}}}}}-1 \right )$ The above answer seems wrong. There is not even a summation in it. It is diﬀerent from my hand solution. Look more into it. ### 98 How to make multiple assumptions on a symbol? assume( A::AndProp(NonZero,constant) );  Now can use is(A,constant); ### 99 How to make Maple display diﬀ(y(x),x) as $$y'(x)$$ or as $$y'$$ ? Add this expr:=diff(y(x),x); Typesetting:-Settings(typesetprime=true, prime=x):  The above will display the expression as $$y'(x)$$. To make it now show the $$x$$ do expr:=diff(y(x),x); Typesetting:-Settings(typesetprime=true, prime=x): Typesetting:-Suppress(y(x));  Now it will show the expression as just $$y'$$. For all the above to work, make sure you have Typesetting level set to Extended in the GUI. This is done inside Tools->Options->Display menu. To clear all the above Typesetting, do restart or do Typesetting:-Unsuppress(y(x)) ### 100 How to check if expression is an equation? check for ‘=‘ as follows eq:= x=1; whattype(eq); # = if whattype(eq) = = then print("yes"); else print("no"); fi; "yes"  ### 101 How to check if expression is a set? check for ‘set‘ as follows eq:= {diff(y(x),x)=1,x(0)=1}; if whattype(eq) = set then print("yes"); else print("no"); fi; "yes"  ### 102 How to set boundary conditions for dsolve or pdsolve? The Maple syntax for seeting initial and boundary conditions is very confusing, as compared to Mathematica, which seems to me to be simpler. So I wrote this to remind me of the syntax each time. For PDE, assuming dependent variable is $$u(x,t)$$ then  Conditions Maple code $$u(0,t)=0$$ u(0,t)=0 $$\frac {\partial u}{\partial x}=0$$ at $$x=0$$ D(u)(0,t)=0 $$\frac {\partial ^2 u}{\partial x^2}=0$$ at $$x=0$$ D[1,1](u)(0,t)=0 $$\frac {\partial ^3 u}{\partial x^3}=0$$ at $$x=0$$ D[1,1,1](u)(0,t)=0 $$\frac {\partial u}{\partial t}=0$$ at $$t=0$$ D(u)(x,0)=0 $$\frac {\partial ^2 u}{\partial t^2}=0$$ at $$t=0$$ D[2,2](u)(x,0)=0 $$\frac {\partial ^3 u}{\partial t^3}=0$$ at $$t=0$$ D[2,2,2](u)(x,0)=0 Notice the syntax for the last one above. It is (D@@2)(u)(0,t)=0 and not (D@@2)(u)(0,t)=0 For an ODE, assuming dependent variable is $$y(x)$$ then the syntax is  Conditions Maple code $$y(0)=0$$ y(0)=0 $$\frac {dy}{dx}=0$$ at $$x=0$$ D(y)(0)=0 $$\frac {d^2 y}{d x^2}=0$$ at $$x=0$$ (D@@2)(y)(0)=0 ### 103 How to export a plot to PDF? I could only ﬁnd a way to export to eps plotsetup(default): plotsetup(postscript, plotoutput=t.eps, plotoptions=color,portrait,height=300); plot(sin(x),x=-Pi..Pi,'gridlines'); plotsetup(default):  Make sure not to put : at the end of the plot command! else it will not be exported. It has to end with ; This will same it to t.eps in the currentdir() location. Then used ps2pdf t.eps t.pdf to convert it to PDF. Or just ps2pdf t.eps it will automatically create t.pdf Or ps2pdf -dCompatibilityLevel=1.4 t.eps but may it is best to do ps2pdf -dCompatibilityLevel=1.4 -dEmbedAllFonts=true t.eps Also try adding -dPDFSETTINGS=/printer to the above. This tells it to optimize it for printing. Another example of a direction ﬁeld for an ODE plotsetup(postscript, plotoutput=t0.eps, plotoptions=color,portrait,height=300); ode:= diff(y(x),x) = 3*x^2 - 1; DEtools:-DEplot( ode, y(x), x=-2..2, [y(0) = 0], y=-2..2, linecolour=red, color = blue, stepsize=.05,arrows=MEDIUM ); plotsetup(default);  ### 104 How to ﬁnd all roots of complex number To ﬁnd roots of $$(3+4 i)^{1/3}$$, do fsolve(z^3=(3+4*I),z); #gives -1.26495290635775+1.15061369838445*I, -.363984239564424-1.67078820068900*I, 1.62893714592218+.520174502304545*I  ### 105 How to convert matrix of matrices to a matrix? A:= Matrix(2, 2, {(1, 1) = 0, (1, 2) = 0, (2, 1) = 0, (2, 2) = 2}); f:=x->if(x<>0,x*LinearAlgebra:-IdentityMatrix(2),0*Matrix(2)); B:=map(f,A);  Which gives \left [ \begin {array}{cc} \left [ \begin {array}{cc} 0&0 \\ \noalign {\medskip }0&0\end {array} \right ] & \left [ \begin {array} {cc} 0&0\\ \noalign {\medskip }0&0\end {array} \right ] \\ \noalign {\medskip } \left [ \begin {array}{cc} 0&0 \\ \noalign {\medskip }0&0\end {array} \right ] & \left [ \begin {array} {cc} 2&0\\ \noalign {\medskip }0&2\end {array} \right ] \end {array} \right ] now r:=Matrix(convert(B,listlist))  Gives \left [ \begin {array}{cccc} 0&0&0&0\\ \noalign {\medskip }0&0&0&0 \\ \noalign {\medskip }0&0&2&0\\ \noalign {\medskip }0&0&0&2\end {array} \right ] ### 106 How to do pattern matching in Maple? Maple has a simple but easy to use pattern matching, which works well. Here are some example. For each case, will show what pattern to detect and how to do it. I am still not very good at pattern matching in Maple and will need to make improvement in this with time. #### 106.1 Example 1 Detect $$\sqrt (x y)$$ in expression. restart; expr:= sin(x)*sqrt(x*y); if patmatch(expr,a::anything*(b::anything*x*y)^(c::anything),'la') then assign(la); if c =1/2 or c=-1/2 then print("found sqrt(x*y)"); else print("did not find sqrt(x*y)"); fi; fi;  But if the expression was $$\sin (x)\sqrt {x y}+3$$ then the above would fail, because there are a term after $$\sqrt {x y}$$, so the pattern has to change to restart; expr:= sin(x)*sqrt(x*y)+3; if patmatch(expr,a::anything*(b::anything*x*y)^(c::anything)+d::anything,'la') then assign(la); if c =1/2 or c=-1/2 then print("found sqrt(x*y)"); else print("did not find sqrt(x*y)"); fi; fi;  ### 107 How to ﬁnd trig indetities? use trigsubs, very useful command. For example trigsubs(cos(theta)^3)  Gives $[1/2\,\cos \left ( \theta \right ) +1/2\,\cos \left ( 2\,\theta \right ) \cos \left ( \theta \right ) ,1/4\,\cos \left ( 3\,\theta \right ) +3/4\,\cos \left ( \theta \right ) ]$ ### 108 How to ﬁnd directional derivative of scalar function? Given $$f(x,y,z)=x^2 z+y^3 z^2-xyz$$ we want to ﬁnd its directional derivative along the vector $$n$$. One way n:=<-1,0,3>; g:=VectorCalculus[Gradient](x^2*z+y^3*z^2-x*y*z, [x,y,z]); Student[VectorCalculus][DotProduct](g,n/LinearAlgebra[Norm](n,2))  Gives $-{\frac { \left ( 2\,xz-yz \right ) \sqrt {10}}{10}}+{\frac { \left ( 6\,{y}^{3}z+3\,{x}^{2}-3\,xy \right ) \sqrt {10}}{10}}$ Another is Student[MultivariateCalculus][DirectionalDerivative](x^2*z+y^3*z^2-x*y*z, [x,y,z], [-1,0,3]);  Gives the same result. ### 109 How to check if name is assigned a value? For simple variable, use assigned restart; x:=10: assigned(x) true assigned(y) false  For a ﬁeld in table do restart; A:=table(["x"=10,"y"=20]): assigned(A["x"]) true assigned(A["z"]) false  For ﬁeld in Record, I do not know how yet, other than using try catch, as assigned does not seem to work for Record ﬁelds. restart; A:=Record('x'=10,'y'=20); try assigned(A:-x) catch: print("no such field in record") end try; true try assigned(A:-z) catch: print("no such field in record") end try; "no such field in record"  ### 110 How to use dsolve with Lie? Use dsolve(ode,Lie) To ﬁnd symmetries, do DEtools:-symgen(ode,y(x),HINT=[c__1+c__2*x+c__3*y,c__4+c__5*x+c__6*y]) or just DEtools:-symgen(ode,y(x)) To debug it do stopat(ODEtools/symgen); before calling dsolve or DEtools:-symgen ### 111 How to select terms with sqrt or radical in them from an expression Given $3+x +\sqrt {-4 a c +b^{2}}+\sin \left (y \right )+x^{3} \sqrt {39}+\sqrt {\cos }x$ Find terms that are sqrt. Use indets restart; expr_with_radical:= 3+x+sqrt(b^2-4*a*c)+sin(y)+x^3*sqrt(39)+sqrt(cos(x));| indets(expr_with_radical, algebraic^fraction)  $$\{ \sqrt {39}, \sqrt {-4 a c +b^{2}}, \sqrt {\cos }x \}$$ Alternative is to use type radical restart; expr_with_radical:= 3+x+sqrt(b^2-4*a*c)+sin(y)+x^3*sqrt(39)+sqrt(cos(x));| indets(expr_with_radical, radical)  $$\{ \sqrt {39}, \sqrt {-4 a c +b^{2}}, \sqrt {\cos }x \}$$ ### 112 How to simplify $$e^{\ln (x)+\ln (y)}$$ given ${\mathrm e}^{\frac {2 \ln \left (\sqrt {p^{2}+1}+p \right )+2 \ln \left (a \right )+\ln \left (p^{2}+1\right ) a}{2 a}}+{\mathrm e}^{3 x}$ simplify(expr) does not work. So tried subsindets restart; expr := exp((2*ln(sqrt(p^2 + 1) + p) + 2*ln(a) + ln(p^2 + 1)*a)/(2*a))+ exp(3*x); subsindets(expr,'specfunc( anything, exp )',f->(if(has(op(1,f),'ln'),expand(f),f)))  $\left (\sqrt {p^{2}+1}+p \right )^{\frac {1}{a}} a^{\frac {1}{a}} \sqrt {p^{2}+1}+{\mathrm e}^{3 x}$ It is possible to also try simplify(expr,exp) in some cases, but for the above example, this did not work, i.e. it did not simplify it. ### 113 How to ﬁnd all csgn() and replace them by 1 I wanted to simplify an expression which could have csgn() in it, and ﬁnd all the arguments. $\frac {1+\mathrm {csgn} \left (a \right ) a}{3 \mathrm {csgn} \left (b \right ) b}$ One way is restart; expr:=(1+csgn(a)*a)/(3*csgn(b)*b): expr:=subsindets(expr,'specfunc( anything, csgn )',f->1);  $\frac {1+a}{3 b}$ ### 114 How to ﬁnd symbols inside csgn() in an expression? Given sol:=1/2*2^(1/2)*csgn(x)*x*csgn(y); how to ﬁnd all symbols inside csgn which will be $$x,y$$ in this case? restart; sol:=1/2*2^(1/2)*csgn(x)*x*csgn(y); indets(sol,'specfunc( anything, csgn )'); vars:=subsindets(%,'specfunc( anything, csgn )',f->op(f))  Gives {x, y} Now if we want to simplify the above solution by assuming that all variables inside vars are positive, how to do that? restart; sol:=1/2*2^(1/2)*csgn(x)*x*csgn(y); indets(sol,'specfunc( anything, csgn )'); vars:=subsindets(%,'specfunc( anything, csgn )',f->op(f)); simplify(sol) assuming op(map2(<,0,vars))  Gives $$\frac {\sqrt {2}\, x}{2}$$. Notice in the above the use of op(map2(<,0,vars)), this will generate the sequence 0 < x, 0 < y automatically. op is needed otherwise the result will be {0 < x, 0 < y} which will give syntax error when passed to assuming Ofcourse, it would have been also possible to just write simplify(sol) assuming positive;  And get the same result. But sometimes we might want to specify which variables are to be assumed positive and not all of them at once in the expression. ### 115 How to replace all abs(expr) by expr I wanted to replace |expr| by (expr) One way is restart; expr:=u(x) = _C1*exp(-3*x^(1/3)*sqrt(c))*(3*x^(1/3)*sqrt(c) + 1) + _C2*exp(3*x^(1/3)*sqrt(c))*abs(-1 + 3*x^(1/3)*sqrt(c));  $u \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-3 x^{\frac {1}{3}} \sqrt {c}} \left (3 x^{\frac {1}{3}} \sqrt {c}+1\right )+\textit {\_C2} \,{\mathrm e}^{3 x^{\frac {1}{3}} \sqrt {c}} {| -1+3 x^{\frac {1}{3}} \sqrt {c}|}$ expr:=subsindets(expr,'specfunc( anything, abs )',f->op(f));  $u \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{-3 x^{\frac {1}{3}} \sqrt {c}} \left (3 x^{\frac {1}{3}} \sqrt {c}+1\right )+\textit {\_C2} \,{\mathrm e}^{3 x^{\frac {1}{3}} \sqrt {c}} \left (-1+3 x^{\frac {1}{3}} \sqrt {c}\right )$ ### 116 How to ﬁnd basis for Null space, Row space and column space of matrix? Given $\left [\begin {array}{cccc}1 & -1 & 0 & 2 \\1 & 2 & 2 & -2 \\0 & 2 & 3 & -1 \end {array}\right ]$ Find its Null, Row and Column space basis vectors. restart; A:=Matrix([[1,-1,0,2],[1,2,2,-2],[0,2,3,-1]]); LinearAlgebra:-NullSpace(A)  $\left \{ \left [\begin {array}{c}0 \\2 \\-1 \\1 \end {array}\right ] \right \}$ restart; A:=Matrix([[1,-1,0,2],[1,2,2,-2],[0,2,3,-1]]); LinearAlgebra:-RowSpace(A)  $\left [\left [\begin {array}{cccc}1 & 0 & 0 & 0 \end {array}\right ], \left [\begin {array}{cccc}0 & 1 & 0 & -2 \end {array}\right ], \left [\begin {array}{cccc}0 & 0 & 1 & 1 \end {array}\right ]\right ]$ restart; A:=Matrix([[1,-1,0,2],[1,2,2,-2],[0,2,3,-1]]); LinearAlgebra:-ColumnSpace(A)  $\left [\left [\begin {array}{c}1 \\0 \\0 \end {array}\right ], \left [\begin {array}{c}0 \\1 \\0 \end {array}\right ], \left [\begin {array}{c}0 \\0 \\1 \end {array}\right ]\right ]$ ### 117 How to do Gaussian elimination on a Matrix? Given $\left [\begin {array}{cccc}1 & -4 & -3 & -7 \\2 & -1 & 1 & 7 \\1 & 2 & 3 & 11 \end {array}\right ]$ Find the new form after Gaussian elimination restart; A:=Matrix([[1,-4,-3,-7],[2,-1,1,7],[1,2,3,11]]); LinearAlgebra:-GaussianElimination(A);  $\left [\begin {array}{cccc}1 & -4 & -3 & -7 \\2 & -1 & 1 & 7 \\1 & 2 & 3 & 11 \end {array}\right ]$ ### 118 How to ﬁnd Reduced Echelon form of a Matrix? Given matrix $\left [\begin {array}{ccc}5 & 2 & 18 \\0 & 1 & 4 \\4 & 1 & 12 \end {array}\right ]$ Find its Reduced Echelon form. restart; A:=Matrix([[5,2,18],[0,1,4],[4,1,12]]); Student:-LinearAlgebra:-ReducedRowEchelonForm(A)  $\left [\begin {array}{ccc}1 & 0 & 2 \\0 & 1 & 4 \\0 & 0 & 0 \end {array}\right ]$ Another option is restart; A:=Matrix([[5,2,18],[0,1,4],[4,1,12]]); MTM:-rref(A)  $\left [\begin {array}{ccc}1 & 0 & 2 \\0 & 1 & 4 \\0 & 0 & 0 \end {array}\right ]$ ### 119 How add a new row to bottom of matrix? Given matrix $\left [\begin {array}{cc}1 & 1 \\2 & 3 \\4 & 5 \end {array}\right ]$ How to add row $[a, b]$ to end of the matrix? restart; A:=Matrix([[1,1],[2,3],[4,5]]); the_row:=convert([a,b],Vector['row']); ArrayTools:-Concatenate(1,A,the_row);  $\left [\begin {array}{cc}1 & 1 \\2 & 3 \\4 & 5 \\a & b \end {array}\right ]$ ### 120 How to obtain list of all occurances of some function in an expression? For an example, How to ﬁnd list of all $$\ln$$ functions in this expression? $\ln \left ({| x +1|}\right )+2 x \ln \left (x \right )+\sin \left (x \right )$ restart; expr:=ln(abs(x+1))+2*x*ln(x)+sin(x); tmp := indets(expr,'specfunc(anything,ln)'); # tmp := {ln(x), ln(abs(x + 1))}  To pick only $$\ln$$ functions which has $$abs$$ inside them anywhere, replace the above with restart; expr:=ln(abs(x+1))+2*x*ln(x)+sin(x); lis:=indets(expr,'specfunc(anything,ln)'); select(Z->has(Z,abs),lis) # tmp := {ln(abs(x + 1))}  Or, better alternative to the above is restart; expr:=ln(abs(x+1))+2*x*ln(x)+sin(x); indets(expr,'specfunc( satisfies(u->has(u,abs)) ,ln )'); # tmp := {ln(abs(x + 1))}  ### 121 How to replace $$\ln (|x|)$$ with $$\ln (x)$$ in an expression? Given $\sin \left (x \right )+\ln \left ({| x |}\right )+\ln \left (x +\frac {{| y |}}{\sqrt {{| x +3|}}}\right )+\ln \left (x^{3}\right )+\cos \left ({| x |}\right )$ How to remove the absolute, the ones only inside each $$\ln$$ in the above expression? restart; expr:=sin(x)+ln(abs(x))+ln(x+abs(y)/sqrt(abs(x+3)))+ln(x^3)+cos(abs(x)); expr:=evalindets(expr,'specfunc(ln)',f->evalindets(f,'specfunc(abs)',f->op(1,f))) # sin(x) + ln(x) + ln(x + y/sqrt(x + 3)) + ln(x^3) + cos(abs(x))  $\sin \left (x \right )+\ln \left (x \right )+\ln \left (x +\frac {y}{\sqrt {x +3}}\right )+\ln \left (x^{3}\right )+\cos \left ({| x |}\right )$ ### 122 How to ﬁnd all signum functions in expression and simplify it? Given $-\frac {\left (\ln \left (\frac {\left (b +\sqrt {b^{2}+y \left (x \right )^{2}}\, \mathrm {signum}\left (b \right )\right ) b}{y \left (x \right )}\right )+\ln \left (2\right )\right ) \mathrm {signum}\left (b \right )}{b} = \mathit {\_C1} +\frac {-\ln \left (a \right )+\ln \left (x \right )-\ln \left (a +\sqrt {a^{2}+x^{2}}\, \mathrm {signum}\left (a \right )\right )-\ln \left (2\right )}{{| a |}}$ How to ﬁnd all arguments of signum and simplify the above by assuming they are all positive? restart; expr:=-(ln((b + sqrt(b^2 + y(x)^2)*signum(b))*b/y(x)) + ln(2))*signum(b)/b = _C1 + (-ln(a) + ln(x) - ln(a + sqrt(a^2 + x^2)*signum(a)) - ln(2))/abs(a); lis:=indets(expr,'specfunc(anything,signum)'); assum:=convert(map(x->op(1,x)>0,lis),list); simplify(expr,assume=assum);  $\frac {-\ln \left (b \right )-\ln \left (\frac {b +\sqrt {b^{2}+y \left (x \right )^{2}}}{y \left (x \right )}\right )-\ln \left (2\right )}{b} = \frac {\mathit {\_C1} a -\ln \left (a \right )-\ln \left (a +\sqrt {a^{2}+x^{2}}\right )+\ln \left (x \right )-\ln \left (2\right )}{a}$ ### 123 How to replace all signum functions in expression by 1? Given $-\frac {\left (\ln \left (\frac {\left (b +\sqrt {b^{2}+y \left (x \right )^{2}}\, \mathrm {signum}\left (b \right )\right ) b}{y \left (x \right )}\right )+\ln \left (2\right )\right ) \mathrm {signum}\left (b \right )}{b} = \mathit {\_C1} +\frac {-\ln \left (a \right )+\ln \left (x \right )-\ln \left (a +\sqrt {a^{2}+x^{2}}\, \mathrm {signum}\left (a \right )\right )-\ln \left (2\right )}{{| a |}}$ How to replace all signum by 1? restart; expr:=-(ln((b + sqrt(b^2 + y(x)^2)*signum(b))*b/y(x)) + ln(2))*signum(b)/b = _C1 + (-ln(a) + ln(x) - ln(a + sqrt(a^2 + x^2)*signum(a)) - ln(2))/abs(a); evalindets(expr, 'specfunc(anything,signum)', f -> 1);  $-\frac {\ln \left (\frac {\left (b +\sqrt {b^{2}+y \left (x \right )^{2}}\right ) b}{y \left (x \right )}\right )+\ln \left (2\right )}{b} = \textit {\_C1} +\frac {-\ln \left (a \right )+\ln \left (x \right )-\ln \left (a +\sqrt {a^{2}+x^{2}}\right )-\ln \left (2\right )}{{| a |}}$ ### 124 How to do change of variables on the dependent variable for an ODE? given an ode $y \left (x \right ) = \left (\frac {d}{d x}y \left (x \right )\right )^{3} y \left (x \right )^{2}+2 x \left (\frac {d}{d x}y \left (x \right )\right )$ do change of variable $$u(x)=y(x)^2$$ restart; ode:=y(x)=diff(y(x),x)^3*y(x)^2+2*x*diff(y(x),x); new_ode:=PDEtools:-dchange({y(x)=sqrt(u(x))},ode,{u});  $\sqrt {u \left (x \right )} = \frac {\left (\frac {d}{d x}u \left (x \right )\right )^{3}}{8 \sqrt {u \left (x \right )}}+\frac {x \left (\frac {d}{d x}u \left (x \right )\right )}{\sqrt {u \left (x \right )}}$ ### 125 How to do change of variable on the independent variable for an ODE? given an ode $\frac {d^{2}}{d t^{2}}y \left (t \right )+y \left (t \right ) = 2 t$ do change of variable $$t=\tau + \pi$$ restart; ode:=diff(y(t),t2)+y(t)=2*t;
PDEtools:-dchange({t=tau+Pi},ode,known={t},unknown={tau},params=Pi)



$\frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+y \left (\tau \right ) = 2 \tau +2 \pi$

it is important to use  params=Pi. Watch what happens if we do not do that

restart;
ode:=diff(y(t),t$2)+y(t)=2*t; PDEtools:-dchange({t=tau+Pi},ode,known={t},unknown={tau});  $\frac {d^{2}}{d \tau ^{2}}y \left (\tau , \pi \right )+y \left (\tau , \pi \right ) = 2 \tau +2 \pi$ Which is not what we want. ### 126 ODE change of variable on both dependent and independent variable? This veriﬁes solution given in https://math.stackexchange.com/questions/3477732/can-t-see-that-an-ode-is-equivalent-to-a-bessel-equation Where a change of variables on $\xi ^2 \frac {d^2\eta }{d\xi ^2} + \xi \frac {d\eta }{d\xi } - (\xi ^2+n^2)\eta =0$ Was made using $\eta =\frac {y}{x^\alpha }, \quad \xi =\beta x^\gamma ,$ To produce the ode $\frac {d^2y}{dx^2} - \frac {(2\alpha -1)}{x}\frac {dy}{dx} - (\beta ^2 \gamma ^2 x^{2\gamma -2}+\frac {n^2\gamma ^2-\alpha ^2}{x^2})y=0.$ In Maple the above is done using restart; ode := zeta^2*diff(eta(zeta),zeta$2) + zeta*diff(eta(zeta),zeta) - (zeta^2 + n^2)*eta(zeta) = 0;
the_tr:={zeta=beta*x^gamma,eta(zeta)=y(x)/x^alpha};
PDEtools:-dchange(the_tr,ode,{y(x),x},'known'={eta(zeta)},'uknown'={y(x)},'params'={alpha,beta,gamma});
simplify(%);
numer(lhs(%))=0;
simplify(numer(lhs(%))/(x^(1-alpha)))=0;
numer(lhs(%))=0;
collect(%,[y(x),diff(y(x),x),diff(y(x),x$2)]);  Which gives $\left (-\gamma ^{2} x^{-1+2 \gamma } \beta ^{2} x -\gamma ^{2} n^{2}+\alpha ^{2}\right ) y \left (x \right )+\left (-2 \alpha x +x \right ) \left (\frac {d}{d x}y \left (x \right )\right )+x^{2} \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right ) = 0$ Here is another example. Here to make change of variables to polar coordinates by making $$x=r \cos \theta$$ and $$y=r \sin \theta$$ The ode is $\frac {y-x y'}{\sqrt {1+(y')^2}}=x^2+y^2$ In Maple restart; ode := (y(x)-x*diff(y(x),x))/sqrt(1+ diff(y(x),x)^2) = x^2+y(x)^2; the_tr:={x=r(t)*cos(t),y(x)=r(t)*sin(t)}; PDEtools:-dchange(the_tr,ode,{r(t),t},'known'={y(x)},'uknown'={r(t)});  Which gives $\frac {r \left (t \right ) \sin \left (t \right )-\frac {r \left (t \right ) \cos \left (t \right ) \left (\left (\frac {d}{d t}r \left (t \right )\right ) \sin \left (t \right )+r \left (t \right ) \cos \left (t \right )\right )}{\left (\frac {d}{d t}r \left (t \right )\right ) \cos \left (t \right )-r \left (t \right ) \sin \left (t \right )}}{\sqrt {1+\frac {\left (\left (\frac {d}{d t}r \left (t \right )\right ) \sin \left (t \right )+r \left (t \right ) \cos \left (t \right )\right )^{2}}{\left (\left (\frac {d}{d t}r \left (t \right )\right ) \cos \left (t \right )-r \left (t \right ) \sin \left (t \right )\right )^{2}}}} = r \left (t \right )^{2} \left (\cos ^{2}\left (t \right )\right )+r \left (t \right )^{2} \left (\sin ^{2}\left (t \right )\right )$ Here is another example. Where we want to change $$R(r)$$ to $$y(x)$$ everywhere $\frac {d^{2}}{d r^{2}}R \left (r \right )+\frac {d}{d r}R \left (r \right )+R \left (r \right ) = 0$ restart; ode:=diff(R(r),r$2)+diff(R(r),r)+R(r)=0;
the_tr:={r=x,R(r)=y(x)};
PDEtools:-dchange(the_tr,ode,{y(x),x},'known'={R(r),r},'uknown'={y(x),x});



$\frac {d^{2}}{d x^{2}}y \left (x \right )+\frac {d}{d x}y \left (x \right )+y \! \left (x \right ) = 0$

The format of the transformation is old_independent_variable=new_independent_variable and old_dependent_variable=new_dependent_variable

### 127 How to ﬁnd the cofactor matrix of a matrix?

Use LinearAlgebra:-Adjoint and then transpose the result. Since the Adjoint is the transpose of the cofactor.

Given

$\left [\begin {array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end {array}\right ]$

then

restart;
A:=Matrix([[1,2,3],[4,5,6],[7,8,10]]);



$\left [\begin {array}{ccc} 2 & 2 & -3 \\ 4 & -11 & 6 \\ -3 & 6 & -3 \end {array}\right ]$

### 128 How to make phase plot of second order ODE?

Make a phase plot of $\frac {d^{2}}{d t^{2}}x \left (t \right )+\frac {\frac {d}{d t}x \left (t \right )}{2}+x \left (t \right ) = u \left (t \right )$

By plotting $$x(t)$$ vs $$x'(t)$$ without solving the ODE.

restart;
alias(DS=DynamicSystems):
ode := diff(x(t),t$2) +1/2*diff(x(t),t)+ x(t) = u(t); sys:=DS:-DiffEquation(ode,'outputvariable'=[x(t)],'inputvariable'=[u(t)]); sys0:=DS:-StateSpace(sys); eq1:=diff(x1(t),t)=sys0:-a[1,..].Vector([x1(t),x2(t)]); eq2:=diff(x2(t),t)=sys0:-a[2,..].Vector([x1(t),x2(t)]); DEtools:-DEplot([eq1,eq2],[x1(t),x2(t)],t=0..35,[[x1(0)=1,x2(0)=1]],x1=-2..2,x2=-2..2, numpoints=200, linecolor=black, axes=boxed); ### 129 How to normalize eigenvectors? When ﬁnding eigenvectors of matrix, using LinearAlgebra, the vectors are not normalized. How to normalized them so the length is one? One way is restart; LA:=LinearAlgebra; Sx:=Matrix([[0,1,0],[1,0,1],[0,1,0]]); #this finds eigenvectors in v lam,v:=LA:-Eigenvectors(Sx); #this normalize it B:=map(n -> v[.., n]/norm(v[.., n], 2), [$1..LA:-RowDimension(v)]):
B:=<|>(op(B)); #this converts the list back to matrix.



$v=\left [\begin {array}{ccc} -1 & 1 & 1 \\ 0 & \sqrt {2} & -\sqrt {2} \\ 1 & 1 & 1 \end {array}\right ]$

$B=\left [\begin {array}{ccc} -\frac {\sqrt {2}}{2} & \frac {1}{2} & \frac {1}{2} \\ 0 & \frac {\sqrt {2}}{2} & -\frac {\sqrt {2}}{2} \\ \frac {\sqrt {2}}{2} & \frac {1}{2} & \frac {1}{2} \end {array}\right ]$

### 130 How to ﬁnd if some function is present in an expression

Given expression $$3 \sin \left (x \right )+t +3 f \left (x , t\right ) t +g \left (x , t\right )$$ ﬁnd if it contains function $$f()$$.

Use indets with specfunc(f)

restart;
expr := 3*sin(x)+t+3*f(x,t)*t+g(x,t);
res  := indets(expr, specfunc(f));
if numelems(res)<>0 then
print("Found f(x,t)");
else
print("could not find f(x,t)");
fi;



   "Found f(x,t)"

### 131 How to ﬁnd all functions in an expression?

Given expression $$3 \sin \left (x \right )+t +3 f \left (x , t\right ) t +g \left (x , t\right )$$ ﬁnd all functions, if any, in the expression.

Use indets with function

restart;
expr := 3*sin(x)+t+3*f(x,t)*t+g(x,t);
res  := indets(expr,function);
if numelems(res)<>0 then
print("Found these functions",res);
else
print("could not find any function)");
fi;



   "Found these functions", {f(x, t), g(x, t), sin(x)}

### 132 How to ﬁnd all functions in an expression but exclude all build in math functions?

Given expression $$3 \sin \left (x \right )+t +3 f \left (x , t\right ) t +g \left (x , t\right )$$ ﬁnd all functions, if any, in the expression but exclude the math functions such as $$\sin$$ in the above.

restart;
expr := 3*sin(x)+t+3*f(x,t)*t+g(x,t);
res  := indets(expr, And( function, Not(typefunc(mathfunc))));
if numelems(res)<>0 then
print("Found these functions",res);
else
print("could not find any function)");
fi;



   "Found these functions", {f(x, t), g(x, t)}

### 133 How to obtain a list of all arguments of function?

use op

restart;
op(1..,f(x,t))

x, t



Note that op(0,f(x,t)) ﬁnds the function name.

### 134 How to obtain a list of all functions in expression whose ﬁrst argument is $$z$$?

restart;
expr := 3*sin(z)+t+3*f(z,t,y)*t+g(x,t);
res  := indets(expr, patfunc(identical(z), anything));
if numelems(res)<>0 then
print("Found these functions",res);
else
print("could not find any function)");
fi;



gives

"Found these functions", {f(z, t, y), sin(z)}

### 135 How to obtain a list of all functions in expression whose second argument is $$t$$?

expr := 3*sin(z)+t+3*f(z,t,y)*t+g(x,t);
res  := indets(expr, patfunc(anything, identical(t), anything));
if numelems(res)<>0 then
print("Found these functions",res);
else
print("could not find any function)");
fi;



gives

"Found these functions", {f(z, t, y), g(x, t)}

### 136 How to typeset $$\hslash$$?

expr:=&hbar;*x

gives

$\hslash x$

Notice, the ; is needed. This &hbar*x will not work. It must be &hbar;*x

### 137 How to ﬁnd the Curl of a vector?

First example

restart;
VectorCalculus:-SetCoordinates( 'cartesian'[x,y,z] );
F:=VectorCalculus:-VectorField(<y,-x,0>);



\begin {align*} F &= y \mathbf {\bar {e}_{x}}-x \mathbf {\bar {e}_{y}} \end {align*}

And now

VectorCalculus:-Curl(F);



$-2 \mathbf {\bar {e}_{z}}$

Second example

restart;
VectorCalculus:-SetCoordinates( 'cartesian'[x,y,z] );
F:=VectorCalculus:-VectorField(<y*z^2,x*z^2+2,2*x*y*z-1>);



\begin {align*} F &= y \,z^{2} \mathbf {\bar {e}_{x}}+(x \,z^{2}+2) \mathbf {\bar {e}_{y}}+(2 x y z -1) \mathbf {\bar {e}_{z}} \end {align*}

And now

VectorCalculus:-Curl(F);



$0$ Since Curl is zero, ﬁeld is conservative.

Third example, in cylinderical coodinates

restart;
VectorCalculus:-SetCoordinates( 'cylindrical'[rho,phi,z] );
F:=VectorCalculus:-VectorField(<0,-rho,2>);



\begin {align*} F &= -\rho \mathbf {\bar {e}_{\phi }}+2 \mathbf {\bar {e}_{z}} \end {align*}

And now

VectorCalculus:-Curl(F);



$2 \mathbf {\bar {e}_{z}}$

### 138 How to see all steps in ﬁnding RREF form of an augmented matrix?

Use  Student:-LinearAlgebra:-GaussJordanEliminationTutor( A, output=steps )  Where $$A$$ is your augmented matrix.

### 139 How to ﬁnd column space of matrix?

Do not use the Maple command LinearAlgebra:-ColumnSpace for this. it gives the columns in the RREF. The correct way is to obtain the corresponding columns of the pivot columns in the original matrix $$A$$. Hence use the command Basis like this

A:=Matrix([[1,0,0],[1,1,1]]);
LinearAlgebra:-Basis([seq(A[..,i],i=1..LinearAlgebra:-ColumnDimension(A) )]);



Which gives

$\left [\left [\begin {array}{c} 1 \\ 1 \end {array}\right ], \left [\begin {array}{c} 0 \\ 1 \end {array}\right ]\right ]$

If you use ColumnSpace command you’ll get this

A:=Matrix([[1,0,0],[1,1,1]]);
LinearAlgebra:-ColumnSpace(A);



$\left [\left [\begin {array}{c} 1 \\ 0 \end {array}\right ], \left [\begin {array}{c} 0 \\ 1 \end {array}\right ]\right ]$

These are diﬀerent. Basis is the correct command to use, which matches the standard deﬁnition in textbooks.

### 140 How to use select with own type to ﬁnd subexpressions?

Given expression such as $$3+(1+x)\sin x$$ or $$3+(1+x)\sin ^2 x$$ use select to ﬁnd any  polynomial * sin^n  subexpressions.

restart;
mytype_1 := ''*'( {polynom(And(algebraic, satisfies(u -> not has(u, I))),x),
Or( 'specfunc(sin)'^integer, 'specfunc(sin)')
}
)';
select(type, 3+(1+x)*sin(x),mytype_1);
select(type, 3+(1+x)*sin(x)^2,mytype_1);



Gives

                         (1 + x) sin(x)

2
(1 + x) sin(x)



### 141 How to write structured types to match some expressions?

#### 141.1 type for $$\sin ^m(x)\cos ^n(x)$$

restart;
my_type:=''*'( { Or('specfunc'(sin),'specfunc'(sin)^Or(integer,rational)),
Or('specfunc'(cos),'specfunc'(cos)^Or(integer,rational))})';

type(sin(x)^2*cos(x)^3,my_type);
type(sin(x)^2*cos(x),my_type);
type(sin(x)*cos(x),my_type);
type(cos(x)*sin(x)^(1/2),my_type);

true
true
true
true



I could not ﬁnd a way to avoid writing  Or('specfunc'(sin),'specfunc'(sin)^Or(integer,rational) in order to match both $$\sin x$$ and $$\sin ^2 x$$. For these things, I ﬁnd Mathematica patterns more ﬂexiable. The above can be done as follows in Mathematica


ClearAll[x,n,m,any]
patt=any_.*Sin[_]^n_. * Cos[_]^m_.
MatchQ[Sin[x]^2*Cos[2*x]^3,patt]
MatchQ[Sin[x]^2*Cos[x],patt]
MatchQ[Sin[x]*Cos[x],patt]
MatchQ[Cos[x]*Sin[x],patt]

True
True
True
True



In Mathematica  n_.  says basically to match $$\sin x$$ or $$\sin ^2 x$$ since the dot says to match zero or more. So no need to duplicate things as I did above in Maple. There might be a way to do the same in Maple using structured type, but I could not ﬁnd it. In General, I ﬁnd patterns in Mathematica more ﬂexible and easier to use for this sort of thing. Maple has patmatch command, but not as easy to use as Patterns in Mathematica.

### 142 How to use new object method calling in Maple 2021?

In Maple 2021, it is now possible to use object:-method(arg) notation. This makes is easier to use OOP in maple. To do this, use _self as follows

restart;

person:=module()
option object;
local name::string:="";
local age::integer:=0;

export get_name:=proc(_self,$) return _self:-name; end proc; export set_name:=proc(_self,name::string,$)
_self:-name:=name;
end proc;

export get_age:=proc(_self,$) return _self:-age; end proc; export set_age:=proc(_self,age::integer,$)
_self:-age:=age;
end proc;

end module;



And now make an object and use it as follows

o:=Object(person)
o := Object<<1846759887808>>

o:-get_name();
""

o:-get_age();
0

o:-set_name("joe doe");
o:-get_name();
"joe doe"



### 143 How to make a constructor for an Object?

Add ModuleCopy proc in the class. This will automatically be called to initialize the object.

Here is an example

restart;

ODE:=module()
option object;
local ode:=NULL;
local y::symbol;
local x::symbol;
local sol;

export ModuleCopy::static := proc( _self::ODE, proto::ODE, ode, func, $) print("Initilizing object with with args: ", [args]); _self:-ode:= ode; _self:-y:=op(0,func); _self:-x:=op(1,func); end proc; export dsolve::static:=proc(_self,$)
_self:-sol := :-dsolve(ode,y(x));
end proc;

export get_sol::static:=proc(_self,$) return sol; end proc; end module;  And now make an object and use it as follows o:=Object(ODE, diff(y(x),x)+y(x)=sin(x), y(x)); o:-dsolve(): o:-get_sol(); #y(x) = -1/2*cos(x) + 1/2*sin(x) + exp(-x)*_C1  So a constructor just makes it easier to initialize the object without having to make a number of set() calls to initialize each memeber data. ### 144 How to make diﬀerent constructors for an Object? This is done using overload with diﬀerent ModuleCopy proc in the class. Here is an example. Lets make a constructor that takes an ode and initial conditions, and one that only takes an ode with no initial conditions. restart; ODE:=module() option object; local ode:=NULL; local y::symbol; local x::symbol; local ic:=NULL; local sol; export ModuleCopy::static:= overload( [ proc( _self::ODE, proto::ODE, ode, func,$ ) option overload;
_self:-ode:= ode;
_self:-y:=op(0,func);
_self:-x:=op(1,func);
end proc,

proc( _self::ODE, proto::ODE, ode, func, ic, $) option overload; _self:-ode:= ode; _self:-y:=op(0,func); _self:-x:=op(1,func); _self:-ic :=ic; end proc ] ); export dsolve::static:=proc(_self,$)
if evalb(ic=NULL) then
sol := :-dsolve(ode,y(x));
else
sol := :-dsolve([ode,ic],y(x));
fi;
end proc;

export get_sol::static:=proc(_self,$) return sol; end proc; end module;  And now use it as follows o:=Object(ODE, diff(y(x),x)+y(x)=sin(x), y(x), y(0)=0); o:-dsolve(): o:-get_sol(); #y(x) = -1/2*cos(x) + 1/2*sin(x) + 1/2*exp(-x) o:=Object(ODE, diff(y(x),x)+y(x)=sin(x), y(x)); o:-dsolve(): o:-get_sol(); #y(x) = -1/2*cos(x) + 1/2*sin(x) + exp(-x)*_C1  ### 145 How to do OOP inheritance? In the child class you want to extend from the parent class, add option object(ParentName); Here is an example restart; module ODE() option object; local ode; export set_ode::static:=proc(_self,ode,$)
_self:-ode :=ode;
end proc;

export get_ode::static:=proc(_self,$) return _self:-ode; end proc; end module; #create class/module which extends the above module second_order_ode() option object(ODE); export get_ode_order::static:=proc(_self,$)
return 2;
end proc;
end module;



In the above second_order_ode inherts all local variables and functions in the ODE class and adds new proc. Use as follows

o:=Object(second_order_ode); #create an object  instance
o:-set_ode(diff(y(x),x)=sin(x));
o:-get_ode();
o:-get_ode_order();



Note that the child class can not have its own variable with the same name as the parent class. This is limitation. in C++ for example, local variables in extended class overrides the same named variable in the parent class.

Even if the variable have diﬀerent type, Maple will not allow overriding. For example, this will fail

restart;
module ODE()
option object;
local ode;
local id::integer;

export set_ode::static:=proc(_self,ode,$) print("Enter ode::set_ode"); _self:-ode :=ode; end proc; export get_ode::static:=proc(_self,$)
return _self:-ode;
end proc;
end module;

module second_order_ode()
option object(ODE);
local id::string;
export get_ode_order::static:=proc(_self,$) return 2; end proc; end module; Error, (in second_order_ode) local id is declared more than once  There might be a way to handle this, i.e. to somehow exlicitly tell Maple to override parant class proc or variable name in the child. I do not know now. The above is using Maple 2021.1 ### 146 How to extend a class and call base class function from the extended class?? The trick is not to use static to override the base class function, and to use _self to refer to base class functions as own functions after extending. Also just use _self and not _self::person Here is an example. person:=module() option object; local age::integer:=100; export ModuleCopy::static:= proc(_self,proto::person, age::integer,$)
_self:-age := age;
end proc;

export process::static:=proc(_self,$) local o; print("In base class process..."); if age<20 then o:=Object(young_person,age); o:-process(); else o:=Object(old_person,age); o:-process(); fi; end proc; local print_age::static:=proc(_self,$)
print("age = ",_self:-age);
end proc;

end module;
#---- extend the above class
module young_person()
option object(person);
process:=proc(_self,$) print("In young_person process"); _self:-print_age(); end proc; end module; module old_person() option object(person); process::static:=proc(_self,$)
print("In old_person process");
_self:-print_age();
end proc;
end module;



Now to make 2 new classes that extend the above.

#---- extend the above class
module young_person()
option object(person);
process:=proc(_self,$) print("In young_person process"); _self:-print_age(); end proc; end module; module old_person() option object(person); process::static:=proc(_self,$)
print("In old_person process");
_self:-print_age();
end proc;
end module;



Here is an example of usage

p:=Object(person,80);
p:-process()
"In base class process..."
"In old_person process"
"age = ", 80

p:=Object(person,10);
p:-process()
"In base class process..."
"In young_person process"
"age = ", 10



The above in Maple 2022.1

### 147 How to use object as user deﬁned record inside a proc?

A Maple Object can be used a record type in other languags, such as Ada or Pascal. This example shows how to deﬁne a local type inside a proc and use it as record.

restart;
foo:=proc(name::string,age::integer)::person_type;
local person_type:=module() #this acts as a record type
option object;
export name::string;
export age::integer;
end module;

local person::person_type:=Object(person_type);

person:-name:=name;
person:-age:=age;
return person;
end proc;

o:=foo("joe doe",100);

o:-name;
"joe doe"
o:-age;
100



In the above person is local variable of type person_type. In the above example, the local variable was returned back to user. But this is just an example. One can declare such variables and just use them internally inside the proc only. This method helps one organize related variables into one record type. The type can also be made global if needed.

### 148 Given an expression with indexed variables, how to select only these variables?

use indets with type 'indexed'

expr:=16*a+6*a;
terms:=indets(expr,'indexed');

terms := {a, a}

#to find maximum index, then do

map(x->op(x),terms)

{1, 3}



### 149 How to show step by step for calculus problem and for diﬀerential equations?

For integration do

Student:-Calculus1:-ShowSolution(Int(x*sin(x),x));



The steps are displayed. This does not work all the time. For example

integrand:=x*y(x)*diff(y(x),x$2)+x*(diff(y(x),x))^2-y(x)*diff(y(x),x); Student:-Calculus1:-ShowSolution(Int(integrand,x));  gives Error, (in Student:-Calculus1:-ShowSolution) unable to determine which calculus operation is being applied in this problem; you can provide this information as the 2nd argument on your call to Rule or Hint For diﬀerential equations, support is limited but these are the steps restart; ode:=diff(y(x),x)=sin(x); Student:-ODEs:-ODESteps(ode)  Prints the steps. If IC is there, then restart; ode:=diff(y(x),x)=sin(x); ic:=y(0)=1; Student:-ODEs:-ODESteps([ode,ic])  ### 150 How to obtain list of ﬁles with some extension in folder? Use FileTools:-ListDirectory dir_name:="C:/tmp"; currentdir(dir_name); #cd to directory files_to_process := FileTools:-ListDirectory(dir_name,'all','returnonly'="*.tex"): numelems(files_to_process) 100  In the above, files_to_process is a list of the ﬁles in the current folder with extension .tex ### 151 How to delete lines from text ﬁle that contains some string? There was a case when I needed to delete lines from text ﬁle that contains a say "foo" as an example. This is what I did. use readline to read the lines, check, and if the line contains "foo" skip, else write the line to a temporary ﬁle. At the line, use Rename to rename the temporary ﬁle to the ﬁle being read. dir_name:="C:/tmp"; currentdir(dir_name); tmp_file_name := "TMP.txt"; source_file_name := "source.txt"; file_id := fopen(tmp_file_name,WRITE): line := readline(source_file_name): while line<>0 do if not StringTools:-Has(line,"foo") then fprintf(file_id,"%s\n",line); fi; line := readline(source_file_name): od: fclose(file_id); FileTools:-Rename(tmp_file_name,source_file_name,force=true);  ### 152 Given an expression, how to ﬁnd all variables and functions in it? Given say $$\frac {d^{2}}{d x^{2}}y \left (x \right )+n \left (\frac {d}{d x}y \left (x \right )\right )+3 = \sin \left (x \right )$$ how to ﬁnd all variables and functions in it, not including math functions such as $$\sin x$$? So the result should be $$n,x,y(x)$$. ode:=diff(y(x),x$2)+n*diff(y(x),x)+3=sin(x);
vars:=indets(ode, Or(  And(symbol,Not(constant)), And(function,Not(typefunc(mathfunc)) ) ))

#gives
#           vars := {n, x, diff(y(x), x), diff(y(x), x, x), y(x)}



I still need to work on excluding derivatives from the search.

### 153 How to check if an expression is integer, when it has symbols in it?

I had case where I needed to check if something is integer or not. The problem is that the result had a symbol $$n$$ in it. I need a way to tell Maple that to check if the result can be an integer given that $$n$$ is also an integer.

Using type does not work, since can’t use assumptions. One way is to use coulditbe as follows

restart;
expr:=n-1+2*m;
vars:=indets(expr,And(symbol,Not(constant)));
coulditbe(expr,integer) assuming op(map(Z->Z::integer,vars))

#  true



In the above indets(expr,And(symbol,Not(constant))) picks all variables in the expression, and assuming op(map(Z->Z::integer,vars)) makes assumption that each is integer.

### 154 How to invert roles of dependent variable and independent variable in an ode?

Sometimes it is useful to invert an ode. i.e. make the independent variable the dependent variable, and the dependent variable the independent. For example, given $1+\left (\frac {x}{y \left (x \right )}-\sin \left (y \left (x \right )\right )\right ) \left (\frac {d}{d x}y \left (x \right )\right ) = 0$ We want the ode to become $-\sin \left (y \right ) y +y \left (\frac {d}{d y}x \left (y \right )\right )+x \left (y \right ) = 0$

This can be done as follows

restart;

ode:=1+ (x/y(x)-sin(y(x) ))*diff(y(x),x)=0;
tr:={x=u(t),y(x)=t};
ode:=PDEtools:-dchange(tr,ode);
ode:=eval(ode,[t=y,u=x]);
ode:=simplify(ode);



$\frac {-\sin \left (y \right ) y +y \left (\frac {d}{d y}x \left (y \right )\right )+x \left (y \right )}{y \left (\frac {d}{d y}x \left (y \right )\right )} = 0$

In this case, we can get rid of the denomator, but this is a manual step for now.

ode:=numer(lhs(ode))=0;



$-\sin \left (y \right ) y +y \left (\frac {d}{d y}x \left (y \right )\right )+x \left (y \right ) = 0$

The above can now be solved more easily for $$x(y)$$ than solving the orignal ode for $$y(x)$$.

### 155 How to truncate a polynomial?

Given $$9 x^{5}+4 x^{4}+3 x^{3}+x^{2}+x +1$$ how to truncate it, so that all terms of $$x^3$$ and higher are removed?

This can be done as follows

restart;
p:=1+x+x^2+3*x^3+4*x^4+9*x^5;
simplify(p,{x^3=0})



$x^{2}+x +1$

### 156 How to make a local declare like block inside a proc?

Sometimes it is useful to make a small local piece of code inside a proc, with its own local variables that do not interfer with the variables of the proc. In Ada, this is done using declare clause for example. In Maple on can do the same as follows

restart;

foo:=proc()
local n;
n:=10;
proc()
local n;
n:=99;
print("inside inner proc, n=",n);
end proc();
print("n=",n);
end proc;

foo();



Which prints

                  "inside inner proc, n=", 99
"n=", 10



Notice the end of the inner anonymous proc above. It has end proc(); and not end proc; as normal proc. This deﬁnes the inner proc and calls it at same time. All the local variables inside the anonymous proc only exist inside that proc and go away after the call, and they do not interfer with the outer proc variables. This way we can declare temporary variables and use them where they are needed.

### 157 How to use short name for a proc which is inside nested modules to avoid long name usage?

There was a case where I was making lots of calls from many places to pne speciﬁc proc inside a module. I did not want to keep using the long name each time.

the command alias did not work. After some trial and error, found that using use works. Here is the solution. First this is the original layout

restart;

A:=module()
export B:=module()
export foo:=proc()
print("in A:-B:-foo()");
end proc;
end module;

export C:=module()
export boo:=proc()
print("in A:-C:-boo()");
A:-B:-foo();
end proc;
end module